Differentiation - IB Math AI Question Bank

Question 1

SL • Paper 1

Easy

Calculator Permitted

The table shows values of a function $f$ for values of $x$ close to $2$ . The value of the function at $x=2$ is also shown.

x	f(x)
1.8	4.8
1.9	4.9
1.99	4.99
2	1
2.01	5.01
2.1	5.1
2.2	5.2

A

Estimate $\lim_{x\to 2}f(x)$ .

[1]

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B

State the value of $f(2)$ .

[1]

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C

Explain why the answers to parts (a) and (b) are different.

[2]

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0

Question 2

SL • Paper 1

Easy

Calculator Permitted

The depth, $D$ centimetres, of water in a container is modelled by

D(t)=0.02t^3-0.6t^2+4t+10

where $t$ is the time in hours after measurements begin, for $0\leq t\leq12$ .

A

Find an expression for $\dfrac{dD}{dt}$ .

[2]

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B

Calculate the value of $\dfrac{dD}{dt}$ when $t=5$ .

[1]

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C

Interpret your answer to part (b) in context.

[2]

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0

Question 3

SL • Paper 1

Medium

Calculator Permitted

Let

f(x)=x^3-3x^2-9x+5

where $x\in\mathbb{R}$ .

A

Find $f'(x)$ .

[2]

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B

Determine the intervals on which $f$ is increasing and the interval on which $f$ is decreasing.

[3]

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0

Question 4

SL • Paper 1

Medium

Calculator Permitted

The curve $C$ has equation

y=x^3-4x+1

The point $P$ on $C$ has $x$ -coordinate $2$ .

A

Find $\dfrac{dy}{dx}$ and hence find the gradient of $C$ at $P$ .

[2]

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B

Find the equation of the tangent to $C$ at $P$ .

[2]

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C

Find the equation of the normal to $C$ at $P$ .

[2]

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0

Question 5

SL • Paper 1

Medium

Calculator Permitted

A farmer uses $40\ \text{m}$ of fencing to enclose a rectangular field beside a straight river. No fencing is required along the river. The width perpendicular to the river is $x\ \text{m}$ .

A rectangular field with one side along a straight river. The two sides perpendicular to the river are each labelled $x$ metres, and the fenced side parallel to the river is labelled $y$ metres. The river side is shown without fencing.

A

Show that the area of the field is given by $A(x)=40x-2x^2$ and state the feasible domain of $x$ .

[2]

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B

Find the value of $x$ that maximizes the area.

[2]

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C

Determine the other dimension of the field and the maximum area.

[2]

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0

Question 6

SL • Paper 1

Medium

Calculator Permitted

The total cost, $C(x)$ dollars, of producing $x$ items is modelled by

C(x)=0.02x^3-1.2x^2+30x+500

for $0\leq x\leq40$ . The marginal cost is given by $C'(x)$ .

A

Find an expression for the marginal cost.

[2]

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B

Find the production levels at which the marginal cost is $12$ dollars per item.

[2]

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C

Interpret the two answers from part (b) in context.

[1]

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0

Question 7

HL • Paper 1

Medium

Calculator Permitted

A function is defined by

f(x)=e^{0.3x}\sin(2x)

where angles are measured in radians.

A

Find $f'(x)$ .

[3]

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B

Calculate the gradient of the graph of $f$ at $x=1$ .

[2]

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0

Question 8

SL • Paper 1

Medium

Calculator Permitted

Let

f(x)=x^4-8x^2+3

A

Find the values of $x$ for which the gradient of the graph of $f$ is zero.

[3]

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B

Find the coordinates of the stationary points.

[1]

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C

Classify each stationary point as a local maximum or a local minimum.

[2]

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0

Question 9

HL • Paper 1

Medium

Calculator Permitted

The function $f$ is defined by

f(x)=\frac{\ln(x^2+1)}{x},\qquad x>0

A

Find $f'(x)$ .

[3]

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B

Find the equation of the tangent to the graph of $f$ at $x=1$ .

