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.

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Question 1

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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