AQA Level 2 Further Maths · Level 2 Certificate · Chapter 8: Calculus · Lesson 2

Tangents and normals

About 80 minutesNo calculator85 marks

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Date

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The first questions are what today's lesson needs. The others bring back earlier topics.

A1
Needed today
Find the equation of the line with gradient $- 3$ that passes through $(2, 5)$ . Give it in the form $y = m x + c$ .
[2 marks]
Answer
A2
From chapter 5
Work out the equation of the line through $(6, - 1)$ that is perpendicular to $y = 2 x + 7$ . Give it in the form $a x + b y + c = 0$ .
[2 marks]
Answer
A3
Last lesson
Work out $\frac{d y}{d x}$ when $y = 4 x^{3} - \frac{6}{x}$ .
[2 marks]
Answer
A4
From chapter 4
$(x + 2)$ is a factor of $x^{3} + k x^{2} + x + 6$ . Work out the value of $k$ .
[2 marks]
Answer

Example, then your turn

Level 2

Your teacher works through each example with you. Then try the one beside it.

Example 1

Work out the equation of the tangent to the curve $y = 2 x^{2} - 7 x + 3$ at the point $P$ where $x = 3$ .

Not drawn accurately

Hint Work out

\frac{d y}{d x}

and put

x = 3

in to get the gradient. You also need the

y

-coordinate of

P

B1
Work out the equation of the tangent to the curve $y = x^{3} - 4 x + 1$ at the point $P$ where $x = 2$ . Give your answer in the form $y = m x + c$ .
[4 marks]
Answer $y = \dots\dots\dots\dots$

Example 2

Work out the equation of the normal to the curve $y = x^{3} - 2 x^{2} + 3$ at the point $(2, 3)$ . Give your answer in the form $a x + b y + c = 0$ , where $a$ , $b$ and $c$ are integers.

Not drawn accurately

Hint The normal is perpendicular to the tangent. Its gradient is

- 1

divided by the gradient of the tangent.

B2
Work out the equation of the normal to the curve $y = 2 x^{2} + x - 6$ at the point $(- 2, 0)$ . Give your answer in the form $a x + b y + c = 0$ , where $a$ , $b$ and $c$ are integers.
[5 marks]
Answer $\dots\dots\dots\dots = 0$

Example 3

Work out the equation of the tangent to the curve $y = \frac{x ^{3} - 4}{x}$ at the point $P$ where $x = 2$ .

Hint Split the fraction first:

y = x^{2} - 4 x^{- 1}

. Then differentiate each term.

B3
Work out the equation of the tangent to the curve $y = \frac{x ^{3} + 6}{x ^{2}}$ at the point $P$ where $x = - 1$ . Give your answer in the form $y = m x + c$ .
[5 marks]
Answer $y = \dots\dots\dots\dots$

Example 4

Work out the coordinates of the points on the curve $y = x^{3} - 6 x^{2} + 10 x - 1$ where the tangent is parallel to the line $y = x + 4$ . Hence work out the equation of each of these tangents.

Hint Parallel lines have equal gradients, so solve

\frac{d y}{d x} = 1

B4
Work out the coordinates of the two points on the curve $y = x^{3} + 3 x^{2} - 7 x + 2$ where the tangent is parallel to the line $y = 2 x - 5$ .
[4 marks]
Answer $(\dots\dots, \dots\dots)$ and $(\dots\dots, \dots\dots)$

Example 5

The point $P$ on the curve $y = a x^{3} - 5 x$ has $x$ -coordinate $1$ . The normal to the curve at $P$ has gradient $- \frac{1}{7}$ . Work out the value of $a$ .

Hint If the normal has gradient

- \frac{1}{7}

, the tangent has gradient

7

, because the two gradients multiply to

- 1

B5
The curve $y = x^{2} + b x + c$ passes through the point $(1, 4)$ . The normal to the curve at $(1, 4)$ has gradient $\frac{1}{3}$ . Work out the values of $b$ and $c$ .
[4 marks]
Answer $b = \dots\dots c = \dots\dots$

Practice

Level 2

C1
Show that the line $y = 7 x - 9$ is the tangent to the curve $y = x^{3} - 5 x + 7$ at the point $(2, 5)$ .
[3 marks]
C2
The curve $y = (x + 1) (x - 2) (x - 3)$ crosses the $y$ -axis at $A$ . Work out the equation of the tangent to the curve at $A$ .
[4 marks]
Answer $y = \dots\dots\dots\dots$