[3]

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0

Question 10

HL • Paper 1

Medium

Calculator Permitted

Let

f(x)=xe^{-x^2/2},\qquad x\in\mathbb{R}

A

Find $f'(x)$ .

[2]

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B

Find the coordinates of the stationary points.

[2]

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C

Classify the stationary points.

[2]

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0

Question 11

HL • Paper 1

Medium

Calculator Permitted

A spherical balloon is being inflated. Its volume is increasing at a constant rate of $20\ \text{cm}^3\text{ s}^{-1}$ . At a particular instant, its radius is $1.5\ \text{cm}$ .

The volume and surface area of a sphere are given by

V=\frac43\pi r^3,\qquad A=4\pi r^2

A

Find the rate at which the radius is increasing at this instant.

[4]

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B

Hence find the rate at which the surface area is increasing at this instant.

[2]

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0

Question 12

HL • Paper 1

Medium

Calculator Permitted

Water is poured into an inverted conical tank at a rate of $0.400\ \text{m}^3\text{ min}^{-1}$ . The tank has height $4\ \text{m}$ and upper radius $2\ \text{m}$ . At time $t$ , the water has depth $h\ \text{m}$ and surface radius $r\ \text{m}$ .

A vertical cross-section of an inverted conical tank with total height $4\ \text{m}$ and upper radius $2\ \text{m}$. Show the tank centreline. The inner water cone has its apex exactly coincident with the outer tank apex. The water surface is horizontal, and a vertical dimension arrow from the water surface to the common apex is labelled $h$. At the water surface, show a horizontal radius dimension arrow from the centreline to one edge only, labelled $r$; do not use $r$ for the full surface width.

A

Show that the volume of water is $V=\dfrac{\pi h^3}{12}$ .

[2]

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B

Find the rate at which the water depth is increasing when $h=1.20\ \text{m}$ .

[3]

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0

Question 13

HL • Paper 1

Medium

Calculator Permitted

Let

f(x)=x^5-5x^4

A

Find $f''(x)$ .

[2]

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B

Determine the intervals on which the graph is concave-up and concave-down.

[2]

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C

Determine all points of inflexion of the graph.

[2]

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0

Question 14

SL • Paper 1

Medium

Calculator Permitted

A closed cylindrical container has volume $500\ \text{cm}^3$ . Its radius is $r\ \text{cm}$ and its height is $h\ \text{cm}$ . The container is made using the least possible surface area.

A closed right circular cylinder with radius $r$ marked on a circular face and perpendicular height $h$ marked along the side.

A

Show that the surface area can be written as

S(r)=2\pi r^2+\frac{1000}{r}

[2]

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B

Find the radius and height of the container.

[4]

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C

Justify that these dimensions give a minimum surface area.

[1]

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0

Question 15

HL • Paper 1

Medium

Calculator Permitted

The function $f$ is defined by

f(x)=x^4-4x^3

A

Find the stationary points of the graph of $f$ .

[2]

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B

Find $f''(x)$ and use it to classify the stationary point at $x=3$ .

[2]

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C

Explain why the second derivative test is inconclusive at $x=0$ , and classify the stationary point $(0,0)$ .

[2]

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0

Question 16

HL • Paper 1

Medium

Calculator Permitted

A company models its monthly revenue, $R(x)$ thousand dollars, by

R(x)=5xe^{-0.04x}

where $x>0$ is the number of hundreds of units sold.

A

Find $R'(x)$ .

[2]

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B

Find the value of $x$ at which the revenue is stationary.

[2]

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C

Justify that this stationary value is a maximum.

[2]

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D

Determine the maximum monthly revenue, in dollars.

[1]

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0

Question 17

SL • Paper 2

Medium

Calculator Permitted

A sensor is calibrated using the function

f(x)=\begin{cases}\dfrac{x^2-9}{x-3},&x\ne3,\\10,&x=3.\end{cases}

A table gives values of $f(x)$ for values of $x$ close to $3$ .

x	f(x)
2.9	5.9
2.99	5.99
2.999	5.999
3	10
3.001	6.001
3.01	6.01
3.1	6.1

A

I.

