Topics
8. Differentiation
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AustraliaNSW
Year 11

8.02 Tangents and the derivative

Lesson
Worksheet
Practice
Worksheet
Tangents and the derivative
1

For each of the following, find the gradient of the tangent at the given point:

a
-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
b
1
2
3
4
5
6
7
8
9
10
x
1
2
3
4
5
6
7
8
9
10
y
c
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
d
-4
-3
-2
-1
1
2
3
x
-1
1
2
3
4
y
e
1
2
3
4
5
6
7
8
9
x
1
2
3
4
5
6
7
8
9
y
f
-3
-2
-1
1
2
3
4
5
x
-6
-5
-4
-3
-2
-1
1
2
y
g
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
2

Consider the graph of y = x below:

a

Find the gradient of the line at x = 4.

b

Find the gradient at any value of x.

c

True or false: A linear function has a constant gradient.

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
3

Find the gradient of the tangent to the following functions for any value of x:

a

f(x) = 1

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
b

f(x) = 2x - 4

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
4

For each of the following exponential functions, use the graph to find the gradient of the tangent drawn at x = 0:

a
-3
-2
-1
1
2
3
x
-4
-3
-2
-1
1
2
3
4
5
6
y
b
-4
-3
-2
-1
1
2
x
-5
-4
-3
-2
-1
1
2
3
y
5

Consider the following graph of the function f \left( x \right) = - \left(x - 4\right)^{3} + 7:

a

State the x-value of the stationary point of f \left( x \right).

b

State the region(s) of the domain where f \left( x \right) is decreasing.

2
4
6
8
x
-4
-2
2
4
6
8
10
12
y
6

For each graph of the following functions:

i

Find the x-value of any stationary point(s) of f \left( x \right).

ii

State the region(s) of the domain where f \left( x \right) is increasing.

iii

State the region(s) of the domain where f \left( x \right) is decreasing.

a
-5
-4
-3
-2
-1
1
2
3
x
-4
-3
-2
-1
1
2
3
4
y
b
-5
-4
-3
-2
-1
1
2
3
x
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
y
c
-4
-3
-2
-1
1
2
3
4
x
-8
-6
-4
-2
2
4
6
8
y
d
-10
-8
-6
-4
-2
2
4
6
x
-10
-8
-6
-4
-2
2
4
6
8
y
e
-14
-12
-10
-8
-6
-4
-2
2
4
6
x
-8
-6
-4
-2
2
4
6
8
y
f
-3
-2
-1
1
2
3
4
x
-16
-12
-8
-4
y
g
-5
-4
-3
-2
-1
1
2
3
4
5
x
-6
-4
-2
2
4
6
8
y
7

Consider the graph of function f \left( x \right):

a

Is there a stationary point of f \left( x \right) when x = 0?

b

Find the x-value of the stationary point of f \left( x \right) when x < 0.

c

Over what region in the domain is f \left( x \right) constant?

d

State the region of the domain where f \left( x \right) is increasing.

e

State the region of the domain where f \left( x \right) is decreasing.

-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
x
-3
-2
-1
1
2
3
4
5
6
7
y
8

Consider the following figure. The black secant line through the given point has the equation y = 2.02 x - 9.14, while the purple secant line through the point has the equation y = 2.01 x - 9.07.

a
Find the coordinates of point A
b

Estimate the gradient of the tangent at point A, to the nearest integer.

x
y
The gradient function
9

Consider the graph of the function f \left( x \right) and the tangent at x=0 graphed below:

-4
-3
-2
-1
1
2
3
4
x
-1
1
y

Complete the table of values:

x-2-1012
f'(x)
10

Consider the graph of the cubic function f \left( x \right) and the tangents at x=-3, x=0 and x=3 graphed below:

-5
-4
-3
-2
-1
1
2
3
4
5
x
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
y
a

Complete the following table of values:

x-3-2023
f'(x)
b

Use the set of data points to draw a possible graph of y = f' \left( x \right).

11

Determine whether rate of change of the following functions is positive, negative or zero for all values of x:

a

f(x) = -8

b

f(x) = 4x-1

c

f(x) = -x+10

12

Consider the function f \left( x \right) = x^2 along with its gradient function graphed below:

a

Which feature of the gradient function tells us whether f \left( x \right) = x^{2} is increasing or decreasing?

b

For x > 0, is the gradient of the tangent positive or negative?

c

For x \geq 0, as the value of x increases, does the gradient of the tangent line increase at a constant rate, increase at an increasing rate, or remain constant?

d

For x < 0, is the gradient of the tangent positive or negative?

-4
-3
-2
-1
1
2
3
4
x
-2
-1
1
2
3
4
5
6
y
e

For x < 0, as the value of x increases, does the gradient of the tangent line increase at a constant rate, increase at an increasing rate, or remain constant?

f

For f \left( x \right) = x^{2}, the derivative f' \left( x \right) is what type of function?

13

Consider the function f \left( x \right) = x^3 along with its gradient function graphed below:

a

For x > 0, is the gradient of the tangent positive or negative?

b

For x \geq 0, as the value of x increases, does the gradient of the tangent line increase at a constant rate, increase at an increasing rate, or remain constant?

c

For x < 0, is the gradient of the tangent positive or negative?

d

For x < 0, as the value of x increases, does the gradient of the tangent line decrease at an decreasing rate, decrease at a constant rate, remain constant?

e

For f \left( x \right) = x^{3}, the derivative f' \left( x \right) is what type of function?

-3
-2
-1
1
2
3
x
-8
-4
4
8
12
y
14

Sean draws the graphs of x^{2}, x^{3}, x^{4}, and x^{5} and draws the tangents to each one at the point where x = 1. Sean then notes where each of the tangents cut the y-axis and records this in the table below:

a
Complete the table using the given information to calculate the gradient of each of the tangents at x = 1.
b

Following the pattern in the table, what would be the gradient of the tangent to the graph of f \left( x \right) = x^{n} at the point where x = 1?

c

Could the equation of the derivative of f \left( x \right) = x^{6} be f' \left( x \right) = x^{5}? Explain your answer.

\text{Graph}y\text{-intercept}\\ \text{of tangent}\text{Gradient} \\ \text{of tangent}
f(x)=x^2(0,-1)
f(x)=x^3(0,-2)
f(x)=x^4(0,-3)
f(x)=x^5(0,-4)
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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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