MATH 1004A - lecture#8 - Exponential%2c logarithmic functions and their derivatives

MATH 1004A - lecture#8 - Exponential%2c logarithmic functions and their derivatives

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Exponential, logarithmic functions and their derivatives A function of the form y=f ( x ) =a x Various cases for ‘ x Laws of exponents Logarithmic functions Laws of logarithms Change of base formulas 1 Exponential, logarithmic functions and their derivatives The derivative of y=f ( x ) =a x A word about the number ‘ e The derivative of y=f ( x ) =e x Derivatives of “ log ” functions: y=f ( x ) =log a x and y=f ( x ) =ln x y=f ( x ) =ln g ( x ) Logarithmic differentiation The power rule, revisited The number “ e ”, revisited
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• In general, an exponential function is a function of the form f ( x ) = a x where a is a positive constant. s Let’s recall what this means: EXPONENTIAL FUNCTIONS n factors a a a a a = ⋅ ⋅ ⋅⋅⋅⋅⋅ 1442443 2 Exponential, logarithmic functions and their derivatives - If x = n , a positive integer, then: - If x = 0 , then a 0 =1 , thus: f ( x ) is a constant function - If x = – n , where n is an integer and n > 0 , then: - If x is a rational number, x = p / q , where p and q are integers and q > 0 , then: n factors 1 n n a a - = / ( ) q q x p q p p a a a a = = = Do not confuse with f ( x ) = x a , in which the variable is the base.
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• However, what is the meaning of a x if x is an irrational number? s For instance, what is meant by or ? • To help us answer that question, we first EXPONENTIAL FUNCTIONS 3 2 5 π 3 Exponential, logarithmic functions and their derivatives look at the graph of the function y = 2 x , where x is rational. s A representation of this graph is shown here. s There are holes in this graph and they correspond to irrational values of x .
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• In particular, since the irrational number satisfies , we must have 3 1.7 3 1.8 < < 2 1.7 < 2 3 < 2 1.8 EXPONENTIAL FUNCTIONS 4 Exponential, logarithmic functions and their derivatives s We know what 2 1.7 and 2 1.8 mean because 1.7 and 1.8 are rational numbers.
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• Similarly, if we use better approximations for , we obtain better approximations for : 3 3 2 1.73 3 1.74 732 3 1.733 1.73 3 1.74 2 2 2 < < < < EXPONENTIAL FUNCTIONS 5 Exponential, logarithmic functions and their derivatives 1.732 1.7320 3 1.7321 1.73205 3 1.73206 1.732 3 1.733 2 2 2 1.7320 3 1.7321 2 2 2 1.73205 3 1.73206 2 2 2 < < < < < < < < < < < <
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• The graphs of members of the family of functions y = a x are shown here for various values of the base a . EXPONENTIAL FUNCTIONS 6 Exponential, logarithmic functions and their derivatives
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• You can see that there are basically three kinds of exponential functions y = a x : s If 0 < a < 1 , the exponential function decreases. EXPONENTIAL FUNCTIONS
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MATH 1004A - lecture#8 - Exponential%2c logarithmic functions and their derivatives

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