Use the table to estimate $\displaystyle\lim_{x\to3}f(x)$ .

[1]

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II.

State whether $f$ is continuous at $x=3$ . Give a reason.

[1]

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B

I.

For $h\ne0$ , simplify the secant gradient

\frac{f(3+h)-6}{h}

[2]

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II.

Hence state the gradient approached by the curve as $x$ approaches $3$ .

[1]

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C

Find the equation of the tangent to the branch $y=\dfrac{x^2-9}{x-3}$ at $x=4$ .

[3]

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0

Question 18

SL • Paper 2

Medium

Calculator Permitted

The mass of algae, $M$ kilograms, in an experimental pond is modelled by

M(t)=-t^3+9t^2+24

where $t$ is the number of weeks after the experiment begins and $0\le t\le8$ .

A

I.

Find $M'(t)$ .

[2]

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II.

Calculate $M'(4)$ and interpret the answer in context.

[2]

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B

Find the stationary point of the model in the interior of the given domain.

[2]

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C

Determine the global maximum mass of algae during the eight weeks. Justify your answer.

[2]

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0

Question 19

SL • Paper 2

Medium

Calculator Permitted

The function $f$ is defined by

f(x)=x^3+ax^2-4x+2

where $a$ is a constant. At $x=1$ , the tangent to the graph has gradient $5$ .

A

I.

Find $f'(x)$ in terms of $a$ .

[1]

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II.

Hence find the value of $a$ .

[2]

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B

Find the equation of the tangent at $x=1$ .

[2]

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C

The normal at $x=1$ meets the $y$ -axis at $Q$ . Find the coordinates of $Q$ .

[3]

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0

Question 20

HL • Paper 3

Medium

Calculator Permitted

A sensor records a quantity using the function

F(t)=\begin{cases}\dfrac{t^2-25}{t-5},&t\ne5,\\13,&t=5.\end{cases}

The manufacturer considers recalibrating the sensor at $t=5$ .

Sensor function near t=5, with the recorded value marked.

A

I.

Estimate $\displaystyle\lim_{t\to5}F(t)$ .

[1]

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II.

Explain why $F$ is not continuous at $t=5$ .

[1]

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B

Use the difference quotient to show that $F'(5)$ does not exist.

[2]

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C

The recalibrated function $G$ has the same values as $F$ except that $G(5)=10$ . Find the equation of the tangent to the graph of $G$ at $t=5$ .

[2]

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D

Find the $t$ -intercept of the normal to the graph of $G$ at $t=5$ .

[2]

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0

Question 21

HL • Paper 3

Medium

Calculator Permitted

The flow of vehicles along a road is modelled by

q(v)=2.4v-0.03v^2

where $v$ is the mean speed in $\text{km h}^{-1}$ and $q(v)$ is the flow in hundreds of vehicles per hour.

Concave-down traffic flow model over the operational speed range.

A

I.

Find $q'(v)$ .

[1]

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II.

Determine the speed at which the traffic flow is stationary.

[2]

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B

For the operational domain $10\le v\le70$ , state the intervals on which the flow is increasing and decreasing.

[2]

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C

Find the maximum modelled flow, in vehicles per hour.

[1]

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D

generalized road model is $q_a(v)=av-0.03v^2$ , where $a>0$ . Determine, in terms of $a$ , the stationary speed and the corresponding maximum value of $q_a$ .

[2]

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0

Question 22

SL • Paper 2

Hard

Calculator Permitted

An open box is made from a rectangular sheet of card measuring $30\ \text{cm}$ by $24\ \text{cm}$ . Squares of side $x\ \text{cm}$ are removed from each corner and the remaining sides are folded upwards.

A rectangular sheet labelled 30 cm by 24 cm with congruent corner squares of side x removed, together with the resulting open rectangular box. The base dimensions and height are indicated symbolically without giving the optimum.

A

I.

Show that the volume of the box is

V(x)=x(30-2x)(24-2x)

[1]

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II.

State the feasible domain for $x$ .

[2]

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B

Find the value of $x$ that gives the maximum volume.

[3]

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C

Determine the maximum volume of the box, correct to three significant figures.

[2]

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0

Question 23

SL • Paper 2

Hard

Calculator Permitted

A theatre charges $p$ dollars for a ticket. The number of tickets expected to be sold is modelled by

q=600-20p

where $6\le p\le30$ . The variable cost is $6 per ticket and the fixed cost is $1200.

A

I.

Write down an expression for the revenue in terms of $p$ .

[1]

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II.

Show that the profit is

P(p)=-20p^2+720p-4800

[2]

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B

Find the ticket price that maximizes the profit.

[3]

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C

Determine the expected number of tickets sold and the resulting maximum profit.

[2]

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0

Question 24

SL • Paper 2

Hard

Calculator Permitted

The energy generated by a prototype solar panel during a test is modelled by

E(t)=-\frac12t^4+8t^3-36t^2+64t

where $E$ is measured in kilojoules, $t$ is measured in hours and $0\le t\le8$ .

A

I.

Find $E'(t)$ .

[2]

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II.

Show that $E'(t)=-2(t-2)^2(t-8)$ .

[1]

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B

Explain why the stationary point at $t=2$ is not a local maximum or a local minimum.

[2]

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C

Determine the global maximum value of $E$ during the test and state when it occurs.

[3]

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0

Question 25

SL • Paper 2

Hard

Calculator Permitted

The function $f$ is defined on $-2\le x\le5$ by

f(x)=x^4-4x^3-8x^2+10

A

I.

Show that

f'(x)=4x(x-4)(x+1)

[1]

Write your answer here...

II.

Find the coordinates of all stationary points.

[2]

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B

Classify each stationary point.

[3]

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C

Determine the global maximum and global minimum values of $f$ on the stated domain.

[2]

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0

Question 26

SL • Paper 2

Hard

Calculator Permitted

The height of a water jet above the ground is modelled by

h(x)=-0.02x^3+0.18x^2+0.6x+1

where $h$ and the horizontal distance $x$ are measured in metres, and $0\le x\le12$ . The polynomial is treated as a mathematical model on this interval, including where its extrapolated value is below ground.

Mathematical height model of a water jet over $0\le x\le12$, with the launch point marked.

A

I.

Find $h'(x)$ .

[1]

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II.

Find the gradient of the jet at $x=4$ .

[2]

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B

Find the equation of the tangent to the path of the jet at $x=4$ .

[2]

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C

Determine the maximum height reached by the jet within the stated domain.

[3]

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0

Question 27

HL • Paper 2

Hard

Calculator Permitted

The concentration of a substance in a sample is modelled by

C(t)=t^2e^{-t/2}

where $C$ is measured in milligrams per litre, $t$ is measured in hours and $0\le t\le12$ .

A

I.

Find $C'(t)$ .

[2]

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II.

Find the time at which the concentration is greatest.

[1]

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B

Determine the maximum concentration.

[2]

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C

Find the times at which the graph of $C$ changes concavity.

[3]

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0

Question 28

HL • Paper 2

Hard

Calculator Permitted

The function $f$ is defined by

f(x)=\frac{\ln x}{x},\qquad x>0

A

I.

Find $f'(x)$ .

[2]

Write your answer here...

II.

Find the coordinates of the stationary point.

[1]

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B

Find $f''(x)$ and hence classify the stationary point.

[3]

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C

Determine the point of inflexion of the graph.

[2]

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0

Question 29

HL • Paper 2

Hard

Calculator Permitted

A cube of ice melts while retaining its cubic shape. Its volume decreases at a constant rate of $18\ \text{cm}^3\text{ min}^{-1}$ . Let the side length be $s\ \text{cm}$ , the surface area be $A\ \text{cm}^2$ and the space diagonal be $d\ \text{cm}$ .

