IB · MATH AA HL

Mathematics AA: Higher Level

Calculus — Topic 5

Name: ____________________Date: October 10, 2026

1.
Differentiate y = x²·e^(3x) using the product rule.
[4 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Break the command into its requested parts. For each part, connect a relevant fact or observation to the conclusion it supports. Describing what happens and explaining why it happens are different tasks.
2. Work through this mathematical step: States the product rule dy/dx = u′v + uv′ with u = x², v = e^(3x). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Finds u′ = 2x and v′ = 3e^(3x). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Substitutes into the product rule. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Work through this mathematical step: Obtains dy/dx = 2xe^(3x) + 3x²e^(3x), which may be factorised as xe^(3x)(2 + 3x). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Factorising the final answer is good practice but not always required; check the command term for what is expected.
Marking points
- States the product rule dy/dx = u′v + uv′ with u = x², v = e^(3x).
- Finds u′ = 2x and v′ = 3e^(3x).
- Substitutes into the product rule.
- Obtains dy/dx = 2xe^(3x) + 3x²e^(3x), which may be factorised as xe^(3x)(2 + 3x).
Examiner tip: Factorising the final answer is good practice but not always required; check the command term for what is expected.
2.
Marking analysis: A learner attempts the following task: “Differentiate y = x²·e^(3x) using the product rule.” Their response addresses only this point: “States the product rule dy/dx = u′v + uv′ with u = x², v = e^(3x).” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: States the product rule dy/dx = u′v + uv′ with u = x², v = e^(3x). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: Finds u′ = 2x and v′ = 3e^(3x). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Requirement 3: Identifies the missing requirement: Substitutes into the product rule. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
5. Requirement 4: Identifies the missing requirement: Obtains dy/dx = 2xe^(3x) + 3x²e^(3x), which may be factorised as xe^(3x)(2 + 3x). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: States the product rule dy/dx = u′v + uv′ with u = x², v = e^(3x).
- Identifies the missing requirement: Finds u′ = 2x and v′ = 3e^(3x).
- Identifies the missing requirement: Substitutes into the product rule.
- Identifies the missing requirement: Obtains dy/dx = 2xe^(3x) + 3x²e^(3x), which may be factorised as xe^(3x)(2 + 3x).
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
3.
Use integration by substitution with u = x² + 1 to find ∫2x(x² + 1)⁵ dx.
[5 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Work through this mathematical step: States u = x² + 1 and finds du/dx = 2x, so du = 2x dx. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Rewrites the integral entirely in terms of u: ∫u⁵ du. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Integrates to obtain u⁶/6 + C. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Work through this mathematical step: Substitutes back for u. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
6. Work through this mathematical step: States the final answer (x² + 1)⁶/6 + C. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
7. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Recognising that 2x dx is exactly du is what makes this substitution work cleanly; always check the derivative matches a factor present.
Marking points
- States u = x² + 1 and finds du/dx = 2x, so du = 2x dx.
- Rewrites the integral entirely in terms of u: ∫u⁵ du.
- Integrates to obtain u⁶/6 + C.
- Substitutes back for u.
- States the final answer (x² + 1)⁶/6 + C.
Examiner tip: Recognising that 2x dx is exactly du is what makes this substitution work cleanly; always check the derivative matches a factor present.
4.
Marking analysis: A learner attempts the following task: “Use integration by substitution with u = x² + 1 to find ∫2x(x² + 1)⁵ dx.” Their response addresses only this point: “States u = x² + 1 and finds du/dx = 2x, so du = 2x dx.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[5 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: States u = x² + 1 and finds du/dx = 2x, so du = 2x dx. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: Rewrites the integral entirely in terms of u: ∫u⁵ du. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Requirement 3: Identifies the missing requirement: Integrates to obtain u⁶/6 + C. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
5. Requirement 4: Identifies the missing requirement: Substitutes back for u. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
6. Requirement 5: Identifies the missing requirement: States the final answer (x² + 1)⁶/6 + C. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
7. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: States u = x² + 1 and finds du/dx = 2x, so du = 2x dx.
- Identifies the missing requirement: Rewrites the integral entirely in terms of u: ∫u⁵ du.
- Identifies the missing requirement: Integrates to obtain u⁶/6 + C.
- Identifies the missing requirement: Substitutes back for u.
- Identifies the missing requirement: States the final answer (x² + 1)⁶/6 + C.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
5.
Differentiate y = ln(x)/x using the quotient rule, and hence find the x-coordinate of the stationary point of the curve.