C3
Work out the equations of the two tangents to the curve $y = x^{3} - 12 x + 5$ that are parallel to the $x$ -axis.
[3 marks]
Answer $y = \dots\dots$ and $y = \dots\dots$
C4
The point $P (2, 1)$ lies on the curve $y = x^{2} - 5 x + 7$ . $Q$ is the point on the curve where the normal is parallel to the tangent at $P$ . Work out the coordinates of $Q$ .
[4 marks]
Answer $(\dots\dots, \dots\dots)$
C5
The curve $y = x^{2} - 8 x + 12$ crosses the $x$ -axis at $A$ and $B$ . The tangents to the curve at $A$ and at $B$ meet at $T$ . Work out the coordinates of $T$ .
[5 marks]
Answer $(\dots\dots, \dots\dots)$

Exam-style questions

AQA exam style

D1
Ravi is finding the normal to the curve $y = x^{2} + 5 x - 2$ at the point $(1, 4)$ . He writes: $\frac{d y}{d x} = 2 x + 5 = 7$ , so the normal is $y - 4 = 7 (x - 1)$ .
Explain Ravi's mistake. Work out the correct equation of the normal. Give your answer in the form $a x + b y + c = 0$ , where $a$ , $b$ and $c$ are integers.
[3 marks]
Answer $\dots\dots\dots\dots = 0$

D2
A curve has equation $y = k x^{3} - 6 x$ , where $k$ is a constant. $P$ is the point on the curve where $x = 2$ . The normal to the curve at $P$ is parallel to the line $x + 18 y = 7$ .
(a)
Work out the value of $k$ .
[4 marks]
Answer $k = \dots\dots$
(b)
Hence work out the equation of the tangent to the curve at $P$ . Give your answer in the form $y = m x + c$ .
[2 marks]
Answer $y = \dots\dots\dots\dots$
Total for D2: 6 marks
D3
A curve has equation $y = x^{2} - 4 x - 5$ . $P$ is the point on the curve where $x = 5$ .
Not drawn accurately
(a)
Show that the equation of the normal to the curve at $P$ is $x + 6 y = 5$ .
[4 marks]
(b)
The normal at $P$ meets the curve again at $Q$ . Work out the $x$ -coordinate of $Q$ . You must show your working.
[3 marks]
Answer $x = \dots\dots$
Total for D3: 7 marks
D4
The curve $y = 10 - x^{2}$ passes through the point $P (2, 6)$ .
Not drawn accurately
(a)
Work out the equation of the tangent to the curve at $P$ .
[3 marks]
Answer $y = \dots\dots\dots\dots$
(b)
Work out the equation of the normal to the curve at $P$ .
[2 marks]
Answer $\dots\dots\dots\dots$
(c)
The tangent at $P$ meets the $y$ -axis at $A$ . The normal at $P$ meets the $y$ -axis at $B$ . Work out the area of triangle $A B P$ .
[3 marks]
Answer $\dots\dots$ square units
Total for D4: 8 marks
D5
The sketch shows the curve $y = x^{3} - 3 x^{2} + 4$ . The curve crosses the negative $x$ -axis at $P (- 1, 0)$ .
Not drawn accurately
(a)
Work out the equation of the tangent to the curve at $P$ .
[3 marks]
Answer $y = \dots\dots\dots\dots$
(b)
The tangent to the curve at the point $Q$ is parallel to the tangent at $P$ . Work out the equation of the tangent at $Q$ .
[4 marks]
Answer $y = \dots\dots\dots\dots$
Total for D5: 7 marks

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
The tangent to the curve $y = x^{3} - 2 x + 1$ at the point $P$ where $x = 2$ meets the curve again at $Q$ . Work out the coordinates of $Q$ .
[5 marks]
Hint 1 · What to try
Find the equation of the tangent at $P$ first. Then solve the equations of the curve and the tangent together.
Hint 2 · The first line
$P$ is $(2, 5)$ and $\frac{d y}{d x} = 3 x^{2} - 2 = 10$ there, so the tangent is $y = 10 x - 15$ .
Hint 3 · The full method
$x^{3} - 2 x + 1 = 10 x - 15$ gives $x^{3} - 12 x + 16 = 0$ . The tangent touches the curve at $x = 2$ , so $(x - 2)^{2}$ is a factor: $x^{3} - 12 x + 16 = (x - 2)^{2} (x + 4)$ . So $x = - 4$ and $y = - 64 + 8 + 1 = - 55$ .
Answer $(\dots\dots, \dots\dots)$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can work out the equation of the tangent to a curve at a point.
I can work out the equation of the normal, in the form $a x + b y + c = 0$ .
I can find the points where a tangent or normal has a given gradient.
I can use tangents and normals to find unknown coefficients, crossing points and areas.

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