Section 5.2a class notes_0

Section 5.2a class notes_0 - n ( 29 1 1 n n + 1 2 2 2.25 10...

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Section 5.2a The Natural Exponential Function Objective 1: Understanding the Characteristics of the Natural Exponential Function The number e is an irrational number that is defined as the value of the expression ( 29 1 1 n n + as n approaches infinity. The table below shows the values of the expression ( 29 1 1 n n + for increasingly large values of n. As the values of n get large, the value e (rounded to 6 decimal places) is2.718281. The function ( ) x f x e = is called the natural exponential function. The graph of the natural exponential function ( ) x f x e = Characteristics of the Natural Exponential Function The Natural Exponential Function is the exponential function with base e and is defined as ( ) x f x e = . The domain of ( ) x f x e = is ( 29 , -∞ ∞ and the range is ( 29 0, . The graph of ( ) x f x e = intersects the y- axis at ( 29 0 1 , . as x e x → ∞ → ∞ 0 as x e x → -∞ The line 0 y = is a horizontal asymptote. The function ( ) x f x e = is one-to-one. 5.2.1 and 4 Use a calculator to approximate the exponential expression to 6 decimal places.
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Unformatted text preview: n ( 29 1 1 n n + 1 2 2 2.25 10 2.5937424601 100 2.7048138294 1000 2.7169239322 10,000 2.7181459268 100,000 2.7182682372 1,000,000 2.7182804693 10,000,000 2.7182816925 100,000,00 2.7182818149 2 x y = 3 x y = ( ) x f x e = ( ) x f x e = Objective 2: Sketching the Graphs of Natural Exponential Functions ( ) x f x e = 5.2.8 Use the graph of ( ) x f x e = and transformations to sketch the exponential functions. Determine the domain and range. Also, determine the y-intercept and find the equation of the horizontal asymptote. Objective 3: Solving Natural Exponential Equations by Relating the Bases The Method of Relating the Bases for Solving Exponential Equations If an exponential equation can be written in the form u v b b = , then u v = . 5.2.14 Solve the exponential equation using the method of relating the bases by first rewriting the equation in the form u v b b = ....
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Section 5.2a class notes_0 - n ( 29 1 1 n n + 1 2 2 2.25 10...

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