IB Maths AA HL Topic 5: Calculus questions

Exam-style questions in the shape of Paper 1 and Paper 2, with full markschemes. Read the markscheme, or type, photograph or draw your answer and get it marked.

Paper
Paper 1, no calculatorPaper 2, calculator
Difficulty
EasyMediumHard

Question 1

Paper 2, calculatorHard6 marks

Let nn be a fixed positive integer and let f(x)=xnf(x)=x^n where x∈Rx\in\mathbb R

(a)

Using the definition of the derivative from first principles, show that

f′(x)=nxn−1.f'(x)=nx^{n-1}.
[3]
(b)

For an integer rr with 1≤r≤n1\le r\le n prove that

f(r)(x)=n!(n−r)!xn−r.f^{(r)}(x)=\frac{n!}{(n-r)!}x^{n-r}.
[3]

Question 2

Paper 2, calculatorEasy6 marks

Consider the functions f(x)=x3−3xf(x)=x^3-3x and g(x)=x3g(x)=x^3

The stationary points of ff are (−1,2)(-1,2) and (1,−2)(1,-2)

(a)

Use the second derivative test to classify each stationary point of ff

[2]
(b)

Determine whether (0,0)(0,0) is a point of inflexion of ff and state the gradient there.

[2]
(c)

Determine whether (0,0)(0,0) is a point of inflexion of gg and state the gradient there.

[2]

Question 3

Paper 2, calculatorHard6 marks

A manufacturer models the permitted side lengths xx and yy in metres, of a rectangular mounting panel by

x2+xy+2y2=36.x^2+xy+2y^2=36.

The positive value of yy is used, and installation requirements restrict xx to 4≤x≤54\leq x\leq 5 The area of the panel is A=xyA=xy

(a)

Determine dydx\dfrac{dy}{dx} in terms of xx and yy

[2]
(b)

Hence, or otherwise, determine the maximum possible area of the panel on the allowed interval.

[4]

Question 4

Paper 2, calculatorHard6 marks

For a real parameter k>0k>0 let

fk(x)=x2−kx2+1,−3≤x≤3.f_k(x)=\frac{x^2-k}{x^2+1},\qquad -3\leq x\leq 3.

It is known that

∫−33fk(x) dx=0.\int_{-3}^{3} f_k(x)\,dx=0.
(a)

Determine the exact value of kk

[2]
(b)

Hence determine the total area between the graph of y=fk(x)y=f_k(x) and the xx-axis for −3≤x≤3-3\leq x\leq 3 Show a definite-integral expression before evaluating. Do not use numerical integration. Give your answer to 3 significant figures.

[4]

Question 5

Paper 1, no calculatorHard6 marks

Figure 1 shows the graph of ff on the interval shown.

0π/4π/23π/4π02468xf(x)f(x)=ex sin x, 0≤x≤πR
(a)

Determine an antiderivative of ff

[4]
(b)

Hence determine, in exact form, the area of the shaded region RR

[2]

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