Mathematical Methods — Question 11

QCAA 2024, Paper 1 · 6 marks

Q11 · 2024 · Technology-freeSimple familiar6 marks

QUESTION 11 (6 marks)

Determine the second derivative of

y=x^3-3x^2

[2 marks]

Use your result from Question 11a) to calculate the value of the second derivative when

x=-1

[1 mark]

Determine the

x

- and

y

-coordinates of the point on the graph of

y=x^3-3x^2

for which the rate of change of the first derivative is zero.

[3 marks]

WORKED SOLUTION

6 marks

ANSWER

\dfrac{d^2y}{dx^2}=6x-6

; b)

-12

; c)

(1,-2)

Worked solution

\frac{dy}{dx}=3x^2-6x,

\frac{d^2y}{dx^2}=6x-6.

b) When

x=-1

\frac{d^2y}{dx^2}=6(-1)-6=-12.

c) The rate of change of the first derivative is zero when

\frac{d^2y}{dx^2}=0.

Thus

6x-6=0

, so

x=1

. Substituting into the graph equation,

y=(1)^3-3(1)^2=-2.

The coordinates are

(1,-2)

correctly determines the first derivative

[1 mark]

determines the second derivative

[1 mark]

determines the value of the second derivative

[1 mark]

equates the second derivative to zero

[1 mark]

determines the x-coordinate of the point

[1 mark]

determines the y-coordinate of the point

[1 mark]

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

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