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Unformatted text preview: MATH 115 FIRST MIDTERM EXAM SOLUTIONS 1. (2 points each) Circle True or False for each of the following problems. Circle True only is the statement is always true. No explanation is necessary. (a) log( 1 A ) = log( A ) . TRUE False (b) If f ( x ) = 5 , then f ( x ) = 5 4 . True FALSE (c) The function y = a b + c e kt for k > 0 and a, b, c constants has a horizontal asymptote of y = a c . True FALSE (d) A degree 7 polynomial must have at least 1 real root but can not have more than 7 real roots. TRUE False (e) f ( a ) is the tangent line of f at the point ( a, f ( a )). True FALSE (f) If f ( x ) = x 2 , then f 1 ( x ) = 1 x 2 . True FALSE (g) Note: This problem was excluded from grading. If f 00 ( a ) = 0, then the point ( a, f ( a )) is an inflection point of f . True FALSE 2 2. (8 points) On the axes below, sketch a graph of a single function , g , with all of the following properties. g (0) = 2 g ( x ) > 0 for x < 5 g 00 ( x ) > 0 for x < g 00 ( x ) < 0 for 0 < x < 5 lim x 5 g ( x ) = 6 and lim x 5 + g ( x ) = 3 g (5) = 4 g ( x ) = 0 for x > 5 x y g ( x ) 3. (1+1+3 points) Upon graduating from the university and landing your first big job, you decide to reward yourself for all the hard work and purchase a brand new sports car. The price of the sports car is $45 , 000. The value of the car depreciates at the rate of 37% per year. Comprehensive insurance costs 10% of the cars value each year. For parts (a) and (b) circle the best choice. (a) The value of the sports car is an exponential function of time. (b) The cost of the comprehensive insurance is a linear function of V , the value of the car. (c) Write a function that gives the cost of the comprehensive insurance policy on the car after the t th year. C ( t ) = 0 . 10(45 , 000(0 . 63) t ) . 3 4. (6 points) A circus is planning to visit your hometown. They claim to have a mummy that is 15,000 years old, but the citizens of your town are suspicious. The town council is given a sample of the mummy for a carbon14 analysis. Your old high school science teacher is able to find that 33% of the mummys carbon14 remains. Using the fact that the halflife of carbon14 is 5,730 years, determine the age of the mummy that the circus is bringing to town. [Show your work!] We know that since the carbon14 decays exponentially, that the formula for the amount of carbon14 left after t years is given by P = P e rt where P is the initial amount and r is the exponential rate of decay....
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This note was uploaded on 09/11/2008 for the course MATH 115 taught by Professor Blakelock during the Fall '08 term at University of Michigan.
 Fall '08
 BLAKELOCK
 Math, Calculus

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