IB Maths AA SL 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, calculatorMedium6 marks

For x∈Rx\in\mathbb{R} let f(x)=(x+2)e−xf(x)=(x+2)e^{-x}

(a)

Find f′(x)f'(x)

[2]
(b)

Hence determine the coordinates of the local maximum of the curve y=f(x)y=f(x) justifying that it is a maximum.

[4]

Question 2

Paper 1, no calculatorMedium5 marks

The curve CC has equation y=f(x)=x3−3xy=f(x)=x^3-3x for −2≤x≤2-2\leq x\leq 2 Figure 1 shows CC

-2-112-3-2-10123xy
(a)

Use Figure 1 to state the xx-coordinates of the stationary points of CC

[1]
(b)

Determine whether each stationary point is a local maximum or a local minimum. Justify your answers.

[2]
(c)

Show that (0,0)(0,0) is a point of inflexion of CC and determine the gradient of CC there. State whether this gradient is zero or non-zero.

[2]

Question 3

Paper 2, calculatorMedium6 marks

The function ff is defined, for x∈Rx\in\mathbb R by

f(x)=14x4−32x2+2x.f(x)=\frac14x^4-\frac32x^2+2x.

Figure 1 shows the graph of f′(x)=(x+2)(x−1)2f'(x)=(x+2)(x-1)^2 for −3≤x≤3-3\leq x\leq 3

-3-2-1123-18-16-14-12-10-8-6-4-20246810121416182022xf'(x)
(a)

Use Figure 1 to state the xx-coordinates of the stationary points of ff and classify each as a local maximum, a local minimum or neither. Justify your classifications using the sign of f′(x)f'(x) on either side.

[3]
(b)

Use Figure 1 to determine the xx-coordinates of the points of inflexion of ff Hence determine the coordinates of both points of inflexion and the gradient of ff at each.

[3]

Question 4

Paper 2, calculatorMedium5 marks

A particle moves along a straight line. Figure 1 shows its velocity, vv during the first 77 seconds of its motion. The graph consists of straight-line segments.

01234567-4-3-2-10123456t / sv / m s-1Figure 1
(a)

Determine the acceleration of the particle during the first 33 seconds. Give your answer with units.

[1]
(b)

Determine the displacement of the particle during the 77 seconds. Give your answer with units.

[2]
(c)

Hence, determine the total distance travelled by the particle during the 77 seconds. Give your answer with units.

[2]

Question 5

Paper 2, calculatorMedium6 marks

The curve CC has equation y=f(x)y=f(x) where

f(x)=x2−x−2.f(x)=x^2-x-2.
(a)

Find the xx-coordinates of the points where CC intersects the xx-axis.

[2]
(b)

Consider the region between CC and the xx-axis from x=−2x=-2 to x=3x=3

(i)

Write down an expression, as a sum or difference of definite integrals of ff for the total area of this region. Do not use absolute-value notation.

[2]
(ii)

Hence determine the total area exactly.

[2]

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