Differential Calculus - IB Math AA Question Bank

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Differential Calculus

Practice exam-style IB Math AA questions for Differential Calculus, aligned with the syllabus and grouped by topic.

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Question 1

SL • Paper 1

Easy

Non Calculator

SL • Paper 1

Easy

Non Calculator

The curve $C$ has equation $y=x^3-3x^2+2x+4$ .

Find $\frac{dy}{dx}$ .

[1]

Find the coordinates of the point on $C$ where $x=1$ , and the gradient of the tangent to $C$ at this point.

[2]

Find the equation of the normal to $C$ at this point.

[2]

Question 2

SL • Paper 2

Easy

Calculator Permitted

SL • Paper 2

Easy

Calculator Permitted

Let $f(x)=\sin x$ . The table shows values of

q(h)=\frac{f(2+h)-f(2)}{h}

for values of $h$ close to $0$ .

h	q(h)
-0.10	-0.370
-0.05	-0.393
-0.02	-0.407
-0.01	-0.412
0.01	-0.421
0.02	-0.425
0.05	-0.439
0.10	-0.461

Estimate $\displaystyle \lim_{h\to 0}q(h)$ .

[2]

Interpret the answer to part (a) in terms of the graph of $f$ .

[1]

Find the equation of the tangent to the graph of $f$ at $x=2$ .

[2]

Question 3

SL • Paper 2

Easy

Calculator Permitted

SL • Paper 2

Easy

Calculator Permitted

The curve $C$ has equation

y=x^3-4x^2+2x+5

The point $P$ on $C$ has $x$ -coordinate $3$ .

Find the coordinates of $P$ .

[1]

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

[2]

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

[2]

Question 4

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

Let $f(x)=x^3+\frac{3}{2}x^2-6x+1$ , for $x\in\mathbb{R}$ .

Find $f'(x)$ in factorized form.

[2]

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

[2]

Find the coordinates and nature of each local extremum of $f$ .

[2]

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Question 5

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

The function $f$ is defined by $f(x)=xe^{2x}+1$ .

Find $f'(x)$ .

[2]

Find the equation of the tangent to the graph of $f$ at the point where $x=0$ .

[3]

Question 6

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

Let $f(x)=x^4-4x^3$ , for $x\in\mathbb{R}$ .

Find $f''(x)$ .

[2]

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

[2]

Find the coordinates of the points of inflexion of the graph of $f$ .

[2]

Question 7

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

The temperature $T$ of a liquid, in degrees Celsius, is modelled by $T(t)=t^2+4t$ , where $t$ is the time in minutes.

Find the average rate of change of $T$ between $t=1$ and $t=1+h$ , where $h\ne0$ .

[2]

Hence find the instantaneous rate of change of $T$ when $t=1$ .

[1]

Find the equation of the tangent to the graph of $T$ at $t=1$ .

[2]

Question 8

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

A spherical balloon has radius $r$ cm, volume $V$ cm $^3$ and surface area $S$ cm $^2$ . At an instant when $r=2$ , the volume is increasing at a rate of $12\pi$ cm $^3$ s $^{-1}$ .

Find $\frac{dr}{dt}$ at this instant.

[3]

Find $\frac{dS}{dt}$ at this instant.

[2]

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Question 9

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

Let

f(x)=xe^{-0.4x},\qquad g(x)=\ln(x+2),\qquad x\geq 0

Find $f'(x)$ .

[3]

Find the value of $x\geq 0$ for which the tangent to $f$ at $x$ is parallel to the tangent to $g$ at $x$ .

[3]

Question 10

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

Let

f(x)=xe^{-\frac{x^2}{4}},\qquad x\in\mathbb{R}

Find $f'(x)$ .

[2]

Show that $f''(x)=e^{-\frac{x^2}{4}}\left(\dfrac{x^3}{4}-\dfrac{3x}{2}\right)$ .

[2]

Find the positive $x$ -coordinate of the point of inflexion of the graph of $f$ .

