Final Exam. V Spring: Calculus I. May 12, 2011

Transcription

1 Name: ID#: Final Exam V Spring: Calculus I May 12, 2011 PLEASE READ THE FOLLOWING INFORMATION. This is a 90-minute exam. Calculators, books, notes, and other aids are not allowed. You may use the backs of the pages or the extra pages for scratch work. Do not unstaple or remove pages as they can be lost in the grading process. Please do not put your name on any page besides the first page.

2 MC Calculus I Final Examination Spring 2011 MC MC (36 points). This part consists of 18 multiple choice problems. Nothing more than the answer is required; consequently no partial credit will be awarded. 1. Find the domain and the range of the function f(x) = 1 + x. A Domain (0, ), range [0, ) B Domain [0, ), range (1, ) C Domain [1, ), range [0, ) D Domain [0, ), range [1, ) E Domain [1, ), range (1, ) ( 1 2. The limit lim t 0 t 1 ) is t 2 + t A 1 B 0 C D E 1 ( ) 1 3. The limit lim x 2 sin is x 0 x A 1 B C D 0 E 1

3 MC Calculus I Final Examination Spring 2011 MC 4. The limit lim x ( 9x 2 + x 3x) is A 1 3 B 1 3 C D E e 1/n 5. The limit lim is n 3 + e2/n A 2 3 B 1 4 C 0 D e E e x 2 6. The limit lim x 0 e x x 1 is A 1 B 1 C 2 D 2 E 0

4 MC Calculus I Final Examination Spring 2011 MC 7. Which of the following must be true? I. If f is differentiable at a, then f is continuous at a. II. If f is continuous at a, then f is differentiable at a. III. If f is differentiable at a, then f is differentiable at a. A I only B II only C III only D I and II only E II and III only 8. If y = 1 ln x 1 + lnx, what is y? 2 A x(1 + lnx) 2 B 2 ln x x(1 + lnx) 2 C 2 ln x (1 + lnx) 2 2 D (1 + lnx) 2 E 2 ln x x(1 + lnx) 2 9. If f(x) = (x 2 + 1) 2 arctanx, what is f (x)? A (1 + x 2 )(1 + 2 arctanx) B 1 + 4x arctanx C 4x(1 + x 2 ) arctanx D (1 + x 2 )(1 + 4x arctan x) E 1 + 2(1 + x 2 ) arctanx

5 MC Calculus I Final Examination Spring 2011 MC 10. If f(x) = A ln x B ln x 2 C ln 2 ln x D ln x 2 x E ln x 2 x x ln tdt, what is f (x)? 11. Suppose that f(x) is a differentiable function. If f(x)y + f(y)x = 10, what is dy/dx? A f(y) + f (x)y f(x) + f (y)x B f(x) + f (y)x f(y) + f (x)y C f(x) + f (y)x + f (y)x f(x) f(x) D f(x) + f (y)x + f (y)x E f(y) f (x)y f(x) + f (y)x 12. Let f(x) = e x2. The linear approximation of f(x) near a = 1 is A e(x 1) + 1 B 2xe x2 (x 1) + e C e(x 1) + e D 2e(x 1) + e E e x2 (x 1) + e

6 MC Calculus I Final Examination Spring 2011 MC 13. If f(x) = x 2, what is the inverse of f? A f 1 (x) = x B f 1 (x) = x 2 2, x 0 C f 1 (x) = x 2 + 2, x 2 D f 1 (x) = x E f 1 (x) = x 2 + 2, x Let f(x) = cosx, 0 x 1. Find the Riemann sum for a regular partition of size 5 taking the sample points to be the left endpoints. A 1 ( ( ) ( ) ( ) ( ) ) cos + cos + cos + cos + cos B 1 ( ( ) ( ) ( ) ( ) ) cos + cos + cos + cos + cos C 1 ( ( ) ( ) ( ) ) cos + cos + cos + cos D 1 ( ( ) ( ) ( ) ( )) cos + cos + cos + cos E 1 ( ( ) ( ) ( ) ( )) cos + cos + cos + cos Let f (x) = 12 sin(2x) + 27e 3x and f(0) = 0, f(π) = 3e 3π. Find the function f(x). A 3 sin(2x) + 3e 3x + 3x π 3 B 3 sin(2x) + 3e 3x + 3x π 3 C 3 sin(2x) + 3e 3x 3x π 3 D 3 sin(2x) + 3e 3x 3x π 3 E 3 sin(2x) + 3e 3x + 3x π + 3

7 MC Calculus I Final Examination Spring 2011 MC 1 + x sin x 16. The indefinite integral dx is x A ln x + cos x + C B ln x cos x + C C ln x + cos x + C D ln x + cos x + C E ln x cos x + C 17. The indefinite integral (2x + 1)5 x2 +x+1 dx is A 5x2 +x+1 ln 5 + C B 5 x2 +x+1 + C C 5x2 +x+1 ln x + C D 5 x2 +x+1 ln 5 + C E 5 x2 +x+1 ln x + C 18. The definite integral π/10 cos(5x)dx is 0 A 1 5 B 1 5 C 0 D 1 E 1

8 FR1 Calculus I Final Examination Spring 2011 FR1 FR (34 points). Problems FR1 FR4 are free response questions. Put your answers in the boxes (where provided) and your work/explanations in the space below the problem. Read and follow the instructions of every problem. Show all of your work for purposes of partial credit. Full credit may not be given for an answer alone. Justify your answers. Full sentences are not necessary, but English words help. When in doubt, do as much as you think is necessary to demonstrate that you understand the problem, keeping in mind that your grader will be necessarily skeptical. FR1 (5 points). Find the derivative of f(x) = 1 using the definition of the derivative. No 1 x credit will be given for shortcut methods such as the quotient rule. f (x) =

9 FR2 Calculus I Final Examination Spring 2011 FR2 FR2 (9 points). Let (a) Find lim f(x) and lim f(x). x 1 + x 1 f(x) = x2 1 x 1 lim f(x) = x 1 + lim f(x) = x 1 (b) Does lim x 1 f(x) exist? Explain why. (c) Sketch the graph of f(x).

10 FR3 Calculus I Final Examination Spring 2011 FR3 FR3 (12 points). Let f(x) = (a) Find the vertical and horizontal asymptote(s). x 2 (x 1) 2. One can see that f (x) = 2x (x 1) 3,f (x) = Vertical asymptote(s) = 2(2x + 1) (x 1) 4. Horizontal asymptote(s) = (b) Find the interval(s) of increase or decrease. Interval(s) of increase = Interval(s) of decrease =

11 FR3 Calculus I Final Examination Spring 2011 FR3 (c) Find the interval(s) of concavity and the inflection point(s). Interval(s) of concavity = Inflection point(s) = (d) Use the information from parts(a)-(c) to sketch the graph of f.

12 FR4 Calculus I Final Examination Spring 2011 FR4 FR4 (8 points). A box with a square base and open top has volume 16 ft 3. The material used to make the base costs 4 dollars per square foot while the material used to make the sides costs 1 dollar per square foot. (a) What are the dimensions of the cheapest such box? Dimensions = (b) How much does it cost to produce? Cost =

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15 Problem Possible Points Number Points Earned MC 36 FR1 5 FR2 9 FR3 12 FR4 8 Total 70