ACSEE (Form 6) Differentiation — practice questions

Explore 3 free Differentiation questions in ACSEE (Form 6) Advanced Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Differentiation questions and exercises with answers

Showing questions 1–3 of 3.

Using the second derivative test on $y=3xe^{-x}$, the stationary point is:Read the answer and worked solution →An air pollution index is $p=x^2+2xy+4xy^2$. At the point $(10,5)$, the partial derivatives $\dfrac{\partial p}{\partial x}$ and $\dfrac{\partial p}{\partial y}$ are:Read the answer and worked solution →Second derivative testRead the answer and worked solution →

Worked practice questions

Using the second derivative test on $y=3xe^{-x}$, the stationary point is:

a minimum at $x=1$, $y=\frac{3}{e}$
a maximum at $x=0$, $y=0$
a maximum at $x=1$, $y=\frac{3}{e}$
a point of inflexion at $x=1$

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Correct answer: C. a maximum at $x=1$, $y=\frac{3}{e}$

1. Differentiate by the product rule: $\dfrac{dy}{dx}=3e^{-x}-3xe^{-x}=3e^{-x}(1-x)$. 2. Setting $\dfrac{dy}{dx}=0$ and noting $e^{-x}\neq 0$ gives $x=1$. 3. Then $y=3(1)e^{-1}=\dfrac{3}{e}$. 4. Second derivative: $\dfrac{d^2y}{dx^2}=-6e^{-x}+3xe^{-x}$. At $x=1$ this is $-6e^{-1}+3e^{-1}=-3e^{-1}<0$. 5. A negative second derivative means a maximum. Writing $y'=3xe^{-x}+3e^{-x}$ loses the minus sign and moves the stationary point. _Source: NECTA ACSEE 2023 Advanced Mathematics 142/1, Question 10: Differentiation_

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An air pollution index is $p=x^2+2xy+4xy^2$. At the point $(10,5)$, the partial derivatives $\dfrac{\partial p}{\partial x}$ and $\dfrac{\partial p}{\partial y}$ are:

420 and 130
120 and 420
130 and 400
130 and 420

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Correct answer: D. 130 and 420

1. Differentiate with respect to $x$, treating $y$ as constant: $\dfrac{\partial p}{\partial x}=2x+2y+4y^2$. 2. At $(10,5)$: $20+10+4(25)=20+10+100=130$. 3. Differentiate with respect to $y$, treating $x$ as constant: $\dfrac{\partial p}{\partial y}=2x+8xy$. 4. At $(10,5)$: $20+8(10)(5)=20+400=420$. 5. Substituting the point before differentiating, a common slip, leaves an $x$ or $y$ still in the answer. _Source: NECTA ACSEE 2023 Advanced Mathematics 142/1, Question 10: Differentiation_

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Second derivative test

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Questions

Statement is true
Yes
Statement
A negative second derivative at a stationary point means a maximum
Statement is true
Yes
Statement
3xe^(−x) differentiates to 3e^(−x)(1 − x)
Statement is true
No
Statement
Setting dy/dx = 0 for 3xe^(−x) gives x = 0
Statement is true
No
Statement
For partial derivatives you substitute the point before differentiating
Statement is true
Yes
Statement
Differentiating p = x² + 2xy + 4xy² by x treats y as a constant

Instructions

Swipe true or false for each claim.

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