[6 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Work through this mathematical step: States the quotient rule dy/dx = (u′v − uv′)/v² with u = ln(x), v = x. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Finds u′ = 1/x and v′ = 1. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Substitutes to obtain dy/dx = (x·(1/x) − ln(x)·1)/x². Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Work through this mathematical step: Simplifies the numerator to 1 − ln(x), giving dy/dx = (1 − ln(x))/x². Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
6. Work through this mathematical step: Sets the numerator equal to zero: 1 − ln(x) = 0. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
7. Work through this mathematical step: Solves to obtain x = e. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
8. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: A fraction is zero only when its numerator is zero (with a nonzero denominator); solve the numerator equation, not the whole derivative expression.
Marking points
- States the quotient rule dy/dx = (u′v − uv′)/v² with u = ln(x), v = x.
- Finds u′ = 1/x and v′ = 1.
- Substitutes to obtain dy/dx = (x·(1/x) − ln(x)·1)/x².
- Simplifies the numerator to 1 − ln(x), giving dy/dx = (1 − ln(x))/x².
- Sets the numerator equal to zero: 1 − ln(x) = 0.
- Solves to obtain x = e.
Examiner tip: A fraction is zero only when its numerator is zero (with a nonzero denominator); solve the numerator equation, not the whole derivative expression.
6.
Marking analysis: A learner attempts the following task: “Differentiate y = ln(x)/x using the quotient rule, and hence find the x-coordinate of the stationary point of the curve.” Their response addresses only this point: “States the quotient rule dy/dx = (u′v − uv′)/v² with u = ln(x), v = x.” Evaluate the response against the complete 6-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[6 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: States the quotient rule dy/dx = (u′v − uv′)/v² with u = ln(x), v = x. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: Finds u′ = 1/x and v′ = 1. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Requirement 3: Identifies the missing requirement: Substitutes to obtain dy/dx = (x·(1/x) − ln(x)·1)/x². Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
5. Requirement 4: Identifies the missing requirement: Simplifies the numerator to 1 − ln(x), giving dy/dx = (1 − ln(x))/x². Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
6. Requirement 5: Identifies the missing requirement: Sets the numerator equal to zero: 1 − ln(x) = 0. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
7. Requirement 6: Identifies the missing requirement: Solves to obtain x = e. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
8. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: States the quotient rule dy/dx = (u′v − uv′)/v² with u = ln(x), v = x.
- Identifies the missing requirement: Finds u′ = 1/x and v′ = 1.
- Identifies the missing requirement: Substitutes to obtain dy/dx = (x·(1/x) − ln(x)·1)/x².
- Identifies the missing requirement: Simplifies the numerator to 1 − ln(x), giving dy/dx = (1 − ln(x))/x².
- Identifies the missing requirement: Sets the numerator equal to zero: 1 − ln(x) = 0.
- Identifies the missing requirement: Solves to obtain x = e.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
7.
The curve x² + y² = 25 passes through the point (3, 4). Use implicit differentiation to find dy/dx at this point.
[4 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Work through this mathematical step: Differentiates both sides with respect to x, treating y as a function of x: 2x + 2y(dy/dx) = 0. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Rearranges to isolate dy/dx: dy/dx = −x/y. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Substitutes x = 3, y = 4. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Work through this mathematical step: Obtains dy/dx = −3/4. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Every time y is differentiated with respect to x, the chain rule contributes a factor of dy/dx — this is the entire idea behind implicit differentiation.
Marking points
- Differentiates both sides with respect to x, treating y as a function of x: 2x + 2y(dy/dx) = 0.
- Rearranges to isolate dy/dx: dy/dx = −x/y.
- Substitutes x = 3, y = 4.
- Obtains dy/dx = −3/4.
Examiner tip: Every time y is differentiated with respect to x, the chain rule contributes a factor of dy/dx — this is the entire idea behind implicit differentiation.
8.
Marking analysis: A learner attempts the following task: “The curve x² + y² = 25 passes through the point (3, 4). Use implicit differentiation to find dy/dx at this point.” Their response addresses only this point: “Differentiates both sides with respect to x, treating y as a function of x: 2x + 2y(dy/dx) = 0.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: Differentiates both sides with respect to x, treating y as a function of x: 2x + 2y(dy/dx) = 0. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: Rearranges to isolate dy/dx: dy/dx = −x/y. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Requirement 3: Identifies the missing requirement: Substitutes x = 3, y = 4. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
5. Requirement 4: Identifies the missing requirement: Obtains dy/dx = −3/4. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: Differentiates both sides with respect to x, treating y as a function of x: 2x + 2y(dy/dx) = 0.
