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Math1011_Prelim1A

Math1011_Prelim1A - Math1011 Fall 2009 Prelim Consider the...

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Math1011 Fall 2009 Prelim I-A1 1 lnx Consider the function f(x) = . What are the domain and range of the function f(x). -1 Write an equation for the inverse function f (x).
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Math1011 Fall 2009 Prelim A1 1 lnx Consider the function f(x) = . What are the domain and range of the function f(x). 1 lnx Domain: ln(x) is defined for all positive numbers, so 0. can take on all values, except when lnx=0, so x 1 ( ) (0,1) (1, ) ln( ) can take on all reall numbers, so when we take the reciprical, we g x D f x > = et all numbers except for 0. ( ) ( ,0) (0, ) f = −∞ -1 Write an equation for the inverse function f (x). 1 ln 1 (1/ ) ln (1/ ) (1/ ) -1 (1/ ) (solve for x in terms of y) (ln ) 1 (cross multiply) ln interchange x and y f ( ) : x y y x y x x y y x x e e x e y e x e Note = = = = = = = 1 1 ( ) ( ,0) (0, ) which = ( ) ( ) (0,1) (1, ) which = ( ) D f f f D f = −∞ = (Doc #011p.16.01t)
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Math1011 Spring 2007 Prelim A2 The table shows the position of a cyclist: t (seconds) 2 3 4 s (meters) 5.1 10.7 17.7 Calculate the average velocity for each time period: i. [2, 3] ii. [3, 4] Use the information above to give an approximate value for the instantaneous velocity when t = 3.
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Math1011 Spring 2007 Prelim A2 The table shows the position of a cyclist: t (seconds) 2 3 4 s (meters) 5.1 10.7 17.7 Calculate the average velocity for each time period: i. [2, 3] (3) (2) 10.7 5.1 5.6 meters/second 3 2 3 2 avg y s s m x Δ = = = = Δ ii. [3, 4] (4) (3) 17.7 10.7 7 meters/second 4 3 4 3 avg y s s m x Δ = = = = Δ Use the information above to give an approximate value for the instantaneous velocity when t = 3. An estimate for the instantaneous velocity is the average of the secant lines from [2, 3] and [3, 4] 5.6 7 12.6 6.3 meters/second 2 2 m + = = = 17.7-5.1 12.6 2 4 2 Other ways are to use the slope of the secant line: from [2, 3] = 5.6 meters/second from [3, 4] = 7 meters/second from [2, 4]= 6.3 meters/second = = (Doc #1011p.21.02)
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Math1011 Fall 2009 Prelim A3 Consider the function whose graph is shown below. Sketch the graph of the function ( ) - ( -2) on the axes provided, and determine its domain and range. g x f x = Domain Range
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Math1011 Fall 2009 Prelim A3 Consider the function whose graph is shown below. Sketch the graph of the function ( ) - ( -2) on the axes provided, and determine its domain and range. g x f x = Domain: [1, 4] Range: [-2, 0] g(x) is the V below the x – axis. (Doc #011p.16.02t)
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Math1011 Fall 2007 Prelim A4 Show that for any three positive numbers a, b, c such that a 1, b 1, and c 1, the following equality holds: (log )(log )(log ) 1 a c b b c a = Solve the equation for x: 4 2log 1 10 ln 4 log 100 x x e + =
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Math1011 Fall 2007 Prelim A4 Show that for any three positive numbers a, b, c such that a 1, b 1, and c 1, the following equality holds: (log )(log )(log ) 1 a c b b c a = log log log log (log )(log )(log ) 1 (log )( )( ) 1 (log a a a a a c b c a a b c a b c a b b = = log )( a c log a b log log )( a a a c ) 1 log 1 1 1 true (for a, b, c 1) a a = = = log a log a : laws of logarithms: log a 1,b 1 log 1 b b x a Note x a = = Solve the equation for x: 4 2log 1 10 ln 4 log 100 x x e + = 2 4 2 2 2 2 2 2 2 2 2 2 2 log 2 1 10 2 1 10 1 2 2 1 2 1 2 1 2 2 1 4 log 10 1 2log 1 1 0 0 2 1 0 ( 1) 0 1 x x x x x x x x x x x x x x
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