1110-fa2011-PRELIM-1-SOLUTIONS

Ln x 1 ln 2 x 1 ln 2 we know that ln 2 0 so x 1 ln 2

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ln ( x ) 1 ln ( 2 ) = x 1 ln ( 2 ) . We know that ln ( 2 ) > 0 , so x 1 ln ( 2 ) is a power function with a positive power, and so as x tends to , the x 1 ln ( 2 ) grows without bound. Thus, lim x →∞ e log 2 ( x ) = . (g) If h ( x ) = e 1 x , find an equation for h - 1 ( x ) . We must solve x = e 1 y for y as a function of x . We take the natural log of both sides to get ln ( x ) = ln ( e 1 y ) = 1 y . Taking the reciprocal, we get the function h - 1 ( x ) = 1 ln ( x ) .
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Math 1110 Prelim I (9/27/2011) 3 Question 3. (10 points) Ten years from now, Victor plans to pay $ 30, 000 for a Chevy Camaro. He has saved $ 15, 000 so far, and plans to invest it in an account that earns continuously compounded interest. (a) Write an equation describing how much money Victor will have in t years at an annual interest rate of r . The equation is M ( t ) = $ 15, 000e rt . This uses the exponential model for continuously compounding interest. (b) Please determine what interest rate r he needs to find in order to have saved enough money to buy the car ten years hence. (H INT : Your answer may contain a logarithm.) Victor wants to have $ 30, 000 in 10 years. So we must solve $ 30, 000 = $ 15, 000e r · 10 for r . We divide by $ 15, 000 and take a natural logarithm: ln ( 2 ) = ln ( e r · 10 ) = r · 10, so r = ln ( 2 ) 10 . Thus, he needs to find an interest rate of r = ln ( 2 ) 10 6.9 %. Question 4. (20 points) Determine whether the following statements are (always) true or (at least sometimes) false, and circle your response. Please give a brief explanation (in complete sentences!) – a reason why it’s true, or an example where it fails. (a) Let f be a continuous function on [- 1, 1 ] such that f ( 1 ) = - f (- 1 ) . Then there is a point c in the interval [- 1, 1 ] such that f ( c ) = 0 . Math 1110 Sample True/Fals Presentation problem 1. Determine whet sometimes) false, and circle your respons - a reason why it's true, or an example w (a) If f(x) is an evenfunction, then so i (b) If f (x) and g(x) both one-to-one fun If f ( 1 ) = 0 the we take c = 1 and we are done. Otherwise, f ( 1 ) and f (- 1 ) have opposite signs, so 0 is between f ( 1 ) and f (- 1 ) . By the Inter- mediate Value Theorem, it follows that there is a point c such that f ( c ) = 0 .
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  • Fall '06
  • MARTIN,C.
  • lim, Limit of a function

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