RandolphT1W98

RandolphT1W98 - (a) f ( x ) = sin-1 2 x (b) For f ( x ) =...

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Winter 1998 Math 21 Test #1 Name 1. [5 pts] Sketch the graph of a function which goes through the point (0,1) and has the property that the slope of the curve at any point is equal to the value of the function. Plot at least one other point besides (0,1). 2. [8] Let the function f be defined by f ( x )= e 2 x +3. (a) Give the domain and range of f . (b) Find f - 1 ( x ). Include its domain and range. 3. [14] (a) Explain how you know that f ( x )=2 x + cos x has an inverse. (b) What is f - 1 (1) (for f in part (a))? (c) If f is a one-to-one differentiable function, then the formula for the derivative of the inverse is: d dx ( f - 1 ( x ) ) = Use this to find the derivative of f - 1 ( x )at x = 1, where f is given in part (a). 4. [12] For g ( x )=2 x + 1 and h ( x ) = log 2 ( x - 1), find g ( h ( x )) = Sketch both curves on the same axes: i) on each curve, label at least one point with exact coordinates. ii) Label any asymptotes. 5. [8] Find the exact, simplified, value of each: 3 2 log 3 (5) = sin - 1 (sin( 3 π 4 )) = 6. [10] Find the derivative of each:
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Unformatted text preview: (a) f ( x ) = sin-1 2 x (b) For f ( x ) = sinh(2 x ), nd f (0). Give a numerical value. 7. [10] Carbon-14 decays exponentially and its half-life is about 5700 years. (a) For an initial 5 lb. amount of Carbon-14, nd the function, y ( t ), that describes how much is left after t years. (b) Suppose that charcoal from a tree that was killed during the volcanic eruption when Crater Lake was formed now contains 2/5 (40%) of its original Carbon-14. How long ago did the eruption occur? 8. [8] Find the minimum values of the function f ( x ) = x ln x . 9. [8] Use logarithmic dierentiation (show your work) to nd the derivative of y = (1 + x ) 2 x . 10. [10] Calculate each integral: (a) Z 1 x (ln x ) 2 dx (b) Z 1 1 + 9 x 2 dx 11. [6] Calculate the limit. Show each step: lim x x 2 e-x...
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