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Tangents and Normals

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AQA Level 2 Further Maths practice questions, each with a hint and a full worked solution.

Have a pen and paper handy, to work out your answers.

Question 13 marks

A curve has equation y = 2x^3 - 5x + 1

Work out the equation of the tangent to the curve at the point where x = 2

Give your answer in the form y = mx + c

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

You need a point and a gradient: find y when x=2, then substitute x=2 into \frac{dy}{dx}.

Worked solution
  1. When x=2: y=2(8)-5(2)+1=7, so the point is (2,\ 7)
  2. \dfrac{dy}{dx}=6x^2-5
  3. When x=2: gradient =6(4)-5=19
  4. y-7=19(x-2)
  5. y=19x-31

Question 23 marks

The point (4,\ 1) lies on the curve y = 5 + 3x - x^2

Work out the equation of the normal to the curve at (4,\ 1)

Give your answer in the form ax + by = c, where a, b and c are integers.

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

The normal is perpendicular to the tangent: find the tangent's gradient, then flip it and change its sign.

Worked solution
  1. \dfrac{dy}{dx}=3-2x
  2. At x=4: gradient of the tangent =3-8=-5
  3. Gradient of the normal is the negative reciprocal: \dfrac15
  4. y-1=\dfrac15(x-4)
  5. Multiply by 5: 5y-5=x-4
  6. x-5y=-1

Question 3Challenge6 marks

A curve has equation y = x^2 + 2x - 4

A is the point on the curve where x = 1

B is the point on the curve where the gradient of the curve is 8

(a)

Work out the equation of the tangent to the curve at A

Give your answer in the form y = mx + c

3 marks

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

(b)

The tangent to the curve at A and the tangent to the curve at B meet at the point R

Work out the coordinates of R

3 marks

Write your answer as (x, y)

Hint

For (b), use \frac{dy}{dx} to find where the gradient is 8, then find the tangent there and solve it simultaneously with your tangent from (a).

Worked solution

Part (a)

  1. When x=1: y=1+2-4=-1, so A is (1,\ -1)
  2. \dfrac{dy}{dx}=2x+2
  3. When x=1: gradient =2+2=4
  4. y-(-1)=4(x-1)
  5. y+1=4x-4
  6. y=4x-5

Part (b)

  1. At B the gradient is 8: \;2x+2=8, so x=3
  2. When x=3: y=9+6-4=11, so B is (3,\ 11)
  3. Tangent at B: \;y-11=8(x-3), so y=8x-13
  4. At R the two tangents meet: \;4x-5=8x-13
  5. 8=4x, so x=2
  6. y=4(2)-5=3
  7. (Check in the other tangent: 8(2)-13=3 ✓)
  8. R(2,\ 3)

More on this topic: Tangents and Normals worksheet with full solutions

All AQA Level 2 Further Maths practice questions

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