- Identifies the missing requirement: Rearranges to isolate dy/dx: dy/dx = −x/y.
- Identifies the missing requirement: Substitutes x = 3, y = 4.
- Identifies the missing requirement: Obtains dy/dx = −3/4.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
9.
A spherical balloon is inflated so that its volume increases at a constant rate of 100 cm³/s. Find the rate of increase of the radius when r = 5 cm. (V = (4/3)πr³)
[4 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Work through this mathematical step: Differentiates V = (4/3)πr³ with respect to t using the chain rule: dV/dt = 4πr²(dr/dt). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Substitutes dV/dt = 100 and r = 5. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Obtains 100 = 4π(25)(dr/dt) = 100π(dr/dt). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Work through this mathematical step: Solves for dr/dt = 1/π cm/s (≈ 0.318 cm/s). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Related rates problems always connect two rates through the chain rule via a shared variable and time — differentiate the geometric formula with respect to t before substituting any numbers.
Marking points
- Differentiates V = (4/3)πr³ with respect to t using the chain rule: dV/dt = 4πr²(dr/dt).
- Substitutes dV/dt = 100 and r = 5.
- Obtains 100 = 4π(25)(dr/dt) = 100π(dr/dt).
- Solves for dr/dt = 1/π cm/s (≈ 0.318 cm/s).
Examiner tip: Related rates problems always connect two rates through the chain rule via a shared variable and time — differentiate the geometric formula with respect to t before substituting any numbers.
10.
Marking analysis: A learner attempts the following task: “A spherical balloon is inflated so that its volume increases at a constant rate of 100 cm³/s. Find the rate of increase of the radius when r = 5 cm. (V = (4/3)πr³)” Their response addresses only this point: “Differentiates V = (4/3)πr³ with respect to t using the chain rule: dV/dt = 4πr²(dr/dt).” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: Differentiates V = (4/3)πr³ with respect to t using the chain rule: dV/dt = 4πr²(dr/dt). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: Substitutes dV/dt = 100 and r = 5. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Requirement 3: Identifies the missing requirement: Obtains 100 = 4π(25)(dr/dt) = 100π(dr/dt). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
5. Requirement 4: Identifies the missing requirement: Solves for dr/dt = 1/π cm/s (≈ 0.318 cm/s). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: Differentiates V = (4/3)πr³ with respect to t using the chain rule: dV/dt = 4πr²(dr/dt).
- Identifies the missing requirement: Substitutes dV/dt = 100 and r = 5.
- Identifies the missing requirement: Obtains 100 = 4π(25)(dr/dt) = 100π(dr/dt).
- Identifies the missing requirement: Solves for dr/dt = 1/π cm/s (≈ 0.318 cm/s).
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
11.
Solve the differential equation dy/dx = xy, given that y = 2 when x = 0.
[5 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Work through this mathematical step: Separates variables: (1/y) dy = x dx. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Integrates both sides: ln|y| = x²/2 + C. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Exponentiates both sides to obtain y = Ae^(x²/2), where A = e^C. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Work through this mathematical step: Substitutes the initial condition x = 0, y = 2 to find A = 2. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
6. Work through this mathematical step: States the particular solution y = 2e^(x²/2). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
7. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Always apply the initial condition after finding the general solution, not before — this determines the constant of integration and gives the one specific solution asked for.
Marking points
- Separates variables: (1/y) dy = x dx.
- Integrates both sides: ln|y| = x²/2 + C.
- Exponentiates both sides to obtain y = Ae^(x²/2), where A = e^C.
- Substitutes the initial condition x = 0, y = 2 to find A = 2.
- States the particular solution y = 2e^(x²/2).
Examiner tip: Always apply the initial condition after finding the general solution, not before — this determines the constant of integration and gives the one specific solution asked for.
12.
Marking analysis: A learner attempts the following task: “Solve the differential equation dy/dx = xy, given that y = 2 when x = 0.” Their response addresses only this point: “Separates variables: (1/y) dy = x dx.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[5 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: Separates variables: (1/y) dy = x dx. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: Integrates both sides: ln|y| = x²/2 + C. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Requirement 3: Identifies the missing requirement: Exponentiates both sides to obtain y = Ae^(x²/2), where A = e^C. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
5. Requirement 4: Identifies the missing requirement: Substitutes the initial condition x = 0, y = 2 to find A = 2. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
6. Requirement 5: Identifies the missing requirement: States the particular solution y = 2e^(x²/2). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
7. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: Separates variables: (1/y) dy = x dx.
- Identifies the missing requirement: Integrates both sides: ln|y| = x²/2 + C.
