5.2 Further Differentiation

1a 1b 1c 1d 2a 2b 2c 2d 3 4a 4b 5 6 7a 7b 7c 7d 8 9a 9b 9c 10 11

Question 1a

Marks: 2

Find an expression for the derivative of each of the following functions:

f (x) = (12 x^{2} - 7) e^{- 2 x}

Key Concepts

Product Rule
Differentiating e^x & lnx

Question 1b

Marks: 3

g (x) = \frac{\tan 3 x}{4 - 5 x^{3}}

Key Concepts

Differentiating Trig Functions
Quotient Rule

Question 1c

Marks: 3

h (x) = {(\ln (2 x^{2} - x - 2))}^{5}

Key Concepts

Chain Rule
Differentiating e^x & lnx

Question 1d

Marks: 4

j (x) = \frac{2 x^{- \frac{3}{4}}}{1 - x^{\frac{3}{5}}}

Key Concepts

Differentiating Powers of x
Quotient Rule

Question 2a

Marks: 3

Find an expression for the derivative of each of the following functions:

f (x) = (3 x - 1) e^{\sin x} D

Key Concepts

Product Rule
Chain Rule

Question 2b

Marks: 3

g (x) = \ln (\cos (x^{2} - 1))

Key Concepts

Chain Rule
Differentiating e^x & lnx

Question 2c

Marks: 4

h (x) = \frac{- \sin (e^{- x})}{e^{x \cos x}}

Key Concepts

Quotient Rule
Chain Rule

Question 2d

Marks: 4

j (x) = \tan (\frac{1}{x^{2} \sqrt[3]{x}})

Key Concepts

Chain Rule
Differentiating Powers of x

Question 3

Marks: 1

Consider the function $f$ defined by $f (x) = - x + \frac{2}{3} \sin^{3} x, x \in ℝ$ .

Show that $f$ is decreasing everywhere on its domain.

Key Concepts

Chain Rule
Increasing & Decreasing Functions

Question 4a

Marks: 4

Consider the function $g$ defined by $g (x) = e^{2 x} - 2 x, x \in ℝ$ .

Point A is the point on the graph of $g$ for which the $x$ -coordinate is $\ln \sqrt{3}$ .

Show that the equation of the tangent to the graph of

g

at point A may be expressed in the form

y = 4 x - 3 (\ln 3 - 1)

Key Concepts

Tangents & Normals
Differentiating e^x & lnx

Question 4b

Marks: 5

Point B is the point on the graph of $g$ at which the normal to the graph is vertical.

Show that the coordinates of the point of intersection between the tangent to the graph of

g

at point A and the tangent to the graph of

g

at point B are

(\frac{3 \ln 3 - 2}{4}, 1)

Key Concepts

Tangents & Normals
Intersecting Graphs

Question 5

Marks: 9

Consider the function $h$ defined by $h (x) = \sin 3 x + e^{3 \sqrt{3} x} \cos 3 x, x \in ℝ .$

Show that the normal line to the graph of $h$ at $x = \frac{π}{9}$ intercepts the $y$ -axis at the point

$(0, \frac{2 π}{27} + \frac{\sqrt{3} + e^{\frac{π \sqrt{3}}{3}}}{2})$

Key Concepts

Product Rule
Tangents & Normals

Question 6

Marks: 8

Let $f (x) = g (x) h (x)$ , where $g$ and $h$ are real-valued functions such that

$g (x) = \ln (\frac{x}{3}) h (x)$

for all $x > 0$ .

Given that $h (3) = a$ and $h' (3) = b$ , where $a \neq 0,$ find the distance between the $y$ -intercept of the tangent to the graph of $f$ at $x = 3$ and the $y$ -intercept of the normal to the graph of $f$ at $x = 3$ . Give your answer in terms of $a$ and/or $b$ as appropriate.

Key Concepts

Product Rule
Tangents & Normals

Question 7a

Marks: 1

Consider the curve with equation $y = \cos (k x) e^{\sin (k x)}$ defined for all $x \in ℝ$ , where $k \neq 0$ is a positive integer.