A cube labelled with side length s and a space diagonal d joining opposite vertices.

A

I.

Write down expressions for the volume $V$ , surface area $A$ and diagonal $d$ in terms of $s$ .

[2]

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II.

State the value of $\dfrac{dV}{dt}$ .

[1]

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B

Find the rate of change of $s$ when $s=6$ .

[2]

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C

Hence find the rates of change of the surface area and the space diagonal when $s=6$ .

[3]

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0

Question 30

HL • Paper 2

Hard

Calculator Permitted

The monthly energy output of a system is modelled by

E(t)=40+10\sin\left(\frac{\pi t}{6}\right)

where $E$ is measured in megawatt-hours, $t$ is measured in months and $0\le t\le12$ .

A

I.

Find $E'(t)$ .

[2]

Write your answer here...

II.

Find the maximum and minimum energy outputs during the year.

[1]

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B

Find $E''(t)$ and determine the point of inflexion in the interior of the domain.

[3]

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C

Find the equation of the tangent to the graph at the point of inflexion.

[2]

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0

Question 31

HL • Paper 3

Hard

Calculator Permitted

The electrical power produced by a tilting solar panel is modelled by

P(\theta)=900e^{-0.2\theta}\sin\theta

where $0\le\theta\le\dfrac{\pi}{2}$ is measured in radians and $P$ is measured in watts.

Side-view diagram of a tilting solar panel showing the angle theta between the panel and a horizontal support.

A

I.

Find $P'(\theta)$ .

[2]

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II.

Determine the stationary value of $\theta$ .

[2]

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B

Calculate the power produced at this angle.

[2]

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C

Justify that the value found in part (a)(ii) gives the global maximum power on the stated domain.

[2]

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0

Question 32

HL • Paper 3

Hard

Calculator Permitted

The concentration of a medicine in a patient's blood is modelled by

C(t)=12te^{-0.5t},\qquad t\ge0

where $C$ is measured in $\text{mg L}^{-1}$ and $t$ is measured in hours.

Medicine concentration in blood over time.

A

I.

Find $C'(t)$ .

[2]

Write your answer here...

II.

Determine when the concentration is greatest.

[1]

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B

Calculate the greatest concentration.

[1]

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C

Show that the graph has a point of inflexion at $t=4$ .

[2]

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D

Interpret the point of inflexion in the context of the medicine concentration.

[2]

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0

Question 33

HL • Paper 3

Hard

Calculator Permitted

A lamp is mounted $6\ \text{m}$ above level ground. A person of height $1.8\ \text{m}$ walks directly away from the lamp. At time $t$ , the person is $x\ \text{m}$ from the lamp and the length of the person's shadow is $y\ \text{m}$ .

Side-view related-rates diagram showing a vertical lamp, a shorter person, the distance x from the lamp to the person, and the shadow length y beyond the person.

A

I.

Use similar triangles to show that $y=\dfrac37x$ .

[2]

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II.

The person walks at $1.40\ \text{m s}^{-1}$ . Find the rate at which the shadow length is increasing.

[1]

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B

Find the speed of the tip of the shadow.

[2]

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C

For a lamp of height $H$ and a person of height $h$ , where $H>h$ , derive the speed of the shadow tip when the person walks at constant speed $u$ .

[2]

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D

different lamp is used with the same person. Determine its height if the tip of the shadow always moves at twice the person's speed.

[1]

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0

Question 34

HL • Paper 3

Hard

Calculator Permitted

A cube of ice melts while retaining its cubic shape. Its side length is $s\ \text{cm}$ , its volume is $V\ \text{cm}^3$ and its surface area is $A\ \text{cm}^2$ . At one instant, $s=4\ \text{cm}$ and $\dfrac{dV}{dt}=-24\ \text{cm}^3\text{ min}^{-1}$ .

Diagram of a cube with side length labelled by lowercase italic $s$, used to relate its volume and surface area during melting.