- Identifies the missing requirement: Exponentiates both sides to obtain y = Ae^(x²/2), where A = e^C.
- Identifies the missing requirement: Substitutes the initial condition x = 0, y = 2 to find A = 2.
- Identifies the missing requirement: States the particular solution y = 2e^(x²/2).
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
13.
Use integration by parts to find ∫x cos x dx.
[5 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Work through this mathematical step: States the integration by parts formula ∫u dv = uv − ∫v du. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Chooses u = x, dv = cos x dx, so du = dx and v = sin x. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Substitutes into the formula: x sin x − ∫sin x dx. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Work through this mathematical step: Integrates the remaining term: −∫sin x dx = cos x. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
6. Work through this mathematical step: States the final answer x sin x + cos x + C. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
7. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Choose u as the factor that simplifies when differentiated (here x becomes 1); choosing it the other way around makes the resulting integral harder, not easier.
Marking points
- States the integration by parts formula ∫u dv = uv − ∫v du.
- Chooses u = x, dv = cos x dx, so du = dx and v = sin x.
- Substitutes into the formula: x sin x − ∫sin x dx.
- Integrates the remaining term: −∫sin x dx = cos x.
- States the final answer x sin x + cos x + C.
Examiner tip: Choose u as the factor that simplifies when differentiated (here x becomes 1); choosing it the other way around makes the resulting integral harder, not easier.
14.
Marking analysis: A learner attempts the following task: “Use integration by parts to find ∫x cos x dx.” Their response addresses only this point: “States the integration by parts formula ∫u dv = uv − ∫v du.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[5 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: States the integration by parts formula ∫u dv = uv − ∫v du. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: Chooses u = x, dv = cos x dx, so du = dx and v = sin x. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Requirement 3: Identifies the missing requirement: Substitutes into the formula: x sin x − ∫sin x dx. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
5. Requirement 4: Identifies the missing requirement: Integrates the remaining term: −∫sin x dx = cos x. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
6. Requirement 5: Identifies the missing requirement: States the final answer x sin x + cos x + C. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
7. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: States the integration by parts formula ∫u dv = uv − ∫v du.
- Identifies the missing requirement: Chooses u = x, dv = cos x dx, so du = dx and v = sin x.
- Identifies the missing requirement: Substitutes into the formula: x sin x − ∫sin x dx.
- Identifies the missing requirement: Integrates the remaining term: −∫sin x dx = cos x.
- Identifies the missing requirement: States the final answer x sin x + cos x + C.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
15.
Find the Maclaurin series for f(x) = eˣ up to and including the term in x³.
[4 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Work through this mathematical step: States the Maclaurin series formula f(x) = f(0) + f′(0)x + f″(0)x²/2! + f‴(0)x³/3! + … Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Notes that every derivative of eˣ is eˣ, so f(0) = f′(0) = f″(0) = f‴(0) = 1. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Substitutes these values into the series formula. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Work through this mathematical step: States the series 1 + x + x²/2 + x³/6 (equivalently x²/2! + x³/3!). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: For eˣ every derivative equals the function itself, which is exactly why every coefficient in its Maclaurin series is simply 1/n! — no differentiation errors are possible if this is remembered.
Marking points
- States the Maclaurin series formula f(x) = f(0) + f′(0)x + f″(0)x²/2! + f‴(0)x³/3! + …
- Notes that every derivative of eˣ is eˣ, so f(0) = f′(0) = f″(0) = f‴(0) = 1.
- Substitutes these values into the series formula.
- States the series 1 + x + x²/2 + x³/6 (equivalently x²/2! + x³/3!).
Examiner tip: For eˣ every derivative equals the function itself, which is exactly why every coefficient in its Maclaurin series is simply 1/n! — no differentiation errors are possible if this is remembered.
16.
Marking analysis: A learner attempts the following task: “Find the Maclaurin series for f(x) = eˣ up to and including the term in x³.” Their response addresses only this point: “States the Maclaurin series formula f(x) = f(0) + f′(0)x + f″(0)x²/2! + f‴(0)x³/3! + …” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: States the Maclaurin series formula f(x) = f(0) + f′(0)x + f″(0)x²/2! + f‴(0)x³/3! + … Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: Notes that every derivative of eˣ is eˣ, so f(0) = f′(0) = f″(0) = f‴(0) = 1. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Requirement 3: Identifies the missing requirement: Substitutes these values into the series formula. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
5. Requirement 4: Identifies the missing requirement: States the series 1 + x + x²/2 + x³/6 (equivalently x²/2! + x³/3!). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: States the Maclaurin series formula f(x) = f(0) + f′(0)x + f″(0)x²/2! + f‴(0)x³/3! + …
- Identifies the missing requirement: Notes that every derivative of eˣ is eˣ, so f(0) = f′(0) = f″(0) = f‴(0) = 1.