For the case where

k = 1

, find the number of points in the interval

- \frac{π}{2} \leq x < \frac{3 π}{2}

at which the curve has a horizontal tangent.

Key Concepts

Key Features of Graphs

Question 7b

Marks: 7

Show algebraically that in general the

x

-coordinates of the points at which the curve has horizontal tangents will be the solutions to the equation

\sin^{2} (k x) + \sin (k x) - 1 = 0

ii)

Hence, for the case where

k = 1

, find the

x

-coordinates of the points identified in part (a).

Key Concepts

Chain Rule
Tangents & Normals

Question 7c

Marks: 8

By considering

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

, show algebraically that in general the

x

-coordinates of the points at which the curve is neither concave up nor concave down will be the solutions to the equation

$\sin (k x) \cos (k x) = 0$

ii)

Hence, for the case where

k = 1

, find the -coordinates of the points of inflection on the curve in the interval

- \frac{π}{2} \leq x < \frac{3 π}{2} .

Key Concepts

Second Order Derivatives
Concavity of a Function

Question 7d

Marks: 2

In terms of

k

, state in general how many (i) turning points and (ii) points of inflection the curve will have in the interval

- \frac{π}{2} \leq x < \frac{3 π}{2}

. Give a reason for your answers.

Key Concepts

Using the Unit Circle

Question 8

Marks: 4

Let $f (x) = \frac{g (x)}{h (x)}$ , where $g$ and $h$ are well-defined functions with $h (x) \neq 0$ anywhere on their common domain.

By first writing $f (x) = g (x) {[h (x)]}^{- 1}$ , use the product and chain rules to show that

$f (x) = \frac{h (x) g' (x) - g (x) h' (x)}{{[h (x)]}^{2}}$

Key Concepts

Chain Rule
Product Rule

Question 9a

Marks: 2

Consider the function $f$ defined by $f (x) = e^{x^{k}}$ , $x \in ℝ$ , where $k \geq 1$ is a positive integer.

Show that the graph of

f

will have no points of inflection in the case where

k = 1

Key Concepts

Second Order Derivatives
Points of Inflection

Question 9b

Marks: 5

Show that, for

k \geq 2

, the second derivative of is given by

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

Key Concepts

Chain Rule
Product Rule

Question 9c

Marks: 7

Hence show that the graph of

f

will only have points of inflection in the case where

k

is an odd integer greater than or equal to 3. In that case, give the exact coordinates of the points of inflection, giving your answer in terms of

k

where appropriate. In your work you may use without proof the fact that for odd integers

k

with

k \geq 3

- 1 < \sqrt[k]{- \frac{k - 1}{k}} < - \frac{1}{2}

Key Concepts

Points of Inflection

Question 10

Marks: 7

A small conical flask, in the shape of a right cone stood on its flat base, is being filled with perfume via a small hole at its vertex. The cone has a height of 6 cm and a radius of 2 cm.

Perfume is being poured into the flask at a constant rate of 0.3 cm³s^-1.

Find the rate of change of the depth of the perfume in the flask at the instant when the flask is half full by volume.

Key Concepts

Volume of 3D Shapes
Related Rates of Change

Question 11

Marks: 8

A large block of ice is being prepared for use by a team of ice sculptors. The block is in the shape of a cuboid with the ratio of its length to width to height being equal to 1 : 2 : 5. The block melts uniformly such that its surface area decreases at a constant rate, losing $k m^{2}$ of surface area every hour. You may assume that as the block melts, its shape remains a cuboid with the dimensions in the same ratio to each other as in the original cuboid.

The block of ice is considered stable enough to be sculpted so long as the loss of volume due to melting does not exceed a rate $0.05 m^{3}$ per hour.

Find, in terms of $k$ , the volume of the largest block of ice that can be used for ice sculpting under such conditions.

Key Concepts

Related Rates of Change

DP IB Maths: AI HL

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