A-Level Topic: Differentiating Logarithms 8 Starter and ...

A-Level Starter Activity

Topic: Differentiating Logarithms

and Exponentials

Chapter Reference: Pure 2, Chapter 9

8 minutes

1. Find the value of f'(x) when,

a. f(x) = 3x + ex

(1)

1

b. f(x) = 2 + 2 ln

(1)

c.

f(x)

=

4

+

1 4

ln

(1)

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2.

Find

the

value

of

x

for

which

f'(x)

=

-1

when

f(x)

=

2 8

-

2

+

ln

(3)

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3. Find the equation for the normal to the curve y = 4 + 3ex when x = 0

(4)

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1

4. Find he equation for the tangent to the curve y = 3 - 3 at the point when x = 1.

(4)

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Solutions

1a.

f'(x) = 3 + ex

M1

1b.

f'(x)

=

1 2

-12

+

2

M1

1c.

f'(x)

=

2 -12

+

1 4

M1

2.

f'(x)

=

1 4

?

2

+

1

1 4

?

2

+

1

=

-1

M1

x2 ? 4x + 4 = 0 (x ? 2)2 = 0

M1

x = 2

M1

3.

When x = 0, y = 7

M1

=

3x

M1

Gradient = 3

Gradient of normal = -1

3

M1

y

=

7

-

1 3

M1

4.

x = 1 y = 1 ? 3e

M1

=

1 3

-23

-

3

Gradient

=

1 3

-

3

M1

y

?

(1

?

3e)

=

(1

3

-

3)(

-

1)

M1

y

=

(13

-

3)

+

2 3

M1

A-Level Starter Activity

Topic: Differentiating sin and cos 8

Chapter Reference: Pure 2, Chapter 9

minutes

1. Differentiate:

a. y = cos (3x ? 2)

(1)

b.

y

=

4

sin

(

3

-

)

(1)

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2. Find the coordinate of any stationary points on the curve y = x + 2 sin x in the interval 0 x 2

(4)

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3. Find the equation for the tangent to the curve y = cos x at the point x =

3

(5)

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Solutions

1a.

=

-3 sin(3 - 2)

M1

1b.

=

-4

cos

(3

-

x)

M1

2.

=

1

+

2

cos

M1

S.P: 1 + 2 cos x = 0

cos x = -12

M1

x

=

-

3

=

2 3

y

=

2 3

+

3

M1

x

=

+

3

=

4 3

y

=

4 3

-

3

M1

3.

=

3

y = 1

M1

2

=

- sin

M1

gradient = -sin (3) = -23

M1

y

-

1 2

=

-

3 2

(

-

)

3

M1

33 + 6y ? 3 - 3 = 0

M1

A-Level Starter Activity

Topic: Differentiating tan, sec,

cosec, and cot

Chapter Reference: Pure 2, Chapter 9

7 minutes

1. Differentiate 3 cosec (x + )

6

(1)

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2. Differentiate sec22x

(1)

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3. Differentiate ln (tan 4x)

(2)

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4. Differentiate 2 cot x2

(1)

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5. Find the equation for the tangent to the curve y = cosec x ? 2 sin x at the point x =

6

(4)

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