Mathematical Methods — Question 13

QCAA 2022, Paper 1 · 9 marks

Q13 · 2022 · Technology-freeSimple familiar9 marks

QUESTION 13 (9 marks)

Determine the derivative of

f(x)=3e^{2x+1}

[1 mark]

Given that

g(x)=\dfrac{\ln(x)}{x}

, determine the simplest value of

g'(e)

[3 marks]

Determine the second derivative of

h(x)=x\sin(x)

. (Give your answer in simplest form.)

[5 marks]

WORKED SOLUTION

9 marks

ANSWER

f'(x)=6e^{2x+1}

; b)

g'(e)=0

; c)

h''(x)=2\cos x-x\sin x

Worked solution

f'(x)=6e^{2x+1}.

b) Using the quotient rule,

g'(x)=\frac{x(1/x)-\ln(x)}{x^2}=\frac{1-\ln(x)}{x^2},

g'(e)=\frac{1-\ln e}{e^2}=0.

c) Using the product rule,

h'(x)=x\cos x+\sin x.

Differentiating again,

h''(x)=-x\sin x+\cos x+\cos x=2\cos x-x\sin x.

correctly determines the derivative

[1 mark]

correctly identifies the use of the product or quotient rule

[1 mark]

correctly determines the derivative

[1 mark]

determines the derivative at the given value

[1 mark]

correctly identifies the use of the product rule

[1 mark]

correctly determines the derivative

[1 mark]

correctly identifies the use of the product rule and that the derivative of a sum is the sum of the derivatives

[1 mark]

determines the second derivative

[1 mark]

simplifies the second derivative

[1 mark]

One QCAA sample method typeset for web; criterion wording is adapted from the official marking guide.

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