A

I.

Find $\dfrac{ds}{dt}$ at this instant.

[2]

Write your answer here...

II.

Hence find $\dfrac{dA}{dt}$ at this instant.

[2]

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B

Show that, for any positive value of $s$ , the instantaneous rates satisfy

\frac{dA}{dt}=\frac4s\frac{dV}{dt}

[2]

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C

When $s=5\ \text{cm}$ , the surface area is decreasing at $30\ \text{cm}^2\text{ min}^{-1}$ . Find the rate of change of the volume.

[2]

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0

Question 35

HL • Paper 3

Hard

Calculator Permitted

An open box is made from a square sheet of card of side $30\ \text{cm}$ by cutting a square of side $x\ \text{cm}$ from each corner and folding up the sides. The feasible domain is $0<x<15$ .

Net of a square sheet with four corner squares of side x removed, together with the resulting open box whose base dimensions are 30 minus 2x.

A

I.

Show that the volume is $V(x)=x(30-2x)^2$ .

[1]

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II.

Show that $V'(x)=(30-2x)(30-6x)$ .

[2]

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B

Determine the dimensions of the box with maximum volume and find this volume.

[2]

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C

manufacturer instead maximizes the score

J(x)=V(x)-kx^2

where $k>0$ represents a waste penalty. For dimensional consistency, treat $J$ as volume-valued: $V$ is measured in $\text{cm}^3$ and $x$ in $\text{cm}$ , so $k$ has units of $\text{cm}$ . Determine $k$ if the optimal cut size is $x=4\ \text{cm}$ .

[2]

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D

Justify that $x=4$ gives a local maximum of $J$ for this value of $k$ .

[1]

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0

Question 36

HL • Paper 3

Hard

Calculator Permitted

The daily demand for a product is modelled by

D(p)=1000e^{-0.05p}

where $p$ is the selling price in dollars. The production cost is $8 per item, so the daily profit is

P(p)=(p-8)D(p),\qquad p\ge8

Daily profit as a function of selling price.

A

I.

Find $P'(p)$ .

[2]

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II.

Determine the price that maximizes daily profit.

[1]

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B

Calculate the corresponding maximum daily profit.

[2]

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C

tax of $\tau$ dollars is imposed on each item sold. Show that the new profit-maximizing price is $p=28+\tau$ .

[2]

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D

Interpret the result from part (c) in terms of how the model allocates the tax.

[1]

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0

Question 37

HL • Paper 3

Hard

Calculator Permitted

A rotating beacon is located $2\ \text{km}$ from the nearest point $O$ on a straight coastline. Its beam makes an angle $\theta$ with the line perpendicular to the coast at $O$ . The illuminated point is $x\ \text{km}$ from $O$ along the coast. The beacon rotates so that $\theta$ is increasing at a constant rate of $0.300\ \text{rad s}^{-1}$ .

Plan-view related-rates diagram showing a beacon two kilometres from a straight coastline, perpendicular point O, illuminated point at distance x along the coast, and angle theta at the beacon.

A

I.

Show that $x=2\tan\theta$ .

[1]

Write your answer here...

II.

Find $\dfrac{dx}{dt}$ when $\theta=\dfrac\pi4$ .

[2]

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B

Find $\dfrac{d^2x}{dt^2}$ when $\theta=\dfrac\pi4$ .

[2]

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C

For a beacon at distance $d$ from the coast rotating so that $\theta$ increases at a constant angular speed $\omega>0$ , derive the speed of the illuminated point.

[2]

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D

Determine the positive angle at which the illuminated point moves at twice its minimum speed.

[1]

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Question 38

HL • Paper 2

Hard

Calculator Permitted

A spotlight is located $30\ \text{m}$ from a straight wall. Its beam rotates so that the angle $\theta$ between the beam and the perpendicular to the wall increases at a constant rate of $0.080\ \text{rad s}^{-1}$ . The illuminated point is $x\ \text{m}$ from the point on the wall nearest the spotlight.

