Gradient of a curve

The Maths Habit Handout to print

A cyclist rides up this hill.
Where is it hardest to pedal?

ABC

At B, where the hill is steepest

Find the gradient of the line through these points.

  1. (1, 2) and (3, 10)

    4

  2. (0, 5) and (5, 0)

    −1

Tangent
A straight line that just touches a curve at one point.

What are the units of the gradient?

Distance (m) against time (s) m/s

Estimate the speed at 3 seconds.

0123450510152025Time (s)Distance (m)212
12÷2=6m/s

Estimate the gradient at x = 2.

0123456048121620xy36
6÷3=2

Estimate the speed at 3 seconds.

0123450510152025Time (s)Distance (m)212
12÷2=6m/s

Estimate the speed at 2 seconds.

0123450510152025Time (s)Distance (m)28
8÷2=4m/s

  1. A tangent goes through (1, 3) and (3, 11). Find its gradient.

    4

  2. A tangent goes through (2, 20) and (6, 12). Find its gradient.

    −2

  3. On a distance (m) against time (s) graph, a tangent goes through (0, 4) and (4, 16). Find the speed.

    3 m/s

  4. A chord joins (1, 2) and (5, 22). Find the average rate of change.

    5

  5. A curve is getting steeper. What is happening to the rate of change?

    It is increasing

Gradient of a curve worksheet, page 1

Answer sheet

  1. A ball is 20 m high after 1 second and 30 m high after 2 seconds. Find its average speed between these times.

    10 m/s

  2. A tangent is drawn at the highest point of a curve. What is its gradient?

    0

  1. Find the gradient of the curve at x = 2.

    4
    0123448120

  2. Distance (m) against time (s): a tangent goes through (1, 5) and (4, 23). Find the speed.

    6 m/s

  3. Ana says the tangent through (0, 9) and (3, 0) has gradient 3. What is wrong?

    It slopes down: −3
Exit tickets to print