IB Maths AI 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 Paper 2

Difficulty

Easy Medium Hard

Question 1

Paper 2Medium14 marks

A small mass moves vertically on a spring. For $t\geq 0$ its displacement $y$ metres from its equilibrium position is modelled by

y(t)=0.32\sin(1.4t+0.30),

where $t$ is measured in seconds and angles are in radians. This is a model of simple harmonic motion.

(a)

State the amplitude and period of the motion. Calculate the displacement of the mass at $t=0$ giving your answer to three significant figures.

[3]

(b)

Show that

\frac{d^2y}{dt^2}=-0.6272\sin(1.4t+0.30),

and hence show that

\frac{d^2y}{dt^2}=-1.96y(t).

[3]

(c)

Determine the time of the first stationary point after $t=0$ Use the second derivative test to classify this point.

[4]

(d)

Using your result from part (b), determine the time of the first point of inflexion after $t=0$ State the concavity immediately before and after this point, and interpret the point in terms of the mass's motion.

[4]

Question 2

Paper 2Medium13 marks

A landscape designer models the vertical profile of a section of a footpath by

h(x)=0.009x^3-0.25x^2+1.7x+2.5,\qquad 0\leq x\leq20,

where $x$ is the horizontal distance, in metres, from the start of the section and $h(x)$ is the height, in metres, above a reference level.

Consider the point on the profile where $x=8$

(a)

Calculate the height of the profile at this point.

[2]

(b)

Show that

h'(x)=0.027x^2-0.50x+1.7.

[2]

(c)

Determine the equation of the tangent to the profile at $x=8$ giving your answer in the form $h-h(8)=m(x-8)$

[3]

(d)

Determine the equation of the normal to the profile at $x=8$ in the same form. Give its gradient to 3 significant figures.

[3]

(e)

Use your GDC to find the coordinates of the other point, different from the point where $x=8$ at which the tangent in part (c) meets the profile. Give your coordinates to 3 significant figures.

[3]

Question 3

Paper 1Medium6 marks

A nutrient-solution pump feeds a tank. During the first 6 minutes, its delivery rate is modelled by

q(t)=\frac{12t}{t^2+9}\text{ mL min}^{-1},

where $t$ is the time in minutes after the pump starts. Let $V(t)$ be the volume, in mL, delivered by time $t$ with $V(0)=0$

(a)

By evaluating the definite integral for $V(t)$ show that

V(t)=6\ln\left(\frac{t^2+9}{9}\right).

[2]

(b)

Calculate $V(6)$ giving your answer to 3 significant figures.

[1]

(c)

A warning light activates when 8 mL has been delivered. Use your answer to part (b) to establish that the light activates before 6 minutes, and hence determine the time at which it activates. Give your answer to 3 significant figures.

[3]

Question 4

Paper 1Medium6 marks

Figure 1 shows the graph of $y=4x\sin(x^2)$ for $0\leq x\leq\sqrt{\pi}$ Angles are measured in radians. The shaded region is bounded by the curve and the $x$ -axis.

(a)

Write down the upper limit represented by the shaded region.

[1]

(b)

Find $\displaystyle\int 4x\sin(x^2)\,dx$

[2]

(c)

Hence find the exact $x$ -coordinate that divides the shaded region into two equal areas.

[3]

Question 5

Paper 2Medium6 marks

A pump transfers liquid into a container. For $t\geq 0$ its flow rate is modelled by

q(t)=2t e^{-t^2}\ \text{L min}^{-1},

where $t$ is the time in minutes since the pump starts. Let $V(t)$ litres be the total volume transferred by time $t$

(a)

Show that $V(t)=1-e^{-t^2}$

[2]

(b)

Calculate the volume transferred in the first $1.5$ minutes. Give your answer to three significant figures.

[1]

(c)

Determine the time at which $0.80$ L has been transferred. Explain why any other mathematical solution is not relevant.

[3]

Question 6

Paper 2Medium7 marks

A laboratory monitors a culture of cells. Its population, $y$ is measured in thousands of cells, and $t$ is measured in hours. The rate of change of the population is modelled by

\frac{\mathrm{d}y}{\mathrm{d}t}=0.06y(14-y).

The table summarizes the slopes in the slope field for this model. Each entry is the value of $\frac{\mathrm{d}y}{\mathrm{d}t}$ at the point given by its row and column. Slopes are in thousands of cells per hour.

$y$	$t=0$	$t=0.5$	$t=1$
0	0	0	0
7	2.94	2.94	2.94
11	1.98	1.98	1.98
14	0	0	0
18	$-4.32$	$-4.32$	$-4.32$

(a)

Show that the slope of the field at $(t,y)=(0,9)$ is $2.7$

[2]

(b)

A culture has $y=9$ at $t=0$ Use the gradient at this point to estimate its population at $t=0.5$

[2]

(c)

Use the slope field to decide whether your estimate in part (b) is likely to be greater or less than the modelled population at $t=0.5$ Justify your answer, and state the population towards which the field predicts the culture will tend over a long time.

[3]

Question 7

Paper 2Medium7 marks

In a reserve, the prey and predator populations are represented by indices $x$ and $y$ measured in hundreds of animals. The time $t$ is measured in years after monitoring begins. The populations are modelled by

\frac{\mathrm{d}x}{\mathrm{d}t}=0.8x-0.04xy, \qquad \frac{\mathrm{d}y}{\mathrm{d}t}=0.02xy-0.6y,

with $x(0)=20$ and $y(0)=5$ A researcher uses Euler’s method with a step size of $0.5$ years.

(a)

Show that the Euler estimate for $x(0.5)$ is $26$ and find the corresponding estimate for $y(0.5)$

[3]

(b)

Hence, use one further Euler step to estimate $x(1)$ and $y(1)$

[2]

(c)

A researcher claims that the predator population is decreasing at $t=1$ since the estimate for $y(1)$ is lower than that for $y(0.5)$ Use the model to assess this claim.

[2]

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