- Identifies the missing requirement: Substitutes these values into the series formula.
- Identifies the missing requirement: States the series 1 + x + x²/2 + x³/6 (equivalently x²/2! + x³/3!).
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
17.
Evaluate lim(x→0) (sin x)/x using L'Hôpital's rule, verifying that the limit is initially of the indeterminate form 0/0.
[4 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Build a supported judgement: identify the claim, use the question's evidence, consider a relevant limitation or alternative, and make the conclusion depend on that evidence. There may be more than one defensible answer.
2. Work through this mathematical step: States that direct substitution of x = 0 gives sin(0)/0 = 0/0, an indeterminate form. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: States L'Hôpital's rule: for a 0/0 form, lim f(x)/g(x) = lim f′(x)/g′(x). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Differentiates numerator and denominator separately: f′(x) = cos x, g′(x) = 1. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Work through this mathematical step: Substitutes x = 0 into cos x/1 to obtain the limit = 1. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: L'Hôpital's rule only applies once you have confirmed the limit is genuinely in an indeterminate form (0/0 or ∞/∞) — applying it to a limit that already evaluates directly gives a wrong answer.
Marking points
- States that direct substitution of x = 0 gives sin(0)/0 = 0/0, an indeterminate form.
- States L'Hôpital's rule: for a 0/0 form, lim f(x)/g(x) = lim f′(x)/g′(x).
- Differentiates numerator and denominator separately: f′(x) = cos x, g′(x) = 1.
- Substitutes x = 0 into cos x/1 to obtain the limit = 1.
Examiner tip: L'Hôpital's rule only applies once you have confirmed the limit is genuinely in an indeterminate form (0/0 or ∞/∞) — applying it to a limit that already evaluates directly gives a wrong answer.
18.
Marking analysis: A learner attempts the following task: “Evaluate lim(x→0) (sin x)/x using L'Hôpital's rule, verifying that the limit is initially of the indeterminate form 0/0.” Their response addresses only this point: “States that direct substitution of x = 0 gives sin(0)/0 = 0/0, an indeterminate form.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: States that direct substitution of x = 0 gives sin(0)/0 = 0/0, an indeterminate form. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: States L'Hôpital's rule: for a 0/0 form, lim f(x)/g(x) = lim f′(x)/g′(x). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Requirement 3: Identifies the missing requirement: Differentiates numerator and denominator separately: f′(x) = cos x, g′(x) = 1. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
5. Requirement 4: Identifies the missing requirement: Substitutes x = 0 into cos x/1 to obtain the limit = 1. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: States that direct substitution of x = 0 gives sin(0)/0 = 0/0, an indeterminate form.
- Identifies the missing requirement: States L'Hôpital's rule: for a 0/0 form, lim f(x)/g(x) = lim f′(x)/g′(x).
- Identifies the missing requirement: Differentiates numerator and denominator separately: f′(x) = cos x, g′(x) = 1.
- Identifies the missing requirement: Substitutes x = 0 into cos x/1 to obtain the limit = 1.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
19.
The curve y = x³ − 6x² + 9x + 1 has two stationary points. Find their x-coordinates, and use the second derivative test to determine the nature of each.
[5 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Work through this mathematical step: Finds dy/dx = 3x² − 12x + 9. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Sets dy/dx = 0 and factorises: 3(x − 1)(x − 3) = 0, giving x = 1 and x = 3. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Finds the second derivative d²y/dx² = 6x − 12. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Work through this mathematical step: Evaluates at x = 1: d²y/dx² = −6 < 0, concluding a local maximum. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
6. Work through this mathematical step: Evaluates at x = 3: d²y/dx² = 6 > 0, concluding a local minimum. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
7. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: A negative second derivative at a stationary point confirms a local maximum (curve is concave down); a positive value confirms a local minimum (concave up) — this is faster than a sign-change table when the second derivative is easy to evaluate.
Marking points
- Finds dy/dx = 3x² − 12x + 9.
- Sets dy/dx = 0 and factorises: 3(x − 1)(x − 3) = 0, giving x = 1 and x = 3.
- Finds the second derivative d²y/dx² = 6x − 12.
- Evaluates at x = 1: d²y/dx² = −6 < 0, concluding a local maximum.
- Evaluates at x = 3: d²y/dx² = 6 > 0, concluding a local minimum.
