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.

Improving the estimate of the gradient of a curve

Maths

The work on gradient in the prior unit is expanded further in this unit by considering gradient as the rate of change and interpreting this in context. The gradient of the tangent is introduced as a way of estimating the gradient of a curve.

Real-life graphs

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.

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.

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 $$x=0$$

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

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

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

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

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

Match these intervals to the estimated gradient of the curve.

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

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

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