Improving the estimate of the gradient of a curve | Maths

Starter quiz

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

the gradient

the estimated gradient ✓

the rate of difference

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

'-2' ✓
Estimate the gradient of this curve between $x=-2$ and $x=2$

'-4' ✓
Match these intervals to the estimated gradient of the curve.

$x=-2 \ \text{to} \ x=-1$

⇔

$4$ ✓

$x=-1 \ \text{to} \ x=1$

⇔

$1$ ✓

$x=0 \ \text{to} \ x=2$

⇔

$5\over2$ ✓

$x=2 \ \text{to} \ x=3$

⇔

$10$ ✓

$x=-1 \ \text{to} \ x=2$

⇔

$2$ ✓
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 $x=1$
Which of these intervals give this curve an estimated gradient of $0$ ?

$x=1$ and $x=3$

$x=0$ and $x=2$ ✓

$x=2$ and $x=3$

$x=0$ and $x=4$ ✓

$x=2$ and $x=4$ ✓

Exit quiz

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.

gradient ✓

length

radius

shape
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.

reducing ✓

increasing

calculating
If you wanted to calculate the gradient of the curve at a given point, which diagram is likely to be the most helpful?
Use this tangent to calculate the gradient of this curve at $x=0$

'3' ✓
Use this tangent to calculate the gradient of this curve at $x=1$

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

$5.3$

$9$

$9\over5$ ✓

$5\over9$

$-{5\over9}$

Worksheet

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Lesson Details

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.

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.

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.