Finding Gradients of Tangents

Finding Gradients of Tangents

Finding gradients of tangents

How are the gradients of graphs and tangents related?

• The gradient of a graph at a point is equal to the gradient of the tangent to the curve at that point

  • A tangent is a line that touches a curve, but does not cross it

How do I estimate the gradient of a curve using a tangent?

• To find an estimate for the gradient of a curve at a point:

  • Draw a tangent to the curve at the point

  • Find the gradient of the tangent using

    • Gradient = rise ÷ run

    • or difference in y ÷ difference in x

    • In the example below, the gradient of the tangent at x = 4 would be $\frac{-2.5}{4}=-0.625$

    • Remember that the rise is negative if it is going down

    • This means the gradient of the curve at x = 4 is also -0.625

  

• It is an estimate because the tangent has been drawn by eye and is not exact

  • To find the exact gradient we would need to use differentiation

What does the gradient represent?

• The gradient represents the rate of change of y with x

  • I.e. For every increase in x by 1, how much does y increase?

• Consider the quantities used for the axes to determine the meaning of the gradient

  • In general, the units of the gradient will be the units of y divided by the units of x

  • In a distance-time graph, the gradient is the rate of change of distance with time

    • This is the speed,

    • Common units for speed are metres per second or kilometres per hour

  • In a speed-time graph, the gradient is the rate of change of speed with time

    • This is the acceleration

    • Common units for acceleration are metres per second per second (metres per second²)

  • In a graph of volume against radius, e.g. as a balloon is inflated, the gradient is the rate of change of volume as the radius increases

    • This could be measured in m³ per m, which is equivalent to m²