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1.02 Exponential functions and Euler's number

Worksheet
Approximating Euler's number
1

Use a calculator or other technology to approximate the following values correct to four decimal places:

a

e^{4}

b

e^{ - 1 }

c

e^{\frac{1}{5}}

d

5 \sqrt{e}

e

\dfrac{4}{e}

f

\dfrac{8}{9 e^{4}}

2

The natural base e, Euler's number, is defined as: e = \lim_{n \to \infty} \left(1 + \dfrac{1}{n}\right)^{n}

The table shows the values of \left(1 + \dfrac{1}{n}\right)^{n} for certain values of n:

n\left(1 + \dfrac{1}{n}\right)^{n}
12
1001.01^{100} = 2.704\,813 ...
10001.001^{1000} = 2.716\,923 ...
10\,0001.0001^{10000} = 2.718\,145 ...
100\,0001.000\,01^{100000} = 2.718\,268 ...
a

Evaluate \left(1 + \dfrac{1}{n}\right)^{n} for n = 1\,000\,000, correct to six decimal places.

b

Find a decimal approximation of e correct to nine decimal places.

3

It is possible to approximate e^{x} using the following formula:

e^{x} = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \frac{x^{4}}{4!} + \text{. . .} + \frac{x^{n}}{n!} + \text{. . .}

The more terms we use from the formula, the closer our approximation becomes.

a

Use the first five terms of the formula to estimate the value of e^{0.7}.

Round your answer to six decimal places.

b

Use the {e^{x}} key on your calculator to find the value of e^{0.7}.

Round your answer to six decimal places.

c

What is the difference between these two results?

Exponential expressions
4

Simplify the following expressions:

a
3 e^{x} + 4 e^{x}
b
2e^x + 7e^x
c
5 e^{x} - 2 e^{x}
d
-3e^x + 4e^x
e
12 e^{ 2 x} - 5 e^{x} + 7 e^{x}
f
2e^{2x} - 8e^x - 7e^{2x}
g
e^{x} \times e^{2}
h
e^x \times e^{4x}
i
e^{3x} \times \left(e^x\right)^2
j
\dfrac{\left(e^{3x}\right)^{4}}{e^{5x}}
k
\dfrac{4 e^{ 2 x} - 6 e^{x}}{2 e^{x}}
l
\dfrac{e^{2x} + 7e^x}{6e^{-x}}
5

Expand and simplify the following expressions:

a
e^{x} \left(e^{x} + 3\right)
b
e^{2x}\left(e^x - 9\right)
c
\left(e^x + 2\right)\left(e^x + 5\right)
d
\left(e^{x} + 2\right) \left(e^{x} - 5\right)
e
2 e^{3x} \left(e^{ - 2x } + 1\right)
f
3e^x\left(e^{-x} + e^x\right)
g
\left(e^{x} + e^{ - x }\right)^{2}
h
\left(e^x - e^{-2x}\right)^2
6

Fully factorise the following expressions:

a
8 e^{x} + 12
b
8e^{2x} + 12e^x
c
8 e^{ 4 x} - 12 e^{ 3 x}
d
8e^{4x} - e^{7x}
e
e^{ 2 x} - 5 e^{x} + 6
f
e^{2x} + 6e^x + 9
g
e^{3x} + 6e^{2x} + 8e^{x}
h
e^{4x} - 7e^{2x} - 10
Exponential graphs
7

The functions y = 2^{x}, y = e^{x} and y = 3^{x} have been sketched on the same axes:

-2
-1
1
x
0.5
1
1.5
2
2.5
y
a

For what values of x does the inequality 3^x > e^x hold?

b

For what values of x does the inequality 2^x > e^x hold?

8

Consider the function f \left( x \right) = e^{x}.

a

Complete the following table of values. Round each value to two decimal places.

x-5-3-10135
f(x)
b

Sketch the graph of f \left( x \right) = e^{x} for -5\leq x\leq 5.

9

Consider the function f\left(x\right) = e^{ - x }.

a

Can the function ever have a negative value?

b
i

As the value of x gets larger and larger, what value does f\left(x\right) approach?

ii

As the value of x gets smaller and smaller, what value does f\left(x\right) approach?

c

Can the value of f\left(x\right) ever be equal to 0?

d

Find the value of f\left(x\right) when x=0.

e

How many x-intercepts does the function have?

f

Sketch the graph of f\left(x\right) = e^{ - x }.

10

Sketch both y = e^{x} and y = e^{ - x } on the same set of axes. What are the coordinates of their intersection point?

11

Consider the function f(x) = - e^{x}.

a

Complete the following table of values. Round each value to three decimal places.

x-9-6-30369
f(x)
b

Are there any values of x where f\left(x\right) is positive?

c

Are there any values of x where f\left(x\right) is equal to 0?

d

Is the function increasing or decreasing?

e

How would you describe the rate of increase or decrease of the function?

Transformations of exponential graphs
12

Using a graphing calculator, plot each set of three graphs below on the same screen and determine whether or not they share the same:

i

y-intercept

ii

Asymptote

iii

Range

a
y = e^{x}, y = e^{x} + 2, and y = e^{x} - 3
b
y = e^x, y = e^{x - 2}, and y = e^{x + 3}
c
y = e^x, y = e^{2x}, and y = e^{3x}
d
y = e^x, y = 4e^x, and y = -4e^x
13

For each of the following functions, state whether they are increasing or decreasing:

a

f_1(x) = e^{x}

b

f_2(x) = e^{-x}

c

f_3(x) = -e^{x}

d

f_4(x) = -e^{-x}

14

Describe a sequence of transformations to apply to the graph of f\left(x\right) = e^{x} to achieve the graphs of the following:

a
y = e^{ 4 x} + 1
b
y = 5 e^{x - 4}
c
y = 5e^{-x}
15

Find the equation corresponding to the graph of y = e^{x} after it has undergone the following transformations:

a

Translated three units upward and four units to the left.

b

Translated five units downward and two units to the left.

c

Dilated by a factor of two vertically and then translated five units downward.

d

Reflected across the x-axis, translated three units upward and then two units to the right.

e

Dilated by a factor of seven horizontally and then translated two units downward.

f

Dilated by a factor of one third horizontally and then translated two units downward.

16

For each of the following transformations on y=e^x find:

i

The equation of the resulting function.

ii

The equation of the resulting horizontal asymptote.

iii

The value of the resulting y-intercept.

iv

The graph of the resulting function.

a

y is first dilated by a factor of 3 vertically and then translated 2 units upwards.

b

y is first dilated by a factor of 3 vertically and then translated 1 unit to the right.

c

y is first translated 2 units upwards and then dilated by a factor of 3.

d

y is first translated 1 unit to the right and then dilated by a factor of 3 vertically.

e

y is first reflected across the y-axis and then translated 3 units upwards.

f

y is first reflected across the x-axis and then translated 2 units to the right.

17
a

State the equation of the asymptote of:

i
f\left(x\right) = - e^{x}
ii
g\left(x\right) = 4 - e^{x}
iii
h\left(x\right) = 2 + e^{2x}
b
Sketch all three graphs on the same set of axes.
18

Sketch f\left(x\right) = e^x, g\left(x\right) = 2e^x + 1, and h\left(x\right) = e^{2x}-1 on the same set of axes.

19

Sketch the curves y = e^{x}, y = e^{x} + 3, and y = e^{x} - 4 on the same number plane.

20

Sketch the graph of y = 5 - e^{2 - 3x}.

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