[1]

Question 11

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

The function $f$ is defined by

f(x)= \begin{cases} x^2+kx, & x<1,\\ ax+b, & x\geq 1. \end{cases}

The function is differentiable at $x=1$ , and $f(3)=8$ .

Write down three equations involving $a$ , $b$ and $k$ .

[3]

Find the values of $a$ , $b$ and $k$ .

[2]

Question 12

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

The diagram shows the curve $y=6-x^2$ , for $0\le x\le \sqrt{6}$ . A rectangle has one vertex at the origin, sides parallel to the axes, and its upper right vertex on the curve. The width of the rectangle is $x$ and its area is $A$ .

Express $A$ in terms of $x$ .

[2]

Find the value of $x$ for which $A$ is a maximum.

[3]

Find the maximum area of the rectangle.

[1]

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Question 13

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Let $f(x)=x^3-2x$ .

Use first principles to find $f'(x)$ .

[4]

Hence find the equation of the tangent to the graph of $f$ at the point where $x=-1$ .

[2]

Question 14

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Consider the limit

\lim_{x\to0}\frac{e^x-1-x}{x^2}

Show that direct substitution gives an indeterminate form.

[1]

Use l'Hopital's rule to evaluate the limit.

[4]

Question 15

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

The curve $C$ is defined implicitly by

x^2+xy+y^2=7

The point $P(1,2)$ lies on $C$ .

Find $\frac{dy}{dx}$ in terms of $x$ and $y$ .

[3]

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

[2]

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

[1]

Question 16

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

Let

f(x)=3^x+\arcsin(2x-1)

where $\frac{1}{2}\le x\le1$ .

Find $f'(x)$ .

[3]

Find the equation of the tangent to the graph of $f$ at $x=\frac{1}{2}$ , using the right-hand derivative at this endpoint.

[2]

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Question 17

HL • Paper 1

Medium

Non Calculator

HL • Paper 1

Medium

Non Calculator

The function $f$ is defined by

f(x)=x^3-3x,\qquad -2\le x\le1

Find the stationary points of $f$ in the interval $-2\le x\le1$ .

[2]

Determine the absolute maximum and absolute minimum values of $f$ on this interval.

[3]

Question 18

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

Consider the function

h(x)=x^4-6x^2+3x+2,\qquad -3\leq x\leq 3

Find the $x$ -coordinates of the stationary points of $h$ .

[3]

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

[2]

Find the absolute maximum value of $h$ on the interval $[-3,3]$ .

[2]

Question 19

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

A rectangular printed region has area $600\ \text{cm}^2$ . It is surrounded by margins of $2\ \text{cm}$ on the left and right, and $3\ \text{cm}$ at the top and bottom. Let $x$ be the width, in cm, of the printed region.

Show that the total area $A\ \text{cm}^2$ of the paper is

A=624+6x+\frac{2400}{x}

[2]

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

[3]

Find the minimum total area of the paper.

[1]

Question 20

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Let

p(x)=x^3+2x

and

r(x)=p'(x)e^{2x}

Use first principles to show that $p'(x)=3x^2+2$ .

[4]

Find $r''(0)$ .

[2]

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Question 21

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Consider the following limits.

Evaluate

\lim_{x\to0}\frac{e^x-1-x-\frac{x^2}{2}}{x^3}

[4]

Find the horizontal asymptote of

y=\frac{5x+e^x}{2e^x-x}

as $x\to\infty$ .

[2]

Question 22

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Water is poured into a right circular cone. At all times the radius $r$ of the water surface and the depth $h$ of the water satisfy

r=\frac{2}{5}h

The volume of water is increasing at a rate of $0.080\ \text{m}^3\text{ min}^{-1}$ .

Show that the volume of water is

V=\frac{4\pi}{75}h^3

[1]

Find an expression for $\dfrac{dV}{dt}$ in terms of $h$ and $\dfrac{dh}{dt}$ .

[2]

Calculate $\dfrac{dh}{dt}$ when $h=6$ .

