IB Maths AI SL Topic 2: Functions 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 1Hard8 marks

An urban greenhouse produces and sells seedlings in complete batches of 1000. Let $q$ be the number of batches produced in a year, where $q$ is an integer and $0\leq q\leq80$

The selling price per seedling is modelled by

p(q)=96-1.2q

dollars. The annual operating cost is modelled by

C(q)=0.4q^2+12q+500

thousand dollars. The greenhouse also pays an annual permit fee of $F$ thousand dollars. Assume that all seedlings produced are sold.

Let $P_F(q)$ denote the annual profit, in thousands of dollars. The greenhouse considers a year acceptable if its profit is at least $350 000.

(a)

Write an expression for $P_F(q)$

[1]

(b)

When $F=200$ use your graphic display calculator to determine all allowed values of $q$ for which the year is acceptable.

[2]

(c)

Determine the greatest annual permit fee that would still allow the greenhouse to have an acceptable year, and the corresponding production.

[4]

(d)

State one limitation of this model when it is used to make the decision in part (c).

[1]

A horticulture centre monitors the depth of water in a shallow pond during a controlled fill-and-drain test. Let $t$ be the time, in hours, since the test began, and let $h$ be the depth of water, in centimetres.

$t$	0	2	5	7
$h$	1.3	7.6	10.1	7.0

(a)

Choose the most suitable model from a linear, quadratic or exponential model for these data. Justify your choice.

[2]

(b)

Using only the observations at $t=0$ $t=2$ and $t=5$ fit a quadratic model of the form $h=at^2+bt+c$ Show that

h=-0.463t^2+4.08t+1.30,

where the coefficients are given to 3 significant figures. Retain unrounded values for later parts.

[3]

(c)

Use your fitted model to estimate the depth at $t=7$ Compare your estimate with the recorded depth.

[2]

(d)

Use your fitted model to estimate the time at which the depth is $8$ cm on the falling part of the curve. Give your answer to the nearest $0.01$ hour.

[3]

(e)

State a reasonable domain for using this model, based on the observations. Use the model to estimate the depth at $t=10$ and comment on whether this prediction is reasonable.

[4]

Question 3

Paper 1Medium14 marks

An engineer records the water height, $h$ cm, in a reservoir while a pump fills it. The pump remains on throughout a planned 8-minute run, and there is no outlet. The measurements shown in Figure 1 were taken during the first 5 minutes. Heights are recorded to the nearest $0.1$ cm.

(a)

Using the shape of the data in Figure 1, choose a linear or quadratic model for $h$ in terms of $t$ Justify your choice.

[2]

(b)

Let $h=at^2+bt+c$ Using the measurements at $t=0$ $t=2$ and $t=4$ show that a fitted model is

h=-0.5375t^2+6.975t+3.8.

[4]

(c)

Hence, using this model, determine the first time during the 5-minute measurement interval at which the model predicts $h=25.0$ cm. Give your answer to the nearest $0.01$ minute.

[3]

(d)

At $t=3$ compare the model’s predicted height with the recorded height. State what this comparison suggests about the model’s fit at that time.

[2]

(e)

Use the fitted model to predict the height after 8 minutes. State whether this prediction should be considered reliable, giving two reasons.

[3]

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