Improving the estimate of the gradient of a curve

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Lesson details

Learning outcome

I can improve the estimate of the gradient by considering the gradient of the tangent at a fixed point.

Key learning points

Since the gradient is improved by moving the points closer together, you could consider a point.
By drawing the tangent to the graph at a given point, you can estimate the gradient at that point.
The gradient at that point is estimated by calculating the gradient of the tangent.

Keywords

Tangent - A tangent to a curve at a given point is a line that intersects the curve at that point. Both the tangent and the curve have the same gradient at the given point.

Common misconception

Pupils may think a tangent to a curve at a given point cannot intersect the curve at another point.

Define the tangent at a point as the line which has the same gradient as the curve at that point. If the graph is cubic then it is possible for the tangent at a point to intersect the graph again. There are examples in the lesson that show this.

Teacher tip

This lesson builds directly on the previous lesson. Pupils could revisit their previous examples and improve their estimates using this skill. On a distance-time graph this skill can be used to estimate speed at a given point. On a speed-time graph this can be used to estimate acceleration rates.

Licence

This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0

except where otherwise stated. See Oak's terms & conditions

(Collection 2).

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Prior knowledge starter quiz

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6 Questions

Q1.
$$5$$ is __________ for this curve between $$x=-10$$ and $$x=-8$$

the gradient

Correct answer: the estimated gradient

the rate of difference

the intercept

Q2.
Estimate the gradient of this curve between $$x=-4$$ and $$x=0$$

Correct Answer: -2

Q3.
Estimate the gradient of this curve between $$x=-2$$ and $$x=2$$

Correct Answer: -4

Q4.
Match these intervals to the estimated gradient of the curve.

Correct Answer:$$x=-2 \ \text{to} \ x=-1$$,$$4$$

$$4$$

Correct Answer:$$x=-1 \ \text{to} \ x=1$$,$$1$$

$$1$$

Correct Answer:$$x=0 \ \text{to} \ x=2$$,$$5\over2$$

$$5\over2$$

Correct Answer:$$x=2 \ \text{to} \ x=3$$,$$10$$

$$10$$

Correct Answer:$$x=-1 \ \text{to} \ x=2$$,$$2$$

$$2$$

Q5.
Order the estimated gradients of these intervals from highest to lowest.

1 - $$x=2$$ to $$x=3$$

2 - $$x=1$$ to $$x=3$$

3 - $$x=1$$ to $$x=2$$

4 - $$x=0$$ to $$x=2$$

5 - $$x=-2$$ to $$x=1$$

Q6.
Which of these intervals give this curve an estimated gradient of $$0$$?

$$x=1$$ and $$x=3$$

Correct answer: $$x=0$$ and $$x=2$$

$$x=2$$ and $$x=3$$

Correct answer: $$x=0$$ and $$x=4$$

Correct answer: $$x=2$$ and $$x=4$$

Assessment exit quiz

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6 Questions

Q1.
A tangent to a curve at a given point is a line that intersects the curve at that point. Both the tangent and the curve have the same __________ at the given point.

Correct answer: gradient

length

radius

shape

Q2.
When using two points on a curve to estimate the gradient we can improve the accuracy of our estimate by __________ the distance between the two points.

Correct answer: reducing

increasing

calculating

Q3.
If you wanted to calculate the gradient of the curve at a given point, which diagram is likely to be the most helpful?

Correct Answer: An image in a quiz

Q4.
Use this tangent to calculate the gradient of this curve at $$x=0$$

Correct Answer: 3

Q5.
Use this tangent to calculate the gradient of this curve at $$x=1$$

Correct Answer: 5

Q6.
This tangent was drawn by hand. The triangle enables us to estimate the gradient at $$x=5$$ to be __________.

$$5.3$$

$$9$$

Correct answer: $$9\over5$$

$$5\over9$$

$$-{5\over9}$$

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Estimating the gradient of a curve

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Improving the estimate of the gradient of a curve

Lesson slides

Lesson details

Learning outcome

Key learning points

Keywords

Common misconception

Teacher tip

Licence

Lesson video

Worksheet

Prior knowledge starter quiz

6 Questions

Q1.$$5$$ is __________ for this curve between $$x=-10$$ and $$x=-8$$

Q2.Estimate the gradient of this curve between $$x=-4$$ and $$x=0$$

Q3.Estimate the gradient of this curve between $$x=-2$$ and $$x=2$$

Q4.Match these intervals to the estimated gradient of the curve.

Q5.Order the estimated gradients of these intervals from highest to lowest.

Q6.Which of these intervals give this curve an estimated gradient of $$0$$?

Assessment exit quiz

6 Questions

Q1.A tangent to a curve at a given point is a line that intersects the curve at that point. Both the tangent and the curve have the same __________ at the given point.

Q2.When using two points on a curve to estimate the gradient we can improve the accuracy of our estimate by __________ the distance between the two points.

Q3.If you wanted to calculate the gradient of the curve at a given point, which diagram is likely to be the most helpful?

Q4.Use this tangent to calculate the gradient of this curve at $$x=0$$

Q5.Use this tangent to calculate the gradient of this curve at $$x=1$$

Q6.This tangent was drawn by hand. The triangle enables us to estimate the gradient at $$x=5$$ to be __________.

How to plan a lesson using our resources

Q1.
$$5$$ is __________ for this curve between $$x=-10$$ and $$x=-8$$

Q2.
Estimate the gradient of this curve between $$x=-4$$ and $$x=0$$

Q3.
Estimate the gradient of this curve between $$x=-2$$ and $$x=2$$

Q4.
Match these intervals to the estimated gradient of the curve.

Q5.
Order the estimated gradients of these intervals from highest to lowest.

Q6.
Which of these intervals give this curve an estimated gradient of $$0$$?

Q1.
A tangent to a curve at a given point is a line that intersects the curve at that point. Both the tangent and the curve have the same __________ at the given point.

Q2.
When using two points on a curve to estimate the gradient we can improve the accuracy of our estimate by __________ the distance between the two points.

Q3.
If you wanted to calculate the gradient of the curve at a given point, which diagram is likely to be the most helpful?

Q4.
Use this tangent to calculate the gradient of this curve at $$x=0$$

Q5.
Use this tangent to calculate the gradient of this curve at $$x=1$$

Q6.
This tangent was drawn by hand. The triangle enables us to estimate the gradient at $$x=5$$ to be __________.