IB Maths AI HL 2.1 Functions Sl Content Questions

Practise IB Mathematics AI HL 2.1 by analysing advanced function models, transformations, inverse and regression evidence with technology.

Exam points

Represent functions with correct notation, domain, range and inverse relationships, evaluating and composing them in context.
Construct, sketch and transform function graphs, identifying intercepts, extrema, symmetry, asymptotes and intersections.
Select and use function models, interpret parameters and use technology to fit, solve or predict within a valid domain.
Use graph and model features to answer thresholds, optimisation and applied questions with appropriate units.
Check a function or model using substitution, graph behaviour, data fit, assumptions and contextual reasonableness.

Covered syllabus points

SL 2.1—Straight lines SL 2.2—Functions and inverses SL 2.3—Graphing functions SL 2.4—Key graph features SL 2.5—Modelling with functions SL 2.6—Modelling process

[Maximum number: 3]

The Voronoi diagram below shows four supermarkets represented by points with coordinates A(0,0), B(6,0), C(0,6) and D(2,2). The vertices X, Y, Z are also shown. All distances are measured in kilometres.

The equation of (XY) is y=2-x and the equation of (YZ) is y=0.5x+3.5.
Find the coordinates of X.

identifying the correct equations to use:

y=2-x \text { and } y=2 x-7

evidence of solving their correct equations or finding points of intersection graphically

(3,-1)

Note: Accept an answer expressed as " x=3, y=-1 ".

[Maximum number: 2]

The function f is defined by $f(x)=4 \ln (2 x-3)$ , where $x>\frac{3}{2}$ .
The graph of y=f(x) is obtained from the graph of $y=\ln x$ by a sequence of three transformations.

The graph of y=g(x) is shown.
The graph passes through the points (-2,-2),(-1,0),(0,2),(1,3),(2,5) and (3,6).

Find $(f \circ g)(2)$ .

$\quad g(2)=5$ $f(5)=7.78 \quad(=7.78364 \ldots, 4 \ln 7)$

[Maximum number: 3]

A student investigating the relationship between chemical reactions and temperature finds the Arrhenius equation on the internet.

k=A \mathrm{e}^{-\frac{c}{T}}

This equation links a variable k with the temperature T, where A and c are positive constants and T>0.

Given that $\lim_{T\to\infty}k=A$ and $\lim_{T\to0}k=0$ , sketch the graph of k against T.

[Maximum number: 5]

A suitable site for the landing of a spacecraft on the planet Mars is identified at a point, A. The shortest time from sunrise to sunset at point A must be found.
Radians should be used throughout this question. All values given in the question should be treated as exact.
Mars completes a full orbit of the Sun in 669 Martian days, which is one Martian year.

On day t, where $t \in \mathbb{Z}$ , the length of time, in hours, from the start of the Martian day until sunrise at point A can be modelled by a function, R(t), where

R(t)=a \sin (b t)+c, t \in \mathbb{R} .

The graph of R is shown for one Martian year.

(a)

Use your answers to parts (b) and (c) to find

[ 3 ]

(i)

the maximum value of R(t);

[ 2 ]

$\quad R_{\max }=\frac{1.97684 \ldots}{0.25472 \ldots}$

=7.76 \text { hours }(7.76075 \ldots)

Note: Accept 7.70 from use of 1.98.

(ii)

the minimum value of R(t).

[ 1 ]

$R_{\text {min }}=\frac{1.16474 \ldots}{0.25472 \ldots}$

=4.57 \text { hours }(4.57258 \ldots)

Note: Accept 4.55 and 4.56 from use of rounded values.

(b)

Find, in hours, the shortest time from sunrise to sunset at point A that is predicted by this model.

[ 2 ]

shortest time between sunrise and sunset
$12.4833 \ldots-3.06249 \ldots$
=9.42 hours ( $9.420843 \ldots$ )

Note: Accept 9.44 from use of 3 sf values.

[Maximum number: 1]

This question explores models for the height of water in a cylindrical container as water drains out.
The diagram shows a cylindrical water container of height 3.2 metres and base radius 1 metre. At the base of the container is a small circular valve, which enables water to drain out.

Eva closes the valve and fills the container with water.
At time t=0, Eva opens the valve. She records the height, h metres, of water remaining in the container every 5 minutes.

Eva first tries to model the height using a linear function, h(t)=a t+b, where $a, b \in \mathbb{R}$ .

Interpret the meaning of parameter a in the context of the model.

Eva uses the equation of the regression line of h on t, to predict the time it will take for all the water to drain out of the container.

[Maximum number: 2]

Eva closes the valve and fills the container with water.
At time t=0, Eva opens the valve. She records the height, h metres, of water remaining in the container every 5 minutes.

Eva first tries to model the height using a linear function, h(t)=a t+b, where $a, b \in \mathbb{R}$ .

(a)

Suggest why Eva's use of the linear regression equation in this way could be unreliable.

Eva thinks she can improve her model by using a quadratic function, $h(t)=p t^{2}+q t+r$ , where $p, q, r \in \mathbb{R}$ .

[ 1 ]

(b)

Hence, write down a suitable domain for Eva's function $h(t)=p t^{2}+q t+r$ .

Let V be the volume, in cubic metres, of water in the container at time t minutes.
Let R be the radius, in metres, of the circular valve.
Eva does some research and discovers a formula for the rate of change of V.

\frac{\mathrm{d} V}{\mathrm{~d} t}=-\pi R^{2} \sqrt{70560 h}

[ 1 ]

EITHER
$(0 \leq) t \leq 26.4(t \leq 26.4046 \ldots)$
OR
$(0 \leq) t \leq 20$ (due to range of original data / interpolation)

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Covered syllabus points

Question 1

Question 2

Question 3

Question 4

Question (a)

Question (i)

Question (ii)

Question (b)

Question 5

Question 6

Question (a)

Question (b)