Mathematical Methods — Question 13

QCAA 2021, Paper 1 · 5 marks

Q13 · 2021 · Technology-freeSimple familiar5 marks

QUESTION 13 (5 marks)

Consider the functions

f(x)=x^2

and

g(x)=4x

Determine the

x

-coordinates of the points of intersection of the graphs of the two functions.

[2 marks]

Use the results from Question 13a) to calculate the area enclosed by the graphs of

f(x)

and

g(x)

[3 marks]

WORKED SOLUTION

5 marks

ANSWER

x=0,4

; b)

\dfrac{32}{3}

square units

Worked solution

a) Solving simultaneously,

x^2=4x

, so

x(x-4)=0

and therefore

x=0

and

x=4

\text{Area}=\int_0^4(4x-x^2)\,dx=\left[2x^2-\frac{x^3}{3}\right]_0^4=\frac{32}{3}\text{ square units}.

correctly uses the simultaneous procedure

[1 mark]

correctly determines both the x-intercept ordinates

[1 mark]

correctly determines the integral

[1 mark]

substitutes limits into integral

[1 mark]

determines area

[1 mark]

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

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