Practice Euler's Number in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Worksheet PDF Free Answer-key PDF Family

Quick Recap

Euler's number $e \approx 2.71828$ is the unique base for which the exponential function $e^x$ is its own derivative — the natural base for growth and decay.

The 'natural' base for growth—what you get from continuous compounding.

Showing a random 20 of 50 problems.

Example 1

medium

Why does

e^x

have the special property among exponentials that its slope at any point equals its value?

Example 2

medium

Simplify

(e^2)^3

to a single power of

e

Example 3

challenge

Solve

100e^{0.04t}=200

for

t

. Use

\ln 2\approx 0.693

Example 4

hard

Show that the derivative of

f(x) = e^x

is itself, i.e.,

f'(x) = e^x

, using the limit definition of the derivative.

Example 5

medium

True or false:

e

is rational.

Example 6

hard

Find the equation of the tangent line to

y=e^x

x=1

. Use

e \approx 2.718

Example 7

easy

Using a calculator, compute

e^2

to four decimal places. Then determine whether

e^2 > 7

Example 8

hard

Solve

e^{x} = 2 e^{-x}

for

x

. Use

\ln 2 \approx 0.693

Example 9

medium

Radioactive decay:

A(t)=80e^{-0.1t}

. Find

A(0)

and state the long-run limit.

Example 10

challenge

\ln(A)=\ln(B)+3

, express

A

in terms of

B

Example 11

medium

Simplify

\dfrac{e^7}{e^4}

Example 12

hard

Solve

3 e^{2x} = 24

for

x

. Use

\ln 2 \approx 0.693

Example 13

medium

A population grows as

P(t)=100e^{0.5t}

. Find

P(2)

using

e\approx 2.718

Example 14

challenge

Using the series

e^x = \sum_{k=0}^{\infty} \tfrac{x^k}{k!}

, approximate

e

using the first 5 terms (

k=0

4

Example 15

medium

Find the slope of

f(x)=e^x

x=0

Example 16

medium

Simplify

e^{\ln 5}

Example 17

medium

Simplify

\dfrac{e^{2x}\cdot e^{3}}{e^{x}}

as a single power of

e

Example 18

medium

Continuous growth at rate

r=0.03

for

t=10

multiplies an amount by what factor? Use

e^{0.3}\approx 1.3499

Example 19

easy

e

a variable or a constant?

Example 20

medium

A car loses value continuously:

V(t)=20000 e^{-0.15 t}

dollars. Find

V(5)

. Use

e^{-0.75}\approx 0.4724

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Related Concepts

Exponential Function Natural Logarithm