A right-triangle plan view showing a spotlight, a wall 30 m away along the perpendicular, the rotating beam, angle theta at the spotlight and distance x along the wall.

A

I.

Show that $x=30\tan\theta$ .

[1]

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II.

Find an expression for $\dfrac{dx}{dt}$ in terms of $\theta$ .

[2]

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B

Calculate the speed of the illuminated point when $\theta=0.600$ radians.

[2]

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C

Find the acceleration $\dfrac{d^2x}{dt^2}$ of the illuminated point when $\theta=0.600$ radians.

[3]

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Question 39

HL • Paper 2

Hard

Calculator Permitted

The population of bacteria in a culture is modelled by

P(t)=\frac{1200}{1+5e^{-0.4t}}

where $t$ is measured in hours and $P$ is the number of bacteria.

A

I.

Find $P'(t)$ .

[2]

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II.

Explain why the population is increasing for all $t\ge0$ .

[1]

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B

The rate of growth is greatest when $P=600$ . Find the corresponding value of $t$ .

[2]

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C

Determine the greatest rate of population growth and explain the significance of this time on the graph of $P$ .

[3]

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Question 40

HL • Paper 2

Hard

Calculator Permitted

The function $f$ is defined for $x\ge0$ by

f(x)=(x^2+1)e^{-x}

A

I.

Show that

f'(x)=-(x-1)^2e^{-x}

[2]

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II.

State the stationary point.

[1]

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B

Explain why this stationary point is not a local maximum or local minimum.

[2]

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C

Find all points of inflexion on the domain.

[3]

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Question 41

HL • Paper 2

Hard

Calculator Permitted

A rectangular page must contain a printed area of $384\ \text{cm}^2$ . The side margins are each $2\ \text{cm}$ wide and the top and bottom margins are each $3\ \text{cm}$ wide. The width of the printed area is $x\ \text{cm}$ .

A rectangular page containing a smaller rectangular printed region. The printed width is labelled x, the two side margins are each 2 cm, and the top and bottom margins are each 3 cm.

A

I.

Write down the height of the printed area in terms of $x$ .

[1]

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II.

Show that the total area of the page is

A(x)=408+6x+\frac{1536}{x},\qquad x>0

[2]

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B

Find the value of $x$ that minimizes the total area of the page.

[3]

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C

Determine the dimensions of the page and justify that they give a minimum area.

[2]

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Question 42

HL • Paper 3

Hard

Calculator Permitted

The vertical profile of a designed track is modelled by

f(x)=(x^2+100)e^{-0.1x},\qquad 0\le x\le40

where $x$ and $f(x)$ denote numerical values of distances measured in metres.

A

I.

Show that $f'(x)=-0.1e^{-0.1x}(x-10)^2$ .

[2]

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II.

Determine the stationary point.

[1]

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B

Explain why this stationary point is not a local maximum or minimum.

[1]

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C

Find $f''(x)$ and hence determine the points of inflexion.

[3]

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D

State the equation of the tangent at the stationary point.

[1]

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Question 43

HL • Paper 3

Hard

Calculator Permitted

The displacement of a damped vibrating component is modelled by

A(t)=e^{-0.4t}\cos(2t),\qquad t\ge0

where $t$ is measured in seconds.

Graph of displacement against time for a damped oscillation.

A

I.

Find $A'(t)$ .

[2]

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II.

Find the first stationary time after $t=0$ .

[1]

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B

Calculate the displacement at this stationary time.

[1]

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C

Use the second derivative to classify this stationary point.

[2]

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D

Determine the first positive time at which the graph has a point of inflexion.

[2]

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Question 44

HL • Paper 3

Hard

Calculator Permitted

The mass of algae in a culture is modelled by the Gompertz function

A(t)=500\exp\left(-4e^{-0.3t}\right),\qquad t\ge0

where $A$ is measured in grams and $t$ in days.

Sigmoidal algae mass curve with horizontal upper asymptote.

A

I.

