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

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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]

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