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]

Hence determine the intervals in the model domain for which $\frac{dh}{dt}>0$ and $\frac{dh}{dt}<0$ State when the water level is increasing and when it is decreasing. Give boundary times to 2 decimal places.

[4]

Question 4

Paper 1Medium6 marks

A manufacturer models its profit, $P$ in thousands of dollars, when producing $x$ thousand units, by

P(x)=\frac{x^3}{3000}-0.0204x^2+0.27x+20,\qquad 0\leq x\leq45.

The model is shown in Figure 1.

(a)

Using a GDC, determine the values of $x$ at which the gradient of the curve is zero. Give your answers to 1 decimal place.

[2]

(b)

Hence, find the coordinates of both stationary points and state whether each is a local maximum or a local minimum. Give the profit coordinates to 2 decimal places.

[2]

(c)

The owner claims that the local maximum gives the greatest profit in the model’s domain. Use Figure 1 to decide whether this claim is correct. Justify your answer.

[2]

Question 5

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 6

Paper 1Medium6 marks

A surveyor estimates the area of an irregular lake inlet. A straight survey line of length $220\text{ m}$ is laid along one edge of the inlet. At equally spaced stations, the perpendicular distance from the line to the opposite shoreline is measured. The width increases continuously between consecutive stations.

Distance along survey line (m)	0	55	110	165	220
Width (m)	0	15	28	42	55

(a)

Use the trapezoidal rule to estimate the area of the inlet. Give your answer to the nearest $10\text{ m}^2$

[3]

(b)

Use rectangular strips to find lower and upper estimates for the area. Hence state bounds for the true area and whether your estimate in part (a) is consistent with these bounds.

[3]

Question 7

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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