Find $A'(t)$ .

[2]

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II.

Find $A''(t)$ in factored form.

[1]

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B

Determine the time and mass at the point of inflexion.

[2]

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C

Find the maximum growth rate of the algae culture.

[1]

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D

For the generalized model

M(t)=L\exp\left(-be^{-kt}\right)

where $L,b,k>0$ and $t$ is any real number, state the time, mass and growth rate at its point of inflexion.

[2]

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Question 45

HL • Paper 3

Hard

Calculator Permitted

A lifeguard is $30\ \text{m}$ offshore from point $A$ on a straight beach. A swimmer is at point $B$ , $100\ \text{m}$ along the beach from $A$ . The lifeguard swims to a landing point $X$ , where $AX=x$ , and then runs from $X$ to $B$ . The swimming speed is $2\ \text{m s}^{-1}$ and the running speed is $5\ \text{m s}^{-1}$ .

Plan-view diagram of a straight beach with points A, X and B, a lifeguard 30 metres perpendicular offshore from A, a swimming segment to X, and a running segment from X to B.

A

I.

Show that the total travel time is

T(x)=\frac{\sqrt{x^2+900}}{2}+\frac{100-x}{5},\qquad 0\le x\le100

[1]

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II.

Find $T'(x)$ .

[2]

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B

Determine the value of $x$ that minimizes the travel time.

[2]

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C

Calculate the minimum travel time and justify that it is a global minimum on the stated domain.

[2]

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D

If the optimal landing point is interior, and the swimming and running speeds are $v_s$ and $v_r$ , respectively, derive the condition satisfied by the optimal landing point.

[1]

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0

Question 46

HL • Paper 3

Hard

Calculator Permitted

For the curve $y=\ln x$ , where $x>0$ , let $P(a,\ln a)$ be a point on the curve. The tangent and normal at $P$ meet the coordinate axes.

Curve y=ln x with a tangent and normal at a sample point.

A

I.

Find the equation of the tangent at $P$ .

[2]

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II.

Show that the $x$ -coordinate of the tangent's $x$ -intercept is $X(a)=a(1-\ln a)$ .

[1]

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B

Determine the greatest possible value of $X(a)$ and the corresponding value of $a$ .

[2]

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C

Show that the $y$ -intercept of the normal at $P$ is $N(a)=\ln a+a^2$ .

[1]

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D

Determine the value of $a$ for which the normal passes through the origin, and justify that this value is unique.

[2]

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Question 47

HL • Paper 3

Hard

Calculator Permitted

A conical drinking cup has fixed slant height $12\ \text{cm}$ . Its circular opening has radius $r\ \text{cm}$ and its vertical height is $h\ \text{cm}$ .

Cross-section of a right circular cone showing radius r, vertical height h, and fixed slant height 12 centimetres.

A

I.

Show that $h=\sqrt{144-r^2}$ .

[1]

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II.

Show that the volume is

V(r)=\frac{\pi}{3}r^2\sqrt{144-r^2},\qquad 0<r<12

[2]

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B

Find the value of $r$ that maximizes the volume.

[2]

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C

Determine the corresponding height and maximum volume.

[2]

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D

Justify that this stationary value is the global maximum for $0<r<12$ .

[1]

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Question 48

HL • Paper 3

Hard

Calculator Permitted

Consider the family of functions

f_a(x)=x^4-2ax^2

where $a$ is a real parameter. The shape of the graph changes as $a$ passes through zero.

Representative members of the family f_a(x)=x^4-2ax^2 for negative, zero, and positive a.

A

I.

Find $f_a'(x)$ and $f_a''(x)$ .

[2]

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II.

For $a>0$ , determine the stationary points.

[1]

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B

For $a>0$ , classify each stationary point using the second derivative.

[2]

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C

For $a>0$ , determine the points of inflexion and state the intervals of concavity.

[2]

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D

Describe and justify how the stationary-point structure changes when $a=0$ and when $a<0$ .

[1]

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0