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8.03 Differentiation by first principles

Worksheet
Differentiation by first principles
1

Determine whether the following functions are differentiable over their entire domain:

a
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
b
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
c
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
d
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
2

Consider the graph of the function \\ f \left( x \right) = 2 x + 2 shown:

a

Are there any points within its domain where the function is discontinuous?

b

Is the function continuous over its domain?

c

Is the function smooth?

d

State the domain over which the function is differentiable.

-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
3

For each of the following graphs of the functions:

i

Are there any points within its domain where the function is discontinuous?

ii

Is the function continuous over its domain?

iii

State the domain over which the function is differentiable.

a
f(x) = \begin{cases} -x^2+4 & \text{when } x \geq 0 \\ x^2+4, & \text{when } x <0 \end{cases}
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
x
-2
-1
1
2
3
4
5
6
7
8
9
10
y
b
f(x) = \begin{cases} \dfrac{x^{3} - x}{x^{2} - 1} & \text{ when } x\neq\pm 1 \\ 1 & \text{when } x = -1 \\ 4 & \text{when } x=1 \end{cases}
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
x
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
y
4

For each of the following functions:

i

Sketch the graph of the function.

ii

Using interval notation, state the domain over which the function is differentiable.

a

f \left( x \right) = \sqrt{36 - x^{2}}

b

f \left( x \right) = \left| 2 x + 4\right| + 3

c
f(x) = \begin{cases} - 3 + \dfrac{1}{x + 1} & \text{when} x \neq -1 \\ 0 & \text{when } x = -1 \end{cases}
5

The water level in a dam is measured to be 190 \text{ m}. After two weeks, the water level is measured again and found to be 140 \text{ m}.

a

Is there a time during the two weeks that the water level of the dam is at 160 \text{ m}?

b

To answer part (a) what property do we assume about the water level over the two weeks?

c

Consider a function f \left( x \right) with domain \left[0, 2\right], where f \left( 0 \right) = 190 and f \left( 2 \right) = 140. If the function is known to be discontinuous, does the function take a value of 160 at some point in its domain?

6

Consider the function f \left( x \right) = 4 x.

a

Does the function have a constant gradient?

b

\left(1, 4\right) and \left(7, 28\right) are two points that satisfy the function. Determine the gradient of the function using these two points.

c

Determine the gradient of the function from first principles.

7

For each of the following functions, find:

i

f \left( x + h \right)

ii

f \left( x + h \right) - f \left( x \right)

iii

\dfrac{f \left( x + h \right) - f \left( x \right)}{h}

a

f \left( x \right) = 3 x + 2

b

f \left( x \right) = 8 - x

c

f \left( x \right) = - 5 x + 4

d

f \left( x \right) = x^{2}

e

f \left( x \right) = 5 - x^{2}

f

f \left( x \right) = 3 x^{2} - 4 x + 2

g

f \left( x \right) = x^{2} + 3 x + 6

8

For each of the following functions:

i

Find f \left( x + h \right).

ii

Find f \left( x + h \right) - f \left( x \right).

iii

Find \dfrac{f \left( x + h \right) - f \left( x \right)}{h}.

iv

Hence, find f' \left( x \right) from first principles by evaluating \lim_{h \to 0}\left(\dfrac{f \left( x + h \right) - f \left( x \right)}{h}\right).

a
f(x) = 4x^2
b
f(x)=-6x^2
9

Find the derivative of the following functions from first principles:

a

f \left( x \right) = 6

b

f \left( x \right) = - 6

c

f \left( x \right) = 5 x

d

f \left( x \right) = x^{3}

e

f \left( x \right) = 2 x + 3

f
f \left( x \right) = x^{2} - 3
g

f \left( x \right) = x^{2} - 3 x

h
f \left( x \right) = 4 x^{2} + 5
i

f \left( x \right) = 4 x^{2} - 3 x - 5

Gradients
10

Consider the function f \left( x \right) = 7 - x^{2}.

a

Find an expression for the derivative f' \left( x \right) using first principles.

b

Hence, calculate the gradient of f \left( x \right) at the point where x = -1.

11

Consider the function f \left( x \right) = x^{2} + 3 x + 4.

a

Find an expression for the derivative f' \left( x \right) using first principles.

b

Hence, calculate the gradient of f \left( x \right) at the point where x = 4.

12

Consider the function f \left( x \right) = \left(x + 6\right) \left(x + 2\right).

a

Find an expression for the derivative f' \left( x \right) using first principles.

b

Hence, calculate the gradient at the point where x = 3.

13

Consider the function f \left( x \right) = x^{3}.

a

Find an expression for the derivative f' \left( x \right) using first principles.

b

Hence, calculate the gradient of f \left( x \right) at the point where x = 2.

14

Use first principles to find the gradient of f \left( x \right) = 3 x^{2} at the point \left(4, 48\right).

15

Use first principles to find the gradient of f \left( x \right) = 2 x \left(x - 6\right) at the point \left(2, - 16 \right).

16

Use first principles to find the gradient of f \left( x \right) = 3 x^{3} at the point \left( - 4 , - 192 \right).

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Outcomes

MA11-1

uses algebraic and graphical techniques to solve, and where appropriate, compare alternative solutions to problems

MA11-5

interprets the meaning of the derivative, determines the derivative of functions and applies these to solve simple practical problems

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