[2]

Calculate $\dfrac{dr}{dt}$ when $h=6$ .

[1]

Question 23

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Let

f(x)=\arctan(2x-1)+3^x,\qquad 0\leq x\leq 2

Find $f'(x)$ .

[3]

Find the value of $x$ for which the tangent to the graph of $f$ has gradient $5$ .

[1]

Find the equation of this tangent.

[2]

Question 24

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

Let $f(x)=x^3-6x^2+9x+1$ , where $x\in\mathbb{R}$ .

Part (a)

I.

Find $f'(x)$ .

[2]

II.

Find the $x$ -coordinates of the stationary points of the graph of $f$ .

[2]

Part (b)

I.

Find the coordinates and nature of each stationary point.

[4]

II.

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

[2]

Find the equation of the tangent to the graph of $f$ at its point of inflexion.

[3]

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Question 25

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

The function $f$ is defined by $f(x)=(x+1)e^{-x}$ , for $x\geq -1$ .

I.

Show that $f'(x)=-xe^{-x}$ .

[3]

II.

Find $f''(x)$ .

[2]

Find the coordinates and nature of the stationary point of the graph of $f$ .

[3]

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

[2]

Question 26

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

Let $f(x)=x^2\ln x$ , where $x>0$ .

I.

Find $f'(x)$ .

[3]

I.

Find the $x$ -coordinate of the stationary point of the graph of $f$ .

[2]

II.

Find the coordinates and nature of this stationary point.

[4]

Find the equation of the tangent to the graph of $f$ at the point where $x=e$ .

[2]

Question 27

SL • Paper 1

Medium

Non Calculator

SL • Paper 1

Medium

Non Calculator

The curve $C$ has equation $y=x+\frac{4}{x}$ , where $x>0$ .

I.

Find $\frac{dy}{dx}$ .

[2]

II.

Find the coordinates and nature of the stationary point of $C$ .

[2]

A tangent to $C$ is parallel to the line $y=-3x+5$ . Find the equation of this tangent.

[3]

Determine the range of $y$ for points on $C$ .

[2]

Find the equation of the tangent to $C$ at the point where $x=4$ .

[2]

Question 28

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

The rate $R(t)$ , in litres per hour, at which water flows through a filter is modelled by $R(t)=0.04t^3-0.9t^2+5t+12$ , where $t$ is the time in hours after the filter is switched on and $0\le t\le18$ .

I.

Find $R'(t)$ .

[1]

II.

Interpret $R'(6)$ in the context of the model.

[2]

Find the times at which $R$ has stationary points, and state the nature of each stationary point.

[3]

Determine the maximum flow rate predicted by the model for $0\le t\le18$ .

[2]

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Question 29

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

A closed cylindrical container has radius $r\ \text{cm}$ and height $h\ \text{cm}$ . Its volume is fixed at $750\ \text{cm}^3$ . The total surface area is denoted by $S\ \text{cm}^2$ .

Show that

S=2\pi r^2+\frac{1500}{r}

[2]

I.

Find $\dfrac{dS}{dr}$ .

[1]

II.

Find the value of $r$ which minimizes $S$ .

[3]

Find the height and the minimum total surface area of the container.

[2]

Question 30

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

A company models its weekly profit, in hundreds of dollars, by $P(x)=-0.02x^3+1.2x^2+15x-200$ where $x$ is the number of units produced in one week, in hundreds, and $0\le x\le70$ .

I.

Find $P'(x)$ .

[1]

II.

Interpret the meaning of $P'(30)=33$ .

[1]

Determine the interval on which the profit is increasing.

[3]

Find the maximum weekly profit predicted by the model.

[3]

Question 31

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

The curve $C$ is defined implicitly by

x^2+y^2+xy+e^y=5

The point $P(2,0)$ lies on $C$ .

Show that

\frac{dy}{dx}=-\frac{2x+y}{2y+x+e^y}

[3]

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

[2]

Find the coordinates of the second point where this normal intersects $C$ .

