Gradient of a Curve

Gradient of a Curve

Gradient from curve

Objective

• Estimate the gradient of a curve by drawing a tangent to the curve

Gradient

• The steepness of a line is called its gradient.

• A straight line has a constant gradient as the steepness doesn’t alter along the length of the line.

Sign of the gradient

• A line that slopes up to the right has a positive gradient.

  The y-change is positive

• A line that slopes down to the right has a negative gradient.

  The y-change is negative.

Example 1

• Calculate the gradient of the line segment EF. 

• Gradient = y-changes/x-changes

• Gradients = -4/3

Example 2

• What is the gradient of this straight line?

• Gradient = y-changes/x-changes

• Gradients = 5/10=1/2

Multiple Choice

What is the gradient of this straight line?

• 2/3

• 1

• 3/2

Multiple Choice

What is the gradient of this straight line?

• -9

• -1/9

• 1/9

Multiple Choice

What is the gradient of this straight line?

• -8/3

• -3/8

• 8/3

Curved Lines

• Some parts of this route have positive gradients but some have negative gradients.

• ​The gradient changes all the time.

• ​So if a line curves, it is because the gradient is changing.

Curved Lines

• This means that you cannot find the gradient of a whole curve but you are able to work out the gradient at any point along the curve.

• By zooming in on a point, a line can be added that matches the slope of the curve

• This line is called the tangent to the curve at that point.

Example 3

• Find the gradient at point C

• Gradient = y-changes/x-changes

• Gradient = 6/1=6

Example 4

• Find the gradient at point B

• Gradient = y-changes/x-changes

• Gradient = 4/-2=-2

Multiple Choice

Which image shows the correct tangent at the point on the curve marked by a dot?

• a

• b

• c

Multiple Choice

• a

• b

• c

Multiple Choice

What is the gradient at point A

• 1.6

• -1.6

• 0.625

Multiple Choice

What is the gradient of point B?

• -10

• 0.35

• 2.9

• -2.9

Multiple Choice

What is the gradient of point C?

Round to the 1 decimal place.

• -10

• 0.35

• 2.9

• -2.9