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# lecture4 - e is the number such that the slope of the...

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Lecture 4: New Functions from old, Part 2 Exponential Functions (Chapter 1, Sec. 3 and 4) ex. If f ( x ) = 1 x , g ( x ) = p 2 ¡ x , ﬂnd with domain: 1) ( f + g )( x ) 2) ( fg )( x ) Function Composition Def. ( f g )( x ) Def. ( g f )( x )

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ex. If f ( x ) = 1 x , g ( x ) = p 2 ¡ x , and h ( x ) = x 2 , ﬂnd with domain: 1) ( f g )( x ) 2) ( g f )( x ) 3) ( h g f )( x )
Exponential Functions Def. An exponential function with base a is a function of the form f ( x ) = a x where a is a positive constant. NOTE: For exponential function f ( x ) = a x and positive integer n , 1. If x = n , f ( x ) = 2. f (0) = 3. f ( ¡ n ) = 4. If reduced rational number x = p q for integers p and q , q > 0, then f ( x ) = 5. If x is irrational

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Graphs of Exponential Functions Graph on the same axes: 1) f ( x ) = 1 x 2) f ( x ) = a x , a > 1 3) f ( x ) = a x , 0 < a < 1 6 - ? ±
ex. Sketch the graph: y = 2 x ¡ 2 + 1 6 - ? ± Laws of Exponents 1) a x + y = 2) a x ¡ y = 3) ( a x ) y = 4) ( ab ) x =

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Application ex. The half-life of Strontium-90 is 25 yrs. In a sample of 80 mg, ﬂnd an expression for m ( t ), the amount remaining after t years.
The base e Consider the slope of the tangent line to the graph of f ( x ) = a x at the point (0 ; 1). If a = 2 6 - ? ± If a = 3 6 - ? ±

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Def.

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Unformatted text preview: e is the number such that the slope of the tangent line to y = e x at x = 0 is exactly 1. NOTE: e … ex. Sketch the graph of y = e x . 6-? ± Sketch the graph of the following: ex. y = e ¡ x 6-? ± Try these! ex. y = 2 ¡ e ¡ ( x +1) 6-? ± ex. Suppose you invest \$ 1000 in an account that doubles every 8 years. What will the value of the account be in 24 years? Find a formula for the value in t years. ex. The concentration of a drug in a particular or-gan at any time t (in seconds) is given by x ( t ) = 0 : 08 + 0 : 12(1 ¡ e ¡ : 02 t ) where x ( t ) is mea-sured in grams/cubic centimeter. What is the initial concentration of the drug in the organ? Find the concentration of the drug in the organ in the long run (as t approaches inﬂnity)....
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lecture4 - e is the number such that the slope of the...

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