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MATH 31 LESSONS Chapter 8: Exponential and Log Functions Exponential Functions

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Laws of Exponents

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If the bases are the same... Keep the base the same and add the exponents

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Similarly,

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Multiply the exponents

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Similarly,

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Note: This works only for multiplication and division. It does NOT work for addition or subtraction.

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Anything raised to the zeroeth power is equal to 1.

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Similarly,

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The root is on the bottom.

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Sketching Exponential Functions Exponential functions are of the form where b is a positive constant

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Case 1: b > 1 “Exponential Growth”

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=

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= 0 HA: y = 0

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Case 2: 0 < b < 1 “Exponential Decay”

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= 0 HA: y = 0

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=

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Ex. 1 (include asymptote) Try this question yourself first. Then, check the answer on the following pages.

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Basic function: (decay)

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Reflect about the x-axis

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Translate down 4 units HA: y = 4 44

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Ex. 2Evaluate: Try this question yourself first. Then, check the answer on the following pages.

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Test a number that is on the left side of 2 (i.e. x = 1.9999) = 0

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Derivatives of Exponential Functions Using first principles,

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Using the exponential law:

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Mathematicians could not determine this directly. So, they chose an indirect path. But what is?

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Mathematicians realized that if this would lead to a simple derivative:

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Finding the Ideal Base Mathematicians began to search for a base b that would satisfy

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For inspiration, consider bases 2 and 3 : h 0.1 0.01 0.001 0 0.7177 0.6956 0.6934 0.693 1.1612 1.1047 1.0992 1.099

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h 0.1 0.01 0.001 0 0.7177 0.6956 0.6934 0.693 1.1612 1.1047 1.0992 1.099 Clearly, the ideal base must be between 2 and 3 (and more specifically, closer to 3).

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In fact, the ideal base is known as Euler’s constant, e e 2.71828 Mathematicians showed that

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Derivative of y = e x Recall, the derivative of y = b x was

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However, if we introduce b = e :

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Since, it follows that

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Thus, the derivative of e x is itself !

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For a composite function, Write the exponential function again Don’t forget the derivative of the inside function

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e.g.

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Write the exponential function again

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e.g. Write the exponential function again Don’t forget the derivative of the inside function

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Understanding the Graph for y = e x Does the function y = e x represent exponential growth or decay?

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Since e 2.71828, this would be exponential growth.

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Recall, Since y = y, it follows that the y-coordinate also represents the tangent slope

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The y-coordinate is the tangent slope (0, 1) m t = 1

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The y-coordinate is the tangent slope (2, e 2 ) m t = e 2

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Ex. 3Differentiate Try this question yourself first. Then, check the answer on the following pages.

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Ex. 4 Find y. Try this question yourself first. Then, check the answer on the following pages.

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Product rule e is just a constant. So, you ignore the constant coefficient and take the derivative of x.

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Ex. 5Where (x-values) does the function have a horizontal tangent? Try this question yourself first. Then, check the answer on the following pages.

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Quotient Rule

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A function has a horizontal tangent when its tangent slope (i.e. derivative) is zero. when Note: e 2x > 0 for all x , so it can never equal zero.

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