11. Differentiation

iGCSE (2021 Edition)

Worksheet

1

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

a

b

c

d

e

f

g

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.

3

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

a

f(x) = 1

b

f(x) = 2x - 4

4

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

a

b

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.

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

b

c

d

e

f

g

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

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.

9

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

Complete the table of values:

x | -2 | -1 | 0 | 1 | 2 |
---|---|---|---|---|---|

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:

a

Complete the following table of values:

x | -3 | -2 | 0 | 2 | 3 |
---|---|---|---|---|---|

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?

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?

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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Understand the idea of a derived function.