代做MATH-UA 121: Calculus I代做Statistics统计
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Final examination
Multiple Choice
Shade your answers to the multiple choice questions on the multiple choice answer sheet on page 2. Only multiple choice answers on the multiple choice answer sheet will be graded. No explanation is required to be shown and no partial credit will be given for the multiple choice questions.
1. (2 points) Which of the following is the domain of the function
A. (−∞,−2) ∪ (3,∞)
B. (3,∞)
C. (0,∞)
D. (−∞, 3)
E. (−2, 0)
2. (2 points) Evaluate
A. 2
B. −2
C. 0
D. ∞
E. None of the above
3. (2 points) Find the values of a and b for which f is continuous everywhere:
A. a = 0, b = 0
B. a = 0, b = 4/3
C. a = 1, b = 3
D. a = 1, b = 4
E. a = 3/4 , b = 1
4. (2 points) Given that the equation x3 + 5 = −x has exactly one real solution, in which of the following intervals does the solution lie?
A. (−2,−1)
B. (−1, 0)
C. (0, 1)
D. (1, 2)
E. None of the above
5. (2 points) Evaluate
A. − 2/1
B. −2
C. 2
D. 2/1
E. Limit does not exist (is possibly ±∞)
6. (2 points) Differentiate
7. (2 points) At which one of the following values of x is the tangent line to the curve y = 3+5x − x 2 parallel to the line x + y = 3?
A. 0
B. 1
C. 2
D. 3
E. 4
8. (2 points) Which one of the following is the derivative of cos(x3) + (sin(x))3?
A. −sin(x3) + 3(sin(x))2
B. −3x 2 sin(x3) + 3(sin(x))2 cos(x).
C. 3x 2 cos(x3) − 3 sin(x)(cos(x))2
D. cos(x3) + 3(sin(x))2
E. None of the above
9. (2 points) Suppose f is a differentiable function for all real numbers. Suppose f'(3) = −2 and f (3) = 10. Using linearization or differentials, approximate f (2.5).
A. 8
B. 9
C. 9.5
D. 11
E. 12
10. (2 points) Evaluate the limit
A. 0
B. 1
C. /e
D. ∞
E. −∞
11. (2 points) The horizontal and vertical asymptotes of the function are
A. y = 1 and x = 0
B. y = 0 and x = 0
C. y = 1, y = 0, and x = 0
D. y = −1, y = 0, and x = 0
E. None of the above
12. (2 points) Suppose f is a one-to-one differentiable function with values as shown. What is(f−1)'(−2)?
A. −2
B. −1
C. − 2/1
D. − 3/1
E. − 4/1
13. (2 points) Find an equation for the tangent line to the ellipse x2 + 2xy + 4y2 = 12 at (2, 1).
A. 2y = 4 − x
B. 2y = x
C. 2y = −x
D. y = x − 1
E. y = −2x + 5
14. (2 points) Evaluate f'(3) if f (x) = x 1−x.
A. − 9/ln3 − 27/2
B. − 2/ln2 − 4/1
C. 1
D. ln2 + 1
E. ln3 − 1
15. (2 points) What is the derivative of y = arctan(eu)?
A. 1 + eu/1
B. 1 + e −2u/1
C. 1 + e2u/eu
D. 1 + e −2u/−e−u
E. None of the above
16. (2 points) Find the absolute minimum value of on the interval [1, 3].
A. 0
B. 5/2
C. 13/6
D. 2/1
E. 5/4
17. (2 points) The graph below is f'(x) for some function f . (Note: This is the graph of the derivative of f .)
Which of the following are true?
I. f has a horizontal tangent line when x = 5.
II. f is increasing on (−∞,−2) and (4, 6).
III. f is decreasing on (−∞, 0) and (5,∞).
A. I only
B. II only
C. III only
D. I and II
E. II and III
18. (2 points) What is the value of
A. 6/1
B. 1
C. 3/8
D. 17/6
E. None of the above
19. (2 points) If and then what is
A. 8
B. 10
C. 14
D. 16
E. 17
20. (2 points) True or False: Suppose f is defined at a and limx→a g(x) exists. Then one always has
A. True
B. False
Free Response
Please show and explain your working to receive credit.
21. The Mall of America is setting up a Christmas Village as part of its holiday display. It is to be laid out in a rectangular shape with three 6 foot-wide openings. The rest of the border should use 82 feet of fencing. (Diagram not to scale.)
(a) (3 points) Let the length and the width of the rectangle be x and y, respectively. Using the constraint on the length of the fence, write y as a function of x.
(b) (3 points) Write the area function, A(x), of the holiday display as a function of x.
(c) (3 points) Find x that maximizes A(x). (Make sure to justify why your answer maximizes A(x).)
(d) (1 point) Find the maximum area.
22. Suppose
(a) (2 points) Find g'(x).
(b) (2 points) Is g(x) one-to-one? Justify your reasoning.
(c) (6 points) Find the intervals where g(x) is concave up and concave down and identify the x values for any inflection points.
23. (10 points) Estimate the area under the graph of f (x) = x3 + 1 from x = 0 to x = 4 using four approximating rectangles and right endpoints. Sketch the graph and the rectangles. Is your estimate an underestimate or overestimate?
24. A particle is moving along a line. It starts off at time t = 0 at position s(0) = 2 and has velocity function v(t) = t2 − t.
(a) (4 points) Find the position of the particle at time t = 4.
(b) (6 points) Find the distance traveled by the particle during the time interval [0, 4].