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Name: Instructor:
Math 121 - Sample Final Exam - