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MS125-01 Test 2: Calculus Examination October 2006, Exams of Calculus

A test paper for a calculus course, consisting of multiple-choice questions and problems requiring differentiation and optimization. Students are required to solve each problem, showing all the work to justify their answers. The exam lasts for one and a half hours and covers topics such as differentiation, limits, and optimization.

Typology: Exams

Pre 2010

Uploaded on 08/18/2009

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koofers-user-0yw 🇺🇸

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MS125-01 Test 2 (Oct. 25, 2006)
NAME: __________________________
You have one and half hours to complete this examination. Write your answers directly on the exam paper. If you need extra space, use the
back of the page and indicate on the front that you have done so. Use of calculator is allowed for Part II only. No other reference materials
are allowed. You must give a complete solution to each problem, showing all the work necessary to justify your final answer. The entire solution,
not just the answer, will be graded, and partial credit is possible. A correct final solution will not, of itself, guarantee full credit.
Total Points: 104 points
Part I
Use appropriate rules and formulas to differentiate (or to find
dx
dy
) of each function.(4pts each)
1.
42
)15()( xxxf
2.
2
)31(
)(
x
exf
3.
xxxf 3sin)(
2
4.
xxf 2tan)(
5.
)arctan()(
3
ttg
6.
3
323
yxyx
pf3
pf4
pf5

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MS125-0 1 Test 2 (Oct. 25 , 2006)

NAME: __________________________

You have one and half hours to complete this examination. Write your answers directly on the exam paper. If you need extra space, use the back of the page and indicate on the front that you have done so. Use of calculator is allowed for Part II only. No other reference materials are allowed. You must give a complete solution to each problem, showing all the work necessary to justify your final answer. The entire solution, not just the answer, will be graded, and partial credit is possible. A correct final solution will not, of itself, guarantee full credit. Total Points: 104 points

Part I

Use appropriate rules and formulas to differentiate (or to find dx dy ) of each function.(4pts each)

  1. f (^ x )(^ x^2 ^5 x ^1 )^4
  2. ( 13 )^2 f ( x )  ex
  3. f^ (^ x ) x sin^3 x ^2 4. f^ (^ x ) tan^2 x 5. g^ (^ t )arctan(^ t^3 ) 6. 3 x^3  xy^2  y^3 
  1. f^ (^ x )cosh(sin x )
    1. 1

 (^) x x e e f x

Part II

  1. Let f^ (^ x )^ ex ^^1 ^4 x ^2 be a one-to-one function. a) ( 3 pts)What is^ f^ ^1 (^3 )? b) ( 4 pts)Evaluate (^ )(^3 ) f ^1  ?

2. Let f^ (^ x )^ x ^2 cos x on ^0 ,^2 .

a) ( 4 pts)Find all critical points for f^ ( x ). b) ( 4 pts)Find all inflection points for f^ ( x ), if any.

  1. Let (^ )^398 f x  x^3  x^2  x  . a) ( 4 pts)Determine the increasing interval for f^ ( x ). b) ( 10 pts)Graph f^ ( x )with all important points clearly labeled.
  2. A ladder 13 feet long is leaning against the wall of the house, while the base of the ladder is pulled away from the wall at a rate of 2 feet per second. a) ( 6 pts)How fast is the top moving down the wall when the base of the ladder is 12 feet away from the wall? b) ( 6 pts)How fast is the angle between the ladder and the wall changing when the base of the ladder is 12 feet away from the wall?
  1. ( 6 pts)The radius of a spherical balloon is changing at a rate of 2 inch per second. Find the rate of change of the volume of the balloon when the radius is 8 inches. (Hint: 3 3 4 V   r , Volume of the sphere with radius r )
  2. ( 6 pts) Set up the objective function to be optimized for the following problem : Alaina wants to get to the bus stop as quickly as possible. The bus stop is across a grassy park, 15 00 feet west and 8 00 feet north of her starting position. Alaina can walk west along the edge of the park on the sidewalk at a speed of 6 ft/sec. She can also travel through the grass in the park, but only at a rate of 4ft/sec. What path will get her to the bus stop the fastest?