[2]

Question 32

SL • Paper 1

Hard

Non Calculator

SL • Paper 1

Hard

Non Calculator

A square sheet of card has side length $12\ \text{cm}$ . Squares of side length $x\ \text{cm}$ are cut from each corner, and the remaining card is folded to make an open rectangular box. The volume of the box is $V\ \text{cm}^3$ .

I.

Show that $V=x(12-2x)^2$ .

[3]

II.

Write down the possible values of $x$ .

[1]

I.

Find $\frac{dV}{dx}$ in factorized form.

[3]

II.

Find the value of $x$ for which the volume is a maximum. Justify your answer.

[3]

Find the maximum volume of the box.

[1]

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Question 33

SL • Paper 1

Hard

Non Calculator

SL • Paper 1

Hard

Non Calculator

The function $f$ is defined by $f(x)=x+2\cos x$ , for $0\leq x\leq \pi$ .

I.

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

[3]

Find the $x$ -coordinates of the stationary points of the graph of $f$ .

[2]

Classify the stationary points as local maxima or local minima.

[2]

Determine the absolute maximum and absolute minimum values of $f$ on $0\leq x\leq \pi$ .

[3]

Question 34

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

Let $p(x)=x^3+3x^2-1$ , where $x\in\mathbb{R}$ .

I.

Use first principles to show that $p'(x)=3x^2+6x$ .

[4]

II.

Find the equation of the tangent to the graph of $p$ at the point where $x=-1$ .

[2]

I.

Find $p''(x)$ and $p^{(3)}(x)$ .

[2]

II.

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

[2]

Find the coordinates and nature of the stationary points of the graph of $p$ .

[2]

Question 35

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

For $a\in\mathbb{R}$ , consider $L(a)=\lim_{x\to0}\frac{e^{2x}-1-2x-ax^2}{x^2}$

I.

Show that direct substitution gives an indeterminate form.

[1]

II.

Use l'Hopital's rule to show that $L(a)=2-a$ .

[4]

Find the value of $a$ for which $L(a)=0$ .

[1]

I.

Evaluate $\lim_{x\to\infty}\frac{x^2+e^x}{3e^x-2x}$

[2]

II.

Hence state the horizontal asymptote of $y=\frac{x^2+e^x}{3e^x-2x}$ as $x\to\infty$ .

[2]

Question 36

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

Let $h(x)=\arctan x-\frac{x}{2}$ , where $x\in\mathbb{R}$ .

I.

Find $h'(x)$ .

[2]

II.

Find the $x$ -coordinates of the stationary points of the graph of $h$ .

[2]

Find the coordinates and nature of each stationary point.

[3]

Determine the intervals on which $h$ is increasing and decreasing.

[1]

Find the equation of the normal to the graph of $h$ at the origin.

[2]

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Question 37

SL • Paper 2

Hard

Calculator Permitted

SL • Paper 2

Hard

Calculator Permitted

A sensor reading is modelled by $f(x)=xe^{-0.3x}+0.5\sin x$ for $0\le x\le10$ , where $x$ is measured in seconds.

I.

Show that $f'(x)=e^{-0.3x}(1-0.3x)+0.5\cos x$ .

[2]

II.

Find the equation of the tangent to the graph of $f$ at $x=2$ .

[3]

Find all values of $x$ in the interval $0\le x\le10$ for which the tangent to the graph of $f$ is horizontal.

[2]

Hence state the intervals on which $f$ is increasing.

[1]

Question 38

SL • Paper 2

Hard

Calculator Permitted

SL • Paper 2

Hard

Calculator Permitted

A function $f$ has derivative $f'(x)=0.1(x+3)(x-1)^2(5-x)$ for $-4\le x\le6$ .

I.

Write down the $x$ -values at which $f$ has stationary points.

[1]

II.

Determine the intervals on which $f$ is increasing.

[2]

State the nature of each stationary point of $f$ .

[2]

Find the $x$ -coordinates of the points of inflexion of $f$ .

