IB Maths AI SL AI 5.1 Differentiation 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 1Paper 2
Difficulty
EasyMediumHard

Question 1

Paper 1Easy5 marks

A manufacturer models its monthly profit, PP in thousands of dollars, from producing qq hundreds of units by

P(q)=−0.12q3+1.6q2−3.5q,0≤q≤10.P(q)=-0.12q^3+1.6q^2-3.5q,\qquad 0\leq q\leq 10.

Figure 1 shows the graph of this model.

012345678910-5051015q (hundreds of units)P(q) (thousands of dollars)
(a)

Use your GDC to generate P′(q)P'(q) and find, to two decimal places, the values of qq at which the gradient of the profit curve is zero.

[2]
(b)

For each value of qq 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, PP in thousands of dollars, when it produces qq hundreds of loaves per week, by

P(q)=q3−6q2+9q+2,0≤q≤4.P(q)=q^3-6q^2+9q+2,\qquad 0\leq q\leq 4.

Figure 1 shows the graph of this model.

01234012345678q (hundreds of loaves per week)P (thousands of dollars per week)
(a)

Use a GDC to generate P′(q)P'(q) then solve P′(q)=0P'(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]

Question 3

Paper 1Medium6 marks

A model for the water level in a test tank is

h(t)=0.2t3−1.14t2+1.566t+42,h(t)=0.2t^3-1.14t^2+1.566t+42,

where hh is the water level in centimetres and tt is the time in hours after the start of a test. The model is valid for 0≤t≤40\leq t\leq4

(a)

Show that

dhdt=0.6t2−2.28t+1.566.\frac{dh}{dt}=0.6t^2-2.28t+1.566.
[2]
(b)

Hence determine the intervals in the model domain for which dhdt>0\frac{dh}{dt}>0 and dhdt<0\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, PP in thousands of dollars, when producing xx thousand units, by

P(x)=x33000−0.0204x2+0.27x+20,0≤x≤45.P(x)=\frac{x^3}{3000}-0.0204x^2+0.27x+20,\qquad 0\leq x\leq45.

The model is shown in Figure 1.

0510152025303540451818.51919.52020.52121.522x (thousand units)P(x) (thousand dollars)
(a)

Using a GDC, determine the values of xx 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]

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