Examiner tip: A negative second derivative at a stationary point confirms a local maximum (curve is concave down); a positive value confirms a local minimum (concave up) — this is faster than a sign-change table when the second derivative is easy to evaluate.
20.
Marking analysis: A learner attempts the following task: “The curve y = x³ − 6x² + 9x + 1 has two stationary points. Find their x-coordinates, and use the second derivative test to determine the nature of each.” Their response addresses only this point: “Finds dy/dx = 3x² − 12x + 9.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[5 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: Finds dy/dx = 3x² − 12x + 9. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: Sets dy/dx = 0 and factorises: 3(x − 1)(x − 3) = 0, giving x = 1 and x = 3. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Requirement 3: Identifies the missing requirement: Finds the second derivative d²y/dx² = 6x − 12. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
5. Requirement 4: Identifies the missing requirement: Evaluates at x = 1: d²y/dx² = −6 < 0, concluding a local maximum. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
6. Requirement 5: Identifies the missing requirement: Evaluates at x = 3: d²y/dx² = 6 > 0, concluding a local minimum. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
7. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: Finds dy/dx = 3x² − 12x + 9.
- Identifies the missing requirement: Sets dy/dx = 0 and factorises: 3(x − 1)(x − 3) = 0, giving x = 1 and x = 3.
- Identifies the missing requirement: Finds the second derivative d²y/dx² = 6x − 12.
- Identifies the missing requirement: Evaluates at x = 1: d²y/dx² = −6 < 0, concluding a local maximum.
- Identifies the missing requirement: Evaluates at x = 3: d²y/dx² = 6 > 0, concluding a local minimum.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
21.
Calculate the area enclosed between the curve y = 4 − x² and the x-axis.
[5 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Work through this mathematical step: Finds the roots of 4 − x² = 0: x = −2 and x = 2, giving the limits of integration. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Sets up the definite integral: ∫₋₂² (4 − x²) dx. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Integrates to obtain 4x − x³/3. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Work through this mathematical step: Evaluates at the limits: (8 − 8/3) − (−8 + 8/3). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
6. Work through this mathematical step: Obtains area = 32/3. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
7. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: The roots of the curve give the natural limits of integration when finding the area enclosed with the x-axis — always find these first before setting up the definite integral.
Marking points
- Finds the roots of 4 − x² = 0: x = −2 and x = 2, giving the limits of integration.
- Sets up the definite integral: ∫₋₂² (4 − x²) dx.
- Integrates to obtain 4x − x³/3.
- Evaluates at the limits: (8 − 8/3) − (−8 + 8/3).
- Obtains area = 32/3.
Examiner tip: The roots of the curve give the natural limits of integration when finding the area enclosed with the x-axis — always find these first before setting up the definite integral.
22.
Marking analysis: A learner attempts the following task: “Calculate the area enclosed between the curve y = 4 − x² and the x-axis.” Their response addresses only this point: “Finds the roots of 4 − x² = 0: x = −2 and x = 2, giving the limits of integration.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[5 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: Finds the roots of 4 − x² = 0: x = −2 and x = 2, giving the limits of integration. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: Sets up the definite integral: ∫₋₂² (4 − x²) dx. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Requirement 3: Identifies the missing requirement: Integrates to obtain 4x − x³/3. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
5. Requirement 4: Identifies the missing requirement: Evaluates at the limits: (8 − 8/3) − (−8 + 8/3). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
6. Requirement 5: Identifies the missing requirement: Obtains area = 32/3. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
7. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: Finds the roots of 4 − x² = 0: x = −2 and x = 2, giving the limits of integration.
- Identifies the missing requirement: Sets up the definite integral: ∫₋₂² (4 − x²) dx.
- Identifies the missing requirement: Integrates to obtain 4x − x³/3.
- Identifies the missing requirement: Evaluates at the limits: (8 − 8/3) − (−8 + 8/3).
- Identifies the missing requirement: Obtains area = 32/3.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
23.
The region bounded by y = x², the x-axis, and the line x = 2 is rotated 360° about the x-axis. Calculate the volume of the solid formed, using V = π∫y² dx.
[5 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Work through this mathematical step: States V = π∫y² dx. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Substitutes y² = (x²)² = x⁴. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Sets up V = π∫₀² x⁴ dx. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Work through this mathematical step: Integrates to π[x⁵/5] evaluated from 0 to 2. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
6. Work through this mathematical step: Obtains V = 32π/5. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
7. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Always square the function first (y²), not the volume formula itself, before integrating — squaring x² gives x⁴, not x³ or 2x².
Marking points
- States V = π∫y² dx.
- Substitutes y² = (x²)² = x⁴.