[3]

Question 39

SL • Paper 2

Hard

Calculator Permitted

SL • Paper 2

Hard

Calculator Permitted

The function $f$ is defined by $f(x)=\frac{\ln(x+1)}{x+1},\qquad x>0$

I.

Find $f'(x)$ .

[3]

Find the equation of the tangent to the graph of $f$ at $x=3$ .

[3]

Find the coordinates of the stationary point of the graph of $f$ .

[2]

Question 40

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

A drone is moving in a vertical plane. At time $t$ , its height above level ground is $h\ \text{m}$ and its horizontal distance from an observer is $x\ \text{m}$ . The angle of elevation of the drone from the observer is $\theta$ , where

\tan\theta=\frac{h}{x}

At a particular instant, $h=120$ , $x=160$ , $\dfrac{dh}{dt}=3$ and $\dfrac{dx}{dt}=-4$ , with distances measured in metres and time in seconds.

I.

Show that

\sec^2\theta\frac{d\theta}{dt}=\frac{x\frac{dh}{dt}-h\frac{dx}{dt}}{x^2}

[3]

II.

Calculate $\dfrac{d\theta}{dt}$ at this instant.

[1]

Let $s$ be the distance from the observer to the drone. Find $\dfrac{ds}{dt}$ at this instant.

[3]

Interpret the sign of your answer to part (b).

[1]

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Question 41

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

For $a>0$ , consider the family of functions

f_a(x)=x^4-4ax^3+2a^2x^2,\qquad x\in\mathbb{R}

The diagram shows the graph of $f_a$ for one positive value of $a$ , with its three stationary points indicated.

I.

Use first principles to show that if $p(x)=x^3$ , then $p'(x)=3x^2$ .

[3]

II.

Hence find $f_a'(x)$ and $f_a''(x)$ .

[2]

I.

For $a=1$ , find the $x$ -coordinates of the stationary points of $f_1$ .

[2]

II.

Classify the three stationary points for $a=1$ .

[2]

I.

By writing $x=at$ , show that $f_a(at)=a^4(t^4-4t^3+2t^2)$ .

[1]

II.

Hence describe the geometrical transformation that maps the graph of $f_1$ onto the graph of $f_a$ .

[3]

Question 42

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A student investigates limits of quotients near $x=0$ using a GDC. For real $m$ , define

L_m=\lim_{x\to0}\frac{\ln(1+x)-x+mx^2}{1-\cos x}

The table generated by the student suggests that $L_m=3$ for one value of $m$ .

x	Quotient
-0.100	2.9304
-0.050	2.9660
-0.010	2.9933
-0.001	2.9993
0.001	3.0007
0.010	3.0066
0.050	3.0328
0.100	3.0646

I.

Show that direct substitution in the quotient defining $L_m$ gives an indeterminate form.

[1]

II.

Using Maclaurin series, show that $L_m=2m-1$ .

[3]

Determine the value of $m$ suggested by the table.

[2]

I.

For $m=2$ , verify $L_m=3$ using l'Hopital's rule.

[3]

II.

For $k>0$ , define

M_k=\lim_{x\to0}\frac{e^{kx}-1-kx}{\ln(1+x)-x}

Find $k$ if $M_k=-9$ .

[3]

Question 43

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A differentiable function $F$ is defined on $\mathbb{R}$ and satisfies

F'(x)=e^{-x}(x^2-4),\qquad F(0)=1

The graph of $F'$ is shown.

I.

Determine the intervals on which $F$ is increasing and decreasing.

[2]

II.

Classify the stationary points of $F$ .

[2]

Show that points of inflexion of $F$ occur where $x=1\pm\sqrt5$ . You are not required to find the corresponding $y$ -coordinates.

[4]

I.

For $c>0$ , suppose $G_c'(x)=e^{-x}(x^2-c^2)$ . State the $x$ -coordinates of the stationary points of $G_c$ .

[1]

II.

Show that the horizontal separation between the two points of inflexion of $G_c$ is always $2\sqrt{c^2+1}$ .

