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Mth 103 Chapter 4 Class Notes-1

Mth 103 Chapter 4 Class Notes-1 - 4.1 Exponential...

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Page 1 (4.1) −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 x y 4.1 Exponential Functions (Textbook Homework: 1-17(odd), 19-24, 25, 29, 33, 39-57(odd), 65-73(odd), 87-91) Objectives: evaluating exponential function graphing exponential functions evaluating functions with base e using compound interest formulas An exponential function f with base b is defined by x b x f = ) ( or x b y = , where b > 0, b 1, and x is any real number. (Note: The variable is in the exponent.) Example 1: Determine which functions are exponential functions. For those that are not, explain why they are not exponential functions. (a) 7 2 ) ( + = x x f Yes No ______________________________________________________ (b) 2 ) ( x x g = Yes No _________________________________________________________ (c) x x h 1 ) ( = Yes No __________________________________________________________ (d) x x x f = ) ( Yes No _________________________________________________________ (e) x x h = 10 3 ) ( Yes No ______________________________________________________ (f) 5 3 ) ( 1 + = + x x f Yes No ____________________________________________________ (g) 5 ) 3 ( ) ( 1 + = + x x g Yes No __________________________________________________ (h) 1 2 ) ( = x x h Yes No ______________________________________________________ Example 2: Graph each of the following and find the domain and range for each function. (a) x x f 2 ) ( = domain: __________ range: __________ (b) x x g = 2 1 ) ( domain: __________ range: __________
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Page 2 (4.1) Characteristics of Exponential Functions x b x f = ) ( b > 1 0 < b < 1 Domain: Range: Transformations of g ( x ) = b x ( c > 0) : (H S R V) V ertical: c b x g x + = ) ( (graph moves up c units) c b x g x = ) ( (graph moves down c units) H orizontal: c x b x g + = ) ( (graph moves c units left) c x b x g = ) ( (graph moves c units right) R eflection: x b x g = ) ( (graph reflects over the x -axis) x b x g = ) ( (graph reflects over the y -axis) S tretch/ S hrink: x cb x g = ) ( (graph stretches if c > 1) (Vertical) (graph shrinks if 0 < c < 1) S tretch/ S hrink: cx b x g = ) ( (graph shrinks if c > 1) (Horizontal) (graph stretches if 0 < c < 1)
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Page 3 (4.1) −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 x y −5 −4 −3 −2 −1 1 2 3 4 5 6 −6 −5 −4 −3 −2 −1 1 2 3 4 5 x y Example 3: Use x x f 2 ) ( = to obtain the graph 1 2 ) ( 3 = + x x g . Domain of g : ____________ Range of g : _____________ Equation of any asymptotes of g : __________________________ Example 4: The graph of g below is the transformation of . 2 ) ( x x f = Find the equation of the graph. x e x f = ) ( is called the natural exponential function, where the irrational number e (approximately 2.718282) is called the natural base . (The number e is defined as the value that n n + 1 1 approaches as n gets larger and larger.) Example 5: In 1969, the world population was approximately 3.6 billion, with a growth rate of 1.7% per year. The function x e x f 017 . 0 6 . 3 ) ( = describes the world population, ) ( x f , in billions, x years after 1969 . Use this function to estimate the world population in 1969 ____________________ 2000 ____________________ 2011 ____________________
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Page 4 (4.1) Period Interest Formula Continuous Interest Formula nt n r P A + = 1 rt Pe A = A = balance in the account (A mount after t years) P = p rincipal (beginning amount in the account) r = annual interest r ate (as a decimal) n = n umber of times interest is compounded per year t = time (in years) Example 6: Find the accumulated value of a $5000 investment which is invested for 8 years at an interest rate of 12% compounded: (a) annually (b) semi-annually (c) quarterly (d) monthly (e) continuously
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Page 1 (4.2) 4.2 Logarithmic Functions (Textbook Homework: 1-45(odd), 47-52, 53-109(odd), 119, 135-149) Objectives:
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