- Sets up V = π∫₀² x⁴ dx.
- Integrates to π[x⁵/5] evaluated from 0 to 2.
- Obtains V = 32π/5.
Examiner tip: Always square the function first (y²), not the volume formula itself, before integrating — squaring x² gives x⁴, not x³ or 2x².
24.
Marking analysis: A learner attempts the following task: “The region bounded by y = x², the x-axis, and the line x = 2 is rotated 360° about the x-axis. Calculate the volume of the solid formed, using V = π∫y² dx.” Their response addresses only this point: “States V = π∫y² dx.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[5 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: States V = π∫y² dx. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: Substitutes y² = (x²)² = x⁴. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Requirement 3: Identifies the missing requirement: Sets up V = π∫₀² x⁴ dx. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
5. Requirement 4: Identifies the missing requirement: Integrates to π[x⁵/5] evaluated from 0 to 2. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
6. Requirement 5: Identifies the missing requirement: Obtains V = 32π/5. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
7. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: States V = π∫y² dx.
- Identifies the missing requirement: Substitutes y² = (x²)² = x⁴.
- Identifies the missing requirement: Sets up V = π∫₀² x⁴ dx.
- Identifies the missing requirement: Integrates to π[x⁵/5] evaluated from 0 to 2.
- Identifies the missing requirement: Obtains V = 32π/5.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
25.
Differentiate y = arctan(3x), using the standard result d/dx[arctan(u)] = u′/(1 + u²).
[3 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Break the command into its requested parts. For each part, connect a relevant fact or observation to the conclusion it supports. Describing what happens and explaining why it happens are different tasks.
2. Work through this mathematical step: States the formula d/dx[arctan(u)] = u′/(1 + u²), with u = 3x. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Finds u′ = 3. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Substitutes to obtain dy/dx = 3/(1 + 9x²). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: This is simply the chain rule applied to the standard arctan derivative — treat u as the 'inside function' exactly as with any other composite function.
Marking points
- States the formula d/dx[arctan(u)] = u′/(1 + u²), with u = 3x.
- Finds u′ = 3.
- Substitutes to obtain dy/dx = 3/(1 + 9x²).
Examiner tip: This is simply the chain rule applied to the standard arctan derivative — treat u as the 'inside function' exactly as with any other composite function.
26.
Marking analysis: A learner attempts the following task: “Differentiate y = arctan(3x), using the standard result d/dx[arctan(u)] = u′/(1 + u²).” Their response addresses only this point: “States the formula d/dx[arctan(u)] = u′/(1 + u²), with u = 3x.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] · no calculator
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: States the formula d/dx[arctan(u)] = u′/(1 + u²), with u = 3x. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: Finds u′ = 3. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Requirement 3: Identifies the missing requirement: Substitutes to obtain dy/dx = 3/(1 + 9x²). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: States the formula d/dx[arctan(u)] = u′/(1 + u²), with u = 3x.
- Identifies the missing requirement: Finds u′ = 3.
- Identifies the missing requirement: Substitutes to obtain dy/dx = 3/(1 + 9x²).
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
27.
Given g(x) = ∫₁ˣ (t² + 1) dt, use the fundamental theorem of calculus to find g′(x), and hence evaluate g′(2).
[3 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Build a supported judgement: identify the claim, use the question's evidence, consider a relevant limitation or alternative, and make the conclusion depend on that evidence. There may be more than one defensible answer.
2. Work through this mathematical step: States the fundamental theorem of calculus: if g(x) = ∫ₐˣ f(t) dt, then g′(x) = f(x). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: States g′(x) = x² + 1. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Substitutes x = 2 to obtain g′(2) = 5. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: The fundamental theorem of calculus means differentiating an integral simply 'undoes' the integration, returning the original integrand evaluated at the variable upper limit — no actual integration is needed to answer this type of question.
Marking points
- States the fundamental theorem of calculus: if g(x) = ∫ₐˣ f(t) dt, then g′(x) = f(x).
- States g′(x) = x² + 1.
- Substitutes x = 2 to obtain g′(2) = 5.
Examiner tip: The fundamental theorem of calculus means differentiating an integral simply 'undoes' the integration, returning the original integrand evaluated at the variable upper limit — no actual integration is needed to answer this type of question.
28.