[3]

Question 44

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

Two autonomous vehicles move on straight roads that meet at right angles at a junction $O$ . Vehicle $A$ moves east along the horizontal road and vehicle $B$ moves south along the vertical road. At time $t$ seconds after noon,

x=40+6t,\qquad y=150-8t

where $x$ and $y$ are the distances in metres of $A$ and $B$ from $O$ , respectively. The model is used while $0\le t\le18$ .

I.

Write down an expression for the square of the distance $D$ between the vehicles.

[1]

II.

Find $\dfrac{dD}{dt}$ in terms of $x$ , $y$ and $D$ .

[3]

Find the rate of change of the distance between the vehicles when $t=10$ .

[3]

I.

Show that the vehicles are closest when $t=9.60$ , correct to three significant figures.

[3]

II.

Find the minimum distance between the vehicles.

[2]

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Question 45

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

The curve $C$ is defined implicitly by $x^2+2y^2-xy=8$ The point $P(2,2)$ lies on $C$ .

I.

(a)(i) Show that $\frac{dy}{dx}=\frac{y-2x}{4y-x}$

[3]

II.

(a)(ii) Find the equation of the tangent to $C$ at $P$ .

[2]

I.

(b)(i) Find the points on $C$ where the tangent is horizontal.

[3]

II.

(b)(ii) Find the points on $C$ where the tangent is vertical.

[3]

Explain why no point on $C$ has both a horizontal and a vertical tangent.

[1]

Question 46

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

A point $(x,y)$ moves in the first quadrant on the ellipse $x^2+4y^2=100$ At a certain instant, $x=6$ , $y=4$ and $\frac{dx}{dt}=-2\ \text{cm s}^{-1}$ . A rectangle is formed with vertices $(\pm x,\pm y)$ , so its area is $A\ \text{cm}^2$ .

I.

Find $\frac{dy}{dx}$ in terms of $x$ and $y$ .

[2]

II.

Find $\frac{dy}{dt}$ at the instant when $x=6$ and $y=4$ .

[3]

I.

Express $A$ in terms of $x$ only.

[2]

II.

Find the maximum possible area of the rectangle.

[4]

Find $\frac{dA}{dt}$ at the instant when $x=6$ and $y=4$ .

[1]

Question 47

HL • Paper 1

Hard

Non Calculator

HL • Paper 1

Hard

Non Calculator

The function $f$ is defined by

f(x)=\begin{cases} ax^2+bx+1, & x<1,\\ c\ln x+x, & x\geq 1,\end{cases}

where $a$ , $b$ and $c$ are constants. The function is differentiable at $x=1$ , and $f'(2)=2$ .

I.

Write down three equations involving $a$ , $b$ and $c$ .

[3]

II.

Find the values of $a$ , $b$ and $c$ .

[3]

Using the values found in part (a), find the coordinates and nature of any stationary point of the graph of $f$ .

[3]

Determine whether the graph of $f$ has a point of inflexion at $x=1$ . Justify your answer.

[2]

Question 48

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Let $f(x)=x^4-3x^2$

I.

Use first principles to show that $f'(x)=4x^3-6x$ .

[4]

Find $f^{(3)}(x)$ and $f^{(4)}(x)$ .

[2]

Let $p(x)=x^4-3x^2+ae^{2x}$ Find the value of $a$ for which $p^{(4)}(0)=p''(0)$ .

[2]

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Question 49

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

For $x\ne0$ , define

F(x)=\frac{\sin x-x\cos x}{x^3}

I.

Show that direct substitution in $\displaystyle\lim_{x\to0}F(x)$ gives an indeterminate form.

[1]

II.

Use l'Hopital's rule to evaluate $\displaystyle\lim_{x\to0}F(x)$ .

[3]

A new function $G$ is defined by $G(x)=F(x)$ for $x\ne0$ and $G(0)=k$ . Find the value of $k$ for which $G$ is continuous at $x=0$ .

[1]

Using $G(0)=\dfrac13$ , determine the absolute minimum value of $G$ on $-2\le x\le2$ .

