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9.07 Defining e and the natural logarithm

Worksheet
Euler's number
1

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

a

Complete the following table of values, correct to two decimal places:

x-3-2-10123
f(x)
b

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

2

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

3

Consider the graph of y = e^{x}.

a

Is the function increasing or decreasing?

b

Is the gradient to the curve negative at any point on the curve?

c

What does that tell us about the gradient function?

d

Describe how the gradient is changing along the curve as x increases.

e

What does this tell us about how the gradient function, y', changes as x increases?

f

Hence, what type of function could y' be?

-5
-4
-3
-2
-1
1
2
3
4
5
x
-1
1
2
3
4
5
6
7
8
9
10
y
4

f \left( x \right) = e^{x} and its tangent line at x = 0 are graphed on the coordinate axes:

a

Determine the gradient to the curve at \\ x = 0.

b

Evaluate f \left( 0 \right).

c

What is the relationship between f \left( 0 \right) and f' \left( 0 \right)?

-4
-3
-2
-1
1
2
3
4
x
-3
-2
-1
1
2
3
4
5
y
5

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

a

If f \left( 4 \right) = 54.59815, determine f' \left( 4 \right) correct to five decimal places.

b

If f \left( - 5 \right) = 0.00674, determine f' \left( - 5 \right) correct to five decimal places.

6

Consider the function y = e^{x}-1.

a

Point P lies on the curve y = e^{x}-1. If the x-coordinate of P is 4, find the y-coordinate of P.

b

Determine an expression for the derivative function \dfrac{d y}{d x}.

c

Find the gradient of the tangent at point P.

d

What is the largest interval over which the function is increasing?

7

Consider the curve with equation y = e^{x}.

a

Determine the value of the gradient m of the tangent to y = e^{x} at the point Q \left( - 1 , \dfrac{1}{e}\right).

b

Hence find the equation of the tangent to the curve at point Q.

c

Does this tangent line pass through the point R \left( - 2 , 0\right)?

8

Find the gradient, m of the tangent to each of the following curves, correct to two decimal places:

a

The curve y = 7 e^{x} at the point where x = 1.3.

b

The curve y = - e^{x} at the point \left(1, - e \right).

c
The curve y = e^{x} at the point where x = 7.
d
The curve y = e^{x} at the point where x = \dfrac{1}{2}.
e
The curve y = e^{x} at the point where \left( - 1 , \dfrac{1}{e}\right).
9

State the value of x where the gradient of the tangent to the curve y = e^{x} is \dfrac{1}{e^{3}}.

10

For each of the following points:

i

Find the gradient m of the tangent to y = e^{x} at this point.

ii

Find a, the angle of inclination of the tangent to y = e^{x} at this point. Express your answers in degrees correct to two decimal places if necessary.

a

The point where x = 0

b

The point where x = 5

11

Use a calculator or other technology to approximate the each of 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}}

The natural logarithm
12

Find the value of each of the following correct to four decimal places:

a

\ln 94

b

\ln 0.042

c

\ln 78^{4}

d

\ln \left( 18 \times 35\right)

13

Consider x=\ln 31. Find the value of x, correct to two decimal places.

14

Use the properties of logarithms to express each of the following without any powers or surds:

a
\ln y^{2}
b
\ln \dfrac{1}{y^2}
c
\ln \sqrt{y}
d
\ln \sqrt[3]{y}
15

Use the properties of logarithms to rewrite - 2 \ln x with a positive power.

16

Rewrite each of the following as the sum and difference of logarithms:

a
\ln \left(\dfrac{15}{8}\right)
b
\ln \left(\dfrac{p q}{r}\right)
c
\ln \left(\dfrac{19}{7}\right)
17

Rewrite each of the following expressions as a single logarithm:

a

\ln 3 + \ln 5

b

\ln 24 - \ln 4

c

\ln 8 - \ln 32

d

\ln \left( 3 x\right) + \ln \left( 5 y\right)

18

Which expression is equivalent to 2 \ln \left( 7 x\right) for x > 0?

A

\ln \left( 14 x\right)

B

\ln 14 + \ln x

C

\ln 49 + \ln x

D

\ln \left( 49 x^{2}\right)

19

Simplify:

a
e^{\ln w}
b
\ln\left(e^x\right)
c
2^{\ln e^y}
d
e^{\ln x}\times e^{\ln y}
e
\ln \left( \dfrac{e^{x}}{e} \right)
f
\ln e^x - \ln e^y
g
\ln \left( \dfrac{1}{e^x} \right)
h
\ln \left( \dfrac{e^5}{e^x} \right)
20

Evaluate each of the following expressions:

a

\ln e^{3.5}

b

\ln e^{4}

c

\sqrt{6} \ln \left(e^{\sqrt{6}}\right)

d

\ln \left(\dfrac{1}{e^{2}}\right)

Applications
21

The population P of a city increases according to the formula P = 3000 e^{ k t} where t is measured in years and k is a constant.

a

Find the initial population.

b

Given the population increases to 8000 in 2 years find the exact value of k.

c

How many complete years will it take for the population to at least double?

22

The voltage V in volts across an electrical component decays according to the equation V = A e^{ - \frac{1}{2} t } where A is the initial voltage and t is the time in years. Find the value of t such that V is half of the initial voltage, correct to two decimal places.

23

The spread of a virus through a city is modelled by the function: N = \dfrac{15\,000}{1 + 100 e^{ - 0.5 t }}, where N is the number of people infected by the virus after t days.

a

How many people will have been infected after 3 days? Round your answer to the nearest whole number.

b

How many whole days will it take for at least 4000 people to be infected with the virus?

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MA11-6

manipulates and solves expressions using the logarithmic and index laws, and uses logarithms and exponential functions to solve practical problems

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