Finding Gradients of Tangents (Edexcel IGCSE Maths)

Spec point — 3.4C

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

Exam Hint When drawing a tangent by hand:

• Use a ruler
• Draw the line as long as you can

When finding the gradient of the tangent:

• Pick two points that are far away from one another
• This will reduce the effect of any inaccuracy

Worked Example The graph below shows $y = \sqrt[3]{x}$ for $0\leq x\leq1$.

Find an estimate of the gradient of the curve at the point where $x=0.5$.

Answer:

Draw a tangent to the curve at the point where x= 0.5

Find suitable, easy to read coordinates as far apart as possible and draw a right-angled triangle between them

Find the difference in the y coordinates (rise) and the difference in the x coordinates (run).

Find the gradient by dividing the difference in y(rise) by the difference in x

$\mathrm{Gradient}=\frac{0.3}{0.5}=\frac{3}{5}=0.6$

The gradient of the tangent at x = 0.5 is equal to the gradient of the curve at x= 0.5

Final answer: Estimate of gradient = 0.6