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 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 2

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 3

Paper 1Hard6 marks

A cart moves along a straight track. Figure 1 shows its signed acceleration, $a$ during the first $4$ seconds. At $t=0$ the cart has velocity $4\text{ m s}^{-1}$ in the positive direction.

(a)

Find an expression for the cart’s velocity, $v$ in terms of $t$ for $0\leq t\leq4$

[2]

(b)

Find the total distance travelled by the cart during the first $4$ seconds. Write down a suitable expression before evaluating it with your GDC, and give your answer to 3 significant figures.

[4]

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