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The final exam questions for a mathematics course (mth 229) from spring 2003. The exam covers various topics including differentiation, limits, calculus, and optimization. Students are required to use only a calculator and show their work for full credit. The exam consists of 9 problems with varying points assigned to each problem.
Typology: Exams
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(35) 1. Use the rules of differentiation to find the derivatives of the following functions. (a) † q ( u ) = cot( u ) a^2 + u^2 (b) † g ( x ) = x f ( x ) (treat † f ( x ) as an unknown function) (c) † r ( t ) = 4 sec^3 ( t ) (d) † k ( x ) = ln(sin( ax ) + cos( bx )) (e) † f ( x ) = tan-^1 ( x^1 /^3 ) (or † f ( x ) = arctan( x^1 /^3 ))
( 25 ) 4. Suppose † f ( x ) = x e -^ x^ (^2) / 2 . (a) Use calculus to find the intervals where † f ( x ) is increasing or decreasing, and the location of any local maxima or minima. (b) Use calculus to find the intervals where † f ( x ) is concave up or concave down, and the location of any inflection points. (c) Find the absolute maximum and minimum values on the interval [–2, 2].
( 20 ) 5. A rocket is launched vertically from a point on the ground. An observer, who is 1000 meters away from the base of the launching pad, notices that the angle of elevation of the rocket is increasing at a rate of π/40 radians per second when the angle of elevation is π/3. Find the speed of the rocket at that instant.
(15) 7. Following are some values of the function f ( x ). x 0.8 0.9 1.0 1.1 1. f(x) 0.437412 0.432777 0.420735 0.403261 0. (a) Use the values from the table to approximate † f ¢(0.9), † f ¢( 1 ), and † f ¢(1.1). Be sure to explicitly show the difference quotients you use. (b) Is f ( x ) concave up or concave down near x = 1? Explain your reasoning.
( 25 ) 8. A particle travels along a straight line. Its position at t seconds is given by † p ( t ) = t^3 - 6 t^2 + 9 t + 5 meters from a fixed origin on the line. (a) What is the average velocity of the particle during the interval 3 ≤ t ≤ 5? (b) What is the instantaneous velocity at t = 3 seconds? (c) What is the acceleration at t = 3 seconds? Is the particle speeding up or slowing down? (d) Find the total distance traveled during the interval † 0 £ t £ 5.