[3]

Question 50

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

The curve $C$ is defined implicitly by $x^2y+y^3=10$ The point $P(1,2)$ lies on $C$ .

I.

Show that $\frac{dy}{dx}=-\frac{2xy}{x^2+3y^2}$

[3]

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

[2]

Find $\dfrac{d^2y}{dx^2}$ at $P$ and state whether the curve is concave-up or concave-down at $P$ .

[3]

Question 51

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

The function $f$ is defined by $f(x)=2^x\arctan x+\log_3(x+2),\qquad x>-2$

I.

Find $f'(x)$ .

[3]

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

[2]

The tangent to the graph of $f$ at $x=a$ has gradient $2$ , where $0<a<1$ . Find the equation of this tangent.

[3]

Question 52

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

The function $f$ is defined by

f(x)=\begin{cases}x^2+px+q, & x<2,\\ re^{x-2}+s(x-2), & x\ge2.\end{cases}

The function is differentiable at $x=2$ , and $f(0)=5$ , $f(4)=10$ .

I.

Write down an equation involving $q$ .

[1]

II.

Use continuity and differentiability at $x=2$ to write down two further equations.

[2]

III.

Determine the values of $p$ , $q$ , $r$ and $s$ .

[2]

Find $f'(x)$ .

[2]

Determine whether $f$ has any local minimum points.

[3]

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Question 53

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

The curve $C$ is given in the first quadrant by

x^{\frac23}+y^{\frac23}=a^{\frac23},\qquad a>0

A point $P$ on $C$ may be written as $P(a\cos^3 t,a\sin^3 t)$ , where $0<t<\frac{\pi}{2}$ .

Use implicit differentiation to show that

\frac{dy}{dx}=-\left(\frac{y}{x}\right)^{\frac13}

[3]

I.

Show that the tangent to $C$ at $P(a\cos^3 t,a\sin^3 t)$ has equation

\frac{x}{\cos t}+\frac{y}{\sin t}=a

[3]

II.

Find the intercepts made by this tangent with the coordinate axes.

[1]

I.

A rectangle with sides parallel to the coordinate axes has opposite vertices at the origin and at $P$ . Show that its area is $A(t)=a^2\cos^3t\sin^3t$ .

[1]

II.

Find the maximum possible area of the rectangle, giving the corresponding coordinates of $P$ .

[3]

In the case $a=8$ , a particle moves along $C$ in the first quadrant. At an instant when $x=3\sqrt3$ and $y=1$ , $x$ is decreasing at $0.060\ \text{m s}^{-1}$ . Find $\dfrac{dy}{dt}$ at this instant.

[3]

Question 54

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

For $a>0$ , define

f_a(x)=\arctan(ax)-\ln(x+1),\qquad x>0

This question investigates how the number of stationary points of $f_a$ depends on $a$ .

I.

Find $f_a'(x)$ .

[2]

II.

Show that stationary points satisfy

a^2x^2-ax+1-a=0

[2]

For $a=2$ , find the exact $x$ -coordinate of the stationary point in the domain $x>0$ .

[3]

I.

Show that the discriminant of the quadratic in part (a)(ii) is $a^2(4a-3)$ .

[2]

II.

Determine the number of stationary points of $f_a$ in the domain $x>0$ for all $a>0$ .

[4]

Question 55

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

For a positive integer $n$ , define

f_n(x)=x^ne^{-x},\qquad x>0

These curves are used to model quantities that rise from zero and then decay.

I.

Find $f_n'(x)$ .

[2]

II.

Hence determine the $x$ -coordinate of the maximum point of $f_n$ .

[2]

I.

Show that

f_n''(x)=e^{-x}x^{n-2}\left(x^2-2nx+n(n-1)\right)

[3]

II.

Find the $x$ -coordinates of the points of inflexion of $f_n$ , taking into account the domain $x>0$ .

[2]

For $n=9$ , find the interval on which $f_9$ is concave-down and explain its position relative to the maximum point.

