Topic 5: Differential calculus

After setting the derivative to zero, test each point: the second derivative tells you maximum or minimum.

Quick checks: multiple choice

Question 1Paper 1

To differentiate

f(x)

= 3x^2 - 5x

from first principles,

\frac{f(x + h) - f(x)}{h}

is simplified before

h \to 0

. Which is the simplified expression, before the limit is taken?

Question 2Paper 1

Which is the derivative of

y

= (2x - 5)^4

Question 3Paper 1

Which is the derivative of

y

= x^2e^{3x}

Question 4Paper 1

Which is the derivative of

y

= \frac{2x + 1}{x - 3}

Question 5Paper 1

Which is the derivative of

y

= \ln(x^2 + 4)

Question 6Paper 1

f(x)

= \sin^2 3x

. Find the value of

f'(x)

x

= \frac{\pi}{18}

Question 7Paper 1

Which is the derivative of

y = \tan^{-1}\left(\frac{x}{2}\right)

Question 8Paper 1

Which is the derivative of

y

= \sin^{-1} 3x

, for

-\frac{1}{3}

< x

< \frac{1}{3}

Question 9Paper 1

y

= \frac{4}{\sqrt{x}}

. Find the value of

\frac{dy}{dx}

when

x

= 4

Question 10Paper 1

The function

f(x)

= x^3 - 12x

has a local minimum. What is the minimum value of

f(x)

Question 11Paper 1

Find the equation of the tangent to the curve

y

= x^3 - 2x + 1

at the point where

x

= 2

Question 12Paper 1

Find the coordinates of the point of inflection of the curve

y

= x^3 - 6x^2 + 9x + 9

Question 13Paper 1

For which values of

x

is the function

f(x)

= 2x^3 - 3x^2 - 12x + 5

decreasing?

Question 14Paper 1

A function has derivative

f'(x)

= (x - 1)^2(x + 3)

. Which describes its stationary points?

Question 15Paper 1

Find the slope of the tangent to the circle

x^2 + y^2

= 25

at the point

(3, -4)

Question 16Paper 1

The radius of a spherical balloon increases at 0.5 cm per second. How fast is its volume increasing, in cm

^3

per second, when the radius is 6 cm? (

V

= \frac{4}{3}\pi r^3

)

Question 17Paper 1

A ladder 5 m long leans against a vertical wall on level ground. The foot of the ladder slides away from the wall at 0.3 m per second. How fast, in metres per second, is the top sliding down the wall when the foot is 3 m from the wall?

Question 18Paper 1

A particle moves in a line so that its distance from a fixed point after

t

seconds is

s

= t^3 - 9t^2 + 24t

metres. Find its acceleration, in m/s

^2

, at the first instant it is at rest.

Question 19Paper 1

A closed cylindrical can is to hold

250\pi

^3

. Its surface area is

A

= 2\pi r^2

{}+ \frac{500\pi}{r}

, where

r

cm is the radius. What is the height, in cm, of the can with the least surface area?

Question 20Paper 1

The number of deer in a forest

t

years after a count is modelled by

N

= 800e^{0.05t}

. At what rate, in deer per year, is the number growing at the moment there are 1,200 deer?

Question 21Paper 1

The curve

y

= x - 2\ln x

, for

x

> 0

, has one turning point. What is its

x

-coordinate?

Question 22Paper 1

The height of a plant

t

weeks after it is planted is

h

= 5 + 4\ln(t + 1)

cm. At what rate, in cm per week, is it growing when

t

= 3

Paper-style questions: Section A, then Section B

Turning points and a tangentPaper 1, Section A style · 30 marks

f(x)

= x^3 - 6x^2 + 9x - 2

, for

x \in \mathbb{R}

(a) Find

f'(x)

. [5]

Type fractions as 3/8, powers as x^2, roots as sqrt(5), logs as ln(x) and complex numbers as 3+4i.

(b) Find the coordinates of the local maximum point of

f

. [10]

Type fractions as 3/8, powers as x^2, roots as sqrt(5), logs as ln(x) and complex numbers as 3+4i.

f

. [5]

Type fractions as 3/8, powers as x^2, roots as sqrt(5), logs as ln(x) and complex numbers as 3+4i.

(d) Find the coordinates of the point of inflection of

f

. [5]

Type fractions as 3/8, powers as x^2, roots as sqrt(5), logs as ln(x) and complex numbers as 3+4i.

(e) Find the equation of the tangent to the curve at the point of inflection. [5]

Type fractions as 3/8, powers as x^2, roots as sqrt(5), logs as ln(x) and complex numbers as 3+4i.

An open box from a square cardPaper 1, Section B style · 50 marks

A square sheet of card is 30 cm by 30 cm. A square of side

x

cm is cut from each corner and the sides are folded up to make an open box.

(a) Write the volume of the box,

V

^3

, in terms of

x

. [10]

Type fractions as 3/8, powers as x^2, roots as sqrt(5), logs as ln(x) and complex numbers as 3+4i.

(b) Find

\frac{dV}{dx}

in its simplest factorised form. [15]

Type fractions as 3/8, powers as x^2, roots as sqrt(5), logs as ln(x) and complex numbers as 3+4i.

x

that gives the largest volume. [10]

Type fractions as 3/8, powers as x^2, roots as sqrt(5), logs as ln(x) and complex numbers as 3+4i.

(d) Find the largest volume, in cm

^3

. [10]

Type fractions as 3/8, powers as x^2, roots as sqrt(5), logs as ln(x) and complex numbers as 3+4i.

(e) A box is made with

x

= 2.5

. Find its volume as a fraction of the largest volume, in its simplest form. [5]

Type fractions as 3/8, powers as x^2, roots as sqrt(5), logs as ln(x) and complex numbers as 3+4i.

A cooling cup of coffeePaper 1, Section B style · 50 marks

A cup of coffee is poured and left on a table in a room. Its temperature,

T

^\circ

t

minutes after it is poured is modelled by

T

= 20 + 70e^{-0.08t}

(a) Find the temperature of the coffee when it is poured. [5]

Type fractions as 3/8, powers as x^2, roots as sqrt(5), logs as ln(x) and complex numbers as 3+4i.

(b) Find

\frac{dT}{dt}

. [10]

Type fractions as 3/8, powers as x^2, roots as sqrt(5), logs as ln(x) and complex numbers as 3+4i.

^\circ

C per minute, correct to two decimal places. [10]

Type fractions as 3/8, powers as x^2, roots as sqrt(5), logs as ln(x) and complex numbers as 3+4i.

(d) Find the time it takes the coffee to cool to 50

^\circ

C, in minutes, correct to two decimal places. [10]

Type fractions as 3/8, powers as x^2, roots as sqrt(5), logs as ln(x) and complex numbers as 3+4i.

(e)

\frac{dT}{dt}

= -k(T - 20)

for a constant

k

. Find

k

. [5]

Type fractions as 3/8, powers as x^2, roots as sqrt(5), logs as ln(x) and complex numbers as 3+4i.

(f) Find the time at which the coffee is cooling at 2

^\circ

C per minute, in minutes, correct to two decimal places. [10]

Type fractions as 3/8, powers as x^2, roots as sqrt(5), logs as ln(x) and complex numbers as 3+4i.