IB Maths AI 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 Paper 2

Difficulty

Easy Medium Hard

Question 1

Paper 1Easy5 marks

A manufacturer models its monthly profit, $P$ in thousands of dollars, from producing $q$ hundreds of units by

P(q)=-0.12q^3+1.6q^2-3.5q,\qquad 0\leq q\leq 10.

Figure 1 shows the graph of this model.

(a)

Use your GDC to generate $P'(q)$ and find, to two decimal places, the values of $q$ at which the gradient of the profit curve is zero.

[2]

(b)

For each value of $q$ found in part (a), calculate the corresponding monthly profit. Give your answers in thousands of dollars, to three significant figures.

[2]

(c)

Use Figure 1 to state which stationary point is a local maximum and which is a local minimum.

[1]

Question 2

Paper 1Easy6 marks

A bakery models its weekly profit, $P$ in thousands of dollars, when it produces $q$ hundreds of loaves per week, by

P(q)=q^3-6q^2+9q+2,\qquad 0\leq q\leq 4.

Figure 1 shows the graph of this model.

(a)

Use a GDC to generate $P'(q)$ then solve $P'(q)=0$ Give your answers to three significant figures.

[3]

(b)

Use Figure 1 to write down the coordinates of the local maximum point and the local minimum point.

[2]

(c)

Interpret the local maximum point in the context of the bakery.

[1]

Question 3

Paper 2Easy6 marks

A surveyor estimates the area of an irregular lake by measuring its width at equal intervals along a straight line through the lake. The measured widths are shown below.

Distance along the line, $x$ (m)	0	25	50	75	100
Width of lake (m)	0	18	24	17	0

(a)

Using the trapezoidal rule, show that the estimated area of the lake between $x=0$ and $x=50$ is $750\text{ m}^2$

[3]

(b)

Using your answer to part (a) and the remaining measurements, calculate an estimate of the total area of the lake.

[3]

Question 4

Paper 2Hard8 marks

During a test, a small cart moves along a straight track. For $0\leq t\leq 8$ its acceleration is modelled by

\frac{dv}{dt}=-0.072t^2+0.610t-1.05\quad\text{m s}^{-2},

where $t$ is in seconds and $v$ is in $\text{m s}^{-1}$ The cart’s velocity at $t=0$ is $1.02\text{ m s}^{-1}$

(a)

Show that

v(t)=-0.024t^3+0.305t^2-1.05t+1.02.

[2]

(b)

Use your GDC to find all values of $t$ for which the cart is momentarily at rest. Give your answers to three significant figures.

[2]

(c)

A technician claims that, since the cart starts moving in the positive direction, its forward distance over $0\leq t\leq 8$ is $\int_0^8 v(t)\,dt$

Assess this claim, giving a reason. Then determine the total distance travelled while the cart is moving in the positive direction. Use your results from part (b), write the definite-integral expression(s) before evaluating them with your GDC, and give the distance to three significant figures.

[4]

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