A-Level Maths (Edexcel 9MA0) · Pure Year 1 · Chapter 14: Exponentials and logarithms · Lesson 1

Exponential functions and y = eˣ

71 marks

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Date

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Quick questions to start.

A1
Needed today
Write $(\frac{1}{8})^{x} \times 4^{x + 1}$ as a single power of $2$ .
[2 marks]
Answer
A2
Needed today
The curve $y = f (x)$ passes through $(2, 5)$ . Write down the point it moves to on $y = 3 f (x)$ , and on $y = f (x + 4)$ .
[2 marks]
Answer
A3
Last lesson
Find the area of the finite region enclosed by $y = x^{2}$ and $y = 3 x$ .
[2 marks]
Answer
A4
From chapter 12
Find the equation of the tangent to $y = x^{2} - 3 x$ at the point where $x = 4$ .
[2 marks]
Answer

A-Level

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

Example 1

Sketch the curve $y = 2^{x + 2} - 4$ . Show where it meets the axes and the equation of its asymptote.

Hint Write

2^{x + 2} = 4 \times 2^{x}

. Then it is

y = 2^{x}

stretched by

4

upwards and moved down

4

B1
Sketch the curve $y = 4 - 3 e^{- 2 x}$ . Show where it meets the $y$ -axis and the equation of its asymptote, and explain why $y < 4$ for every $x$ .
[4 marks]

Example 2

The curve $y = k a^{x}$ , where $k$ and $a$ are positive constants, passes through $(- 1, 2)$ and $(2, 250)$ . Find $k$ and $a$ .

Hint Write two equations and divide one by the other:

k

cancels.

B2
The curve $y = p q^{x}$ , where $p$ and $q$ are positive constants, passes through $(- 1, 40)$ and $(2, 0.625)$ . Find $p$ and $q$ .
[4 marks]
Answer

Example 3

$y = 3 e^{2 x} + e^{- x}$ . Find $\frac{d y}{d x}$ , and the equation of the tangent to the curve at the point where $x = 0$ .

Hint The derivative of

e^{k x}

k e^{k x}

. The point is

(0, 3 + 1)

B3
$y = e^{x} (e^{2 x} - 4)$ . Find $\frac{d y}{d x}$ , and the equation of the tangent to the curve at the point where $x = 0$ .
[4 marks]
Answer

A-Level

C1
On one set of axes, sketch $y = 4^{x}$ , $y = 2^{x}$ and $y = (\frac{1}{4})^{x}$ . Show the point they all pass through. Describe the single transformation that maps $y = 4^{x}$ onto $y = (\frac{1}{4})^{x}$ , and explain why none of the three curves meets the $x$ -axis.
[4 marks]

C2
The sketch shows part of the curve $y = A e^{b x} + C$ , where $A$ , $b$ and $C$ are constants. The curve meets the $y$ -axis at $(0, 9)$ , and the line $y = 3$ is an asymptote. Find $A$ and $C$ , and state whether $b$ is positive or negative.
[3 marks]
Answer
C3
Show that the gradient of the chord of $y = 3^{x}$ from $x$ to $x + h$ is $3^{x} \times \frac{3 ^{h} - 1}{h}$ . Hence explain why the gradient of $y = 3^{x}$ at any $x$ is $3^{x}$ times its gradient at $x = 0$ . Using $h = 0.0001$ , estimate the gradient of $y = 3^{x}$ at $x = 2$ , to $3$ significant figures.
[4 marks]
Answer

Edexcel exam style

D2
$f (x) = e^{2 x} - 4 e^{x} + 2 x$ .
(a)
Find $f^{'} (x)$ , and show that $f^{'} (x) = 2 (e^{x} - 1)^{2}$ .
[3 marks]
(b)
Hence show that the curve $y = f (x)$ has exactly one stationary point, and find its coordinates. Find $f^{''} (x)$ , and explain why $f^{''} (0)$ does not show the nature of the stationary point. Determine its nature.
[6 marks]
Answer
Total for D2: 9 marks
D3
Jo writes: if $y = e^{3 x + 1} - 4 x$ , then $\frac{d y}{d x} = (3 x + 1) e^{3 x} - 4$
Jo is wrong. Explain her mistake. Write $e^{3 x + 1}$ in the form $A e^{3 x}$ , where $A$ is a constant, and hence find $\frac{d y}{d x}$ correctly. Show that the curve $y = e^{3 x + 1} - 4 x$ is increasing for all $x \geq 0$ , and find the exact gradient of the curve at the point where it crosses the $y$ -axis.
[7 marks]
Answer
D4
The curve $C$ has equation $y = 4 e^{- \frac{1}{2} x}$ . The point $P$ is where $C$ meets the $y$ -axis. The sketch shows $C$ with its tangent and its normal at $P$ .
(a)
Find $\frac{d y}{d x}$ , and show that the tangent to $C$ at $P$ has equation $y = 4 - 2 x$ .
[3 marks]
(b)
Find an equation of the normal to $C$ at $P$ . The tangent meets the $x$ -axis at $A$ and the normal meets the $x$ -axis at $B$ . Hence find the area of triangle $A B P$ .
[5 marks]
Answer
Total for D4: 8 marks

Stretch

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

Colour one circle on each line.

	Not yet	Nearly	Yes
I can sketch exponential curves, with their intercepts and asymptotes.
I can find k and a when a curve y = kaˣ passes through two points.
I can differentiate e^(kx) and find tangents, normals and stationary points.

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