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Year 11
Higher

Improving the estimate of the gradient of a curve

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

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New
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Year 11
Higher

Improving the estimate of the gradient of a curve

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

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These resources will be removed by end of Summer Term 2025.

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Lesson details

Key learning points

  1. Since the gradient is improved by moving the points closer together, you could consider a point.
  2. By drawing the tangent to the graph at a given point, you can estimate the gradient at that point.
  3. 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.


To help you plan your year 11 maths lesson on: Improving the estimate of the gradient of a curve, download all teaching resources for free and adapt to suit your pupils' needs...

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.
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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).

Lesson video

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Starter quiz

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6 Questions

Q1.
$$5$$ is __________ for this curve between $$x=-10$$ and $$x=-8$$
An image in a quiz
the gradient
Correct answer: the estimated gradient
the rate of difference
the intercept
Q2.
Estimate the gradient of this curve between $$x=-4$$ and $$x=0$$
An image in a quiz
Correct Answer: -2
Q3.
Estimate the gradient of this curve between $$x=-2$$ and $$x=2$$
An image in a quiz
Correct Answer: -4
Q4.
Match these intervals to the estimated gradient of the curve.
An image in a quiz
Correct Answer:$$x=-2 \ \text{to} \ x=-1$$,$$4$$
tick

$$4$$

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

$$1$$

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

$$5\over2$$

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

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

$$2$$

Q5.
Order the estimated gradients of these intervals from highest to lowest.
An image in a quiz
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 $$0$$?
An image in a quiz
$$x=1$$ 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 $$x=4$$

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?
An image in a quiz
Correct Answer: An image in a quiz
An image in a quiz
An image in a quiz
Q4.
Use this tangent to calculate the gradient of this curve at $$x=0$$
An image in a quiz
Correct Answer: 3
Q5.
Use this tangent to calculate the gradient of this curve at $$x=1$$
An image in a quiz
Correct Answer: 5
Q6.
This tangent was drawn by hand. The triangle enables us to estimate the gradient at $$x=5$$ to be __________.
An image in a quiz
$$5.3$$
$$9$$
Correct answer: $$9\over5$$
$$5\over9$$
$$-{5\over9}$$