Marking analysis: A learner attempts the following task: “Given g(x) = ∫₁ˣ (t² + 1) dt, use the fundamental theorem of calculus to find g′(x), and hence evaluate g′(2).” Their response addresses only this point: “States the fundamental theorem of calculus: if g(x) = ∫ₐˣ f(t) dt, then g′(x) = f(x).” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: States the fundamental theorem of calculus: if g(x) = ∫ₐˣ f(t) dt, then g′(x) = f(x). Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: States g′(x) = x² + 1. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Requirement 3: Identifies the missing requirement: Substitutes x = 2 to obtain g′(2) = 5. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: States the fundamental theorem of calculus: if g(x) = ∫ₐˣ f(t) dt, then g′(x) = f(x).
- Identifies the missing requirement: States g′(x) = x² + 1.
- Identifies the missing requirement: Substitutes x = 2 to obtain g′(2) = 5.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
29.
Water enters a recycling tank at 12 L min⁻¹. At time t minutes the tank contains V litres, and water leaves at a rate of 0.08V L min⁻¹. Initially V = 50. (a) Write a differential equation for V. (b) Find the equilibrium volume. (c) Solve for V in terms of t. (d) Determine the first time when V reaches 120 L.
[5 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
2. Work through this mathematical step: Writes dV/dt = 12 − 0.08V. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
3. Work through this mathematical step: Sets dV/dt = 0 and obtains the equilibrium volume V = 150 L. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
4. Work through this mathematical step: Obtains the general form V = 150 + Ce⁻⁰·⁰⁸ᵗ. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
5. Work through this mathematical step: Uses V(0) = 50 to obtain V = 150 − 100e⁻⁰·⁰⁸ᵗ. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
6. Work through this mathematical step: Solves 120 = 150 − 100e⁻⁰·⁰⁸ᵗ to obtain t = −ln(0.3)/0.08 ≈ 15.0 min. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
7. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Check the solution against both the initial condition and the long-term equilibrium; both should agree with the physical model.
Marking points
- Writes dV/dt = 12 − 0.08V.
- Sets dV/dt = 0 and obtains the equilibrium volume V = 150 L.
- Obtains the general form V = 150 + Ce⁻⁰·⁰⁸ᵗ.
- Uses V(0) = 50 to obtain V = 150 − 100e⁻⁰·⁰⁸ᵗ.
- Solves 120 = 150 − 100e⁻⁰·⁰⁸ᵗ to obtain t = −ln(0.3)/0.08 ≈ 15.0 min.
Examiner tip: Check the solution against both the initial condition and the long-term equilibrium; both should agree with the physical model.
30.
Marking analysis: A learner attempts the following task: “Water enters a recycling tank at 12 L min⁻¹. At time t minutes the tank contains V litres, and water leaves at a rate of 0.08V L min⁻¹. Initially V = 50. (a) Write a differential equation for V. (b) Find the equilibrium volume. (c) Solve for V in terms of t. (d) Determine the first time when V reaches 120 L.” Their response addresses only this point: “Writes dV/dt = 12 − 0.08V.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[5 marks]
Answer explanation
Draft walkthroughs are based on marking guidance, not independently verified derivations.
1. Separate the learner's stated response from the complete task. Credit only what their response demonstrates, then identify each missing requirement; do not assume unstated working.
2. Requirement 1: Recognises credit for the stated point: Writes dV/dt = 12 − 0.08V. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
3. Requirement 2: Identifies the missing requirement: Sets dV/dt = 0 and obtains the equilibrium volume V = 150 L. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
4. Requirement 3: Identifies the missing requirement: Obtains the general form V = 150 + Ce⁻⁰·⁰⁸ᵗ. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
5. Requirement 4: Identifies the missing requirement: Uses V(0) = 50 to obtain V = 150 − 100e⁻⁰·⁰⁸ᵗ. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
6. Requirement 5: Identifies the missing requirement: Solves 120 = 150 − 100e⁻⁰·⁰⁸ᵗ to obtain t = −ln(0.3)/0.08 ≈ 15.0 min. Compare this requirement with the supplied learner response; missing evidence cannot earn credit.
7. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.
Marking points
- Recognises credit for the stated point: Writes dV/dt = 12 − 0.08V.
- Identifies the missing requirement: Sets dV/dt = 0 and obtains the equilibrium volume V = 150 L.
- Identifies the missing requirement: Obtains the general form V = 150 + Ce⁻⁰·⁰⁸ᵗ.
- Identifies the missing requirement: Uses V(0) = 50 to obtain V = 150 − 100e⁻⁰·⁰⁸ᵗ.
- Identifies the missing requirement: Solves 120 = 150 − 100e⁻⁰·⁰⁸ᵗ to obtain t = −ln(0.3)/0.08 ≈ 15.0 min.
Examiner tip: Treat each marking point as a separate requirement. Do not award the same idea twice, and do not infer work the learner did not show.