Question 1

Marks: 4

Let $f (x) = 3 x^{2}$ .

By differentiating from first principles, show that $f^{'} (x) = 6 x$ .

Key Concepts

First Principles Differentiation

Question 2a

Marks: 1

Let $f (x) = \sin x$ .

a)

Solve the equation

f' (x) = f''' (x)

in the interval

0 \leq x \leq 2 π

.

Key Concepts

Higher Order Derivatives
Differentiating Trig Functions

Question 2b

Marks: 2

b)

Show that

f^{(4)} (x) = f (x)

.

Key Concepts

Higher Order Derivatives
Differentiating Trig Functions

Question 3a

Marks: 2

Find the derivative of each of the following functions:

a)

f (x) = \cot (x + \frac{π}{3})

Key Concepts

Differentiating Reciprocal Trigonometric Functions

Question 3b

Marks: 2

b)

g (x) = 5^{x} - 3 \log_{3} x

Key Concepts

Differentiating Exponential & Logarithmic Functions

Question 3c

Marks: 3

c)

h (x) = \arcsin 4 x

Key Concepts

Differentiating Inverse Trigonometric Functions

Question 4a

Marks: 4

a)

For the curve defined by

y = \tan 3 x

, show that

$\frac{d^{2} y}{d x^{2}} = 18 (\tan 3 x + \tan^{3} 3 x)$

Key Concepts

Differentiating Reciprocal Trigonometric Functions
Pythagorean Identities

Question 4b

Marks: 2

b)

For the curve defined by

y = \arctan x

, show that

$y^{''} = - \frac{2 x}{x^{4} + 2 x^{2} + 1}$

Key Concepts

Differentiating Inverse Trigonometric Functions
Chain Rule

Question 5a

Marks: 6

Consider the function $f$ defined by $f (x) = 2 x + c o t x$ , $0 < x < π$ .

The following diagram shows the graph of the curve $y = f (x)$ :

q5a_5-8_medium_ib-aa-hl-maths

The points marked $A$ and $B$ are the turning points of the graph.

a)

(i)

Find

f^{'} (x)

.

(ii)

Hence find the coordinates of points

A

and

B

.

Key Concepts

Differentiating Reciprocal Trigonometric Functions
Stationary Points & Turning Points

Question 5b

Marks: 4

b)

Find the equation of the normal to the graph at the point where the

x

-coordinate is equal to

\frac{π}{2}

.

Key Concepts

Tangents & Normals

Question 6a

Marks: 2

For each of the following, find $\frac{d y}{d x}$ by differentiating implicitly with respect to $x$ .

a)

x^{2} + y^{2} = 16

Key Concepts

Implicit Differentiation
Applications of Implicit Differentiation

Question 6b

Marks: 2

b)

4 x^{2} - 3 x = y^{2} + 2 y

Key Concepts

Implicit Differentiation
Applications of Implicit Differentiation

Question 6c

Marks: 3

c)

\frac{{(x + y)}^{2}}{3 x} = 1

Key Concepts

Implicit Differentiation
Applications of Implicit Differentiation

Question 6d

Marks: 4

d)

\sqrt{x^{2} + y^{3}} = 4

Key Concepts

Implicit Differentiation
Applications of Implicit Differentiation

Question 7a

Marks: 2

A curve is described by the equation

$\frac{2}{x} - \frac{1}{y} = 1$

a)

Use implicit differentiation with respect to

x

to show that

$\frac{d y}{d x} = \frac{2 y^{2}}{x^{2}}$

Key Concepts

Implicit Differentiation
Applications of Implicit Differentiation

Question 7b

Marks: 4

b)

Use your result from part (a) to find the equation of the

(i)

tangent

(ii)

normal

to the curve at the point

(1, 1)

.

Key Concepts

Applications of Implicit Differentiation
Tangents & Normals

Question 7c

Marks: 5

(c)

(i)

Rearrange the equation of the curve into the form

y = f (x) .

(ii)

Hence find an expression for

\frac{d y}{d x}

entirely in terms of

x .

Key Concepts

Quotient Rule

Question 7d

Marks: 2

d)

Verify that your answer to part (c)(ii) and the result from part (b)(i) both give the same value for the gradient of the tangent to the curve at the point

(1, 1)

.

Key Concepts

Introduction to Derivatives
Applications of Implicit Differentiation

Question 8a

Marks: 2

An international mission has landed a rover on the planet Mars. After landing, the rover deploys a small drone on the surface of the planet, then rolls away to a distance of 6 metres in order to observe the drone as it lifts off into the air. Once the rover has finished moving away, the drone ascends vertically into the air at a constant speed of 2 metres per second.

Let $D$ be the distance, in metres, between the rover and the drone at time $t$ seconds.

Let $h$ be the height, in metres, of the drone above the ground at time $t$ seconds. The entire area where the rover and drone are situated may be assumed to be perfectly horizontal.

a)

Show that

$D = \sqrt{h^{2} + 36}$

Key Concepts

Pythagoras

Question 8b

Marks: 5

b)

(i)

Explain why

\frac{d h}{d t} = 2 .

(ii)

Hence use implicit differentiation to show that

$\frac{d D}{d t} = \frac{2 h}{\sqrt{h^{2} + 36}}$

Key Concepts

Applications of Implicit Differentiation
Modelling with Differentiation

Question 8c

Marks: 4

c)

Find

(i)

the rate at which the distance between the rover and the drone is increasing at the moment when the drone is 8 metres above the ground.

(ii)

the height of the drone above the ground at the moment when the distance between the rover and the drone is increasing at a rate of

1 {ms}^{- 1}

.

Question 9a

Marks: 3

In the diagram below, $CDEFG$ is the outline of a type of informational signboard that a county council plans to use in one of its parks. The shape is formed by a rectangle $CDEG$ , to one side of which an equilateral triangle $EFG$ has been appended.

q9a_5-8_medium_ib-aa-hl-maths

The signboards will be produced in various different sizes. However because of the cost of the edging that must go around the perimeter of the signboards, the council is eager to design the signboards so that the area of a signboard is the maximum possible for a given perimeter.

Let $| CD | = x cm$ and let $| DE | = y cm$ .

a)

(i)

Write down an expression in terms of

x

and

y

for the perimeter of the signboard,

P

.

(ii)

Hence use implicit differentiation to find

\frac{d P}{d x}

Key Concepts

Implicit Differentiation
Applications of Implicit Differentiation

Question 9b

Marks: 3

b)

Explain why, for a given perimeter, it must be true that

\frac{d P}{d x} = 0

, and use this fact to show that

\frac{d y}{d x} = - \frac{3}{2}

.

Key Concepts

Introduction to Derivatives

Question 9c

Marks: 4

c)

Show that the area

A

, of the signboard is given by

A = x y + \frac{\sqrt{3}}{4} x^{2}

.

Question 9d

Marks: 5

d)

Hence use implicit differentiation to find the ratio of

y

to

x

that gives the maximum area.

Key Concepts

Implicit Differentiation
Applications of Implicit Differentiation

DP IB Maths: AA HL

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5.8 Advanced Differentiation

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