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Exponential Functions

6.1 Antiderivatives of Functions

Antiderivatives of Specific Functions

• We will be looking at antiderivatives of polynomial, exponential and trigonometric functions.
• The formulas are as below:
i) Polynomial functions

\int x^{n} d x=\frac{1}{n+1} x^{n+1}+c \quad,\ n \neq-1

\int x^{-1} d x=\int \frac{1}{x} d x=\ln |x|+c

Note: Notice that in there is an absolute value over to make the result fit the domain for all logarithmic functions is (0,\ +\infty), and x=0 is not possible in the first place as it would make \frac{1}{x} undefined.

ii) Exponential functions
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3.3 Differentiation Rules for Exponentials and Logarithms

Differentiation Rules for Exponentials and Logarithms

• The other rules regarding differentiation such as addition, subtraction, multiplication, division of functions, chain rule etc. applies too.
• All of these rules can be proven via first principles, but it is not required.
• The basic rules in this section are:

i) If f(x)=a^{x}, \text { then } f^{\prime}(x)=a^{x} \ln a

ii) If f(x)=\log _{a} x, \text { then } f^{\prime}(x)=\frac{1}{x}(\ln a)^{-1}

where n \neq 0, n \in R (or n \in R \backslash\{0\})

• Applying the rules to natural exponential and logarithm functions (i.e. a=e), yields:
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3.2 Index Laws

The Exponential Function’s Algebraic Properties

• There are six properties in exponentials to consider:

i) a^{x} \times a^{y} = a^{x+y}

ii) a^{x} {\div} a^{y} = a^{x-y}

iii) (a^{x})^{y}=a^{xy}

5.3 Special Exponential and Logarithmic Functions

Special Exponential Functions

• The exponential function y=e^x is a special exponential function, and is vital in the field of mathematics.
• In particular, e=\lim _{n \rightarrow \infty}\left(1+\frac{1}{n}\right)^{n} is a constant known as the Euler’s number. It is approximately 2.718281459045…, and it is an irrational number like \pi.
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3.6 Transformations of Exponential Functions

Plotting Exponential Functions

Example

Say y=2^{x}. Ideally, you should remember the general shape of the graph, then labelling any important points (i.e. Step 2). You should try yourself for y=0.5^{x}.

Generally, the steps to plot a graph are:

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1.3 Exponential Functions [Free]

The Exponential Function

• The exponential function is defined as follows:

y=a^x, where a \in R^+\backslash {1}.

Note:a \in R^+\backslash {1}’ means that could be any positive number, excluding 1.

Example

9=3^{2}=3 \times 3 is equivalent to log_{3}9=2.

8=16^{\frac{3}{4}}=\sqrt[4]{16^{3}}=\sqrt{\sqrt{16 \times 16 \times 16}}.

Graph & Properties

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Derivatives [Video Tutorial]

This tutorial covers material encountered in chapter 9 of the VCE Mathematical Methods Textbook, namely: The derivative of functions seen previously in tutorial worksheets 1-5… Read More »Derivatives [Video Tutorial]