# Exam1

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Exam 1 Tuesday, September 15 Remember to show an appropriate amount of work for each part. 1. [17 points] This problem has 4 parts, which are completely independent. (a) [4 points] f ( x ) is an exponential function with f (0) = 2 and f (3) = 54. Find a formula for f ( x ). Show your work. (b) [4 points] g ( x ) is a function with g (1) = 2, g (2) = 3 and g (3) = 5. Can g ( x ) be linear? Justify briefly . (c) [4 points] Let h ( x ) be the function ln x x - 5 . What is the domain of h ( x )? (d) [5 points] Suppose that - x 2 + 4 x ' ( x ) x 2 - 4 x + 8, and ' (2) = 4. Is ' ( x ) continuous at x = 2? Justify briefly, clearly stating any theorems you use. 2. [8 points] Let h ( x ) be the function: h ( x ) = p 1 + ln(10 8 x + 7) Find a formula for h - 1 ( x ). Show your work. 3. [10 points] Solve for x : log ( x 1 log e ) + ln( x ) = π Show your work. Hint : You will likely want to use either log x = ln x ln 10 or ln x = log x log e . 4. [10 points] Let f ( x ) be the function: f ( x ) = ax + b if x < 0 a if x = 0 - 5 · 7 x if x > 0 For what value(s) of a and b is f ( x ) continuous? Show your work. 5. [5 points] Let f ( x ) = ( C · g ( x ) if x < 0 ln( x 2 + 1) if x > 0 where lim x 0 - g ( x ) = 1 and lim x 0 + g ( x ) = - 1 For what value(s) of C does lim x 0 f ( x ) exist? Show your work.