[4]

Question 56

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

For $c>0$ , define

s_c(x)=\sqrt{x^2+c^2},\qquad x\in\mathbb{R}

These functions are sometimes used as smooth approximations to $|x|$ .

I.

Find $s_c'(x)$ .

[2]

II.

State $s_c'(0)$ and interpret its meaning geometrically.

[2]

I.

For fixed $x\ne0$ , find $\displaystyle \lim_{c\to0^+}s_c'(x)$ .

[2]

II.

Explain why this does not prove that $|x|$ is differentiable at $x=0$ .

[2]

Find $s_c''(x)$ and hence determine the value of $x$ at which the concavity of $s_c$ is greatest.

[4]

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Question 57

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

Let

f(x)=x+e^x,\qquad x\in\mathbb{R}

The inverse function of $f$ is denoted by $g$ . This question investigates derivatives of $g$ without finding an explicit formula for $g$ .

I.

Show that $f$ has an inverse function.

[2]

II.

Write down the value of $g(1)$ .

[1]

I.

Find $g'(1)$ .

[2]

II.

Show that $g''(1)=-\dfrac18$ .

[2]

Use the tangent line to $y=g(x)$ at $x=1$ to estimate $g(1.1)$ .

[2]

Prove that, for any twice differentiable function $f$ with inverse $g$ and with $f'(x)\ne0$ ,

g''(f(x))=-\frac{f''(x)}{(f'(x))^3}

[4]

Question 58

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

For real $a$ , consider

Q_a(x)=\frac{\cos x-1+\frac{x^2}{2}+ax^4}{x^4},\qquad x\ne0

A GDC graph of $Q_a$ near $x=0$ is used to investigate whether $Q_a$ has a removable discontinuity at $x=0$ .

I.

Use a Maclaurin series to show that

\lim_{x\to0}Q_a(x)=a+\frac{1}{24}

[3]

II.

Find the value of $a$ for which the limit is $0$ .

[2]

For $a=0$ , evaluate $\displaystyle \lim_{x\to0}Q_0(x)$ using repeated l'Hopital's rule.

[4]

I.

Define $Q_a(0)$ so that $Q_a$ is continuous at $x=0$ .

[1]

II.

Explain why this definition removes the discontinuity at $x=0$ .

[2]

Question 59

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

For $0\le x\le3$ , let $C$ be the curve

y=\arctan x

For a fixed real number $a$ , the point $P(a,0)$ lies on the $x$ -axis. A point $Q$ on $C$ has coordinates $(x,\arctan x)$ . The distance $PQ$ is to be minimized.

I.

Show that minimizing $PQ$ is equivalent to minimizing

D(x)=(x-a)^2+(\arctan x)^2

[1]

II.

Show that an interior stationary point satisfies

a=x+\frac{\arctan x}{1+x^2}

[3]

For $a=2$ , find the coordinates of the point $Q$ on $C$ closest to $P$ .

[4]

I.

Let

\phi(x)=x+\frac{\arctan x}{1+x^2}

Show that $\phi$ is increasing on $0\le x\le3$ .

[3]

II.

Deduce the values of $a$ for which the closest point occurs at the endpoint $x=3$ .

[3]

Question 60

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

Let

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

Repeated differentiation of $f$ produces expressions of the form $R_ne^{2x}\sin(x+\alpha_n)$ . This question investigates this pattern.

(a)

I.

Find $f'(x)$ .

[2]

II.

Find $f''(x)$ .

[2]

(b)

I.

Let $\theta=\arctan\left(\dfrac12\right)$ . Show that

f'(x)=\sqrt5e^{2x}\sin(x+\theta)

[2]

II.

Prove by induction that, for all integers $n\ge0$ ,

f^{(n)}(x)=5^{\frac n2}e^{2x}\sin(x+n\theta)

[3]

Use your result to find $f^{(6)}(0)$ in exact form.

[2]

Using a GDC or otherwise, find the least positive integer $n$ for which $|f^{(n)}(0)|>1000$ .

[2]

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