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Winter 2004 Solutions - MATH 115 FIRST MIDTERM EXAM...

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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 ) = π 5 , then f 0 ( x ) = 5 π 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 0 ( 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
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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 0 ( x ) > 0 for x < 5 g 00 ( x ) > 0 for x < 0 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 0 ( x ) = 0 for x > 5 PSfrag replacements 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 car’s 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 ) .
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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 carbon-14 analysis. Your old high school science teacher is able to find that 33% of the mummy’s carbon-14 remains. Using the fact that the half-life of carbon-14 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 carbon-14 decays exponentially, that the formula for the amount of carbon-14 left after t years is given by P = P 0 e rt where P 0 is the initial amount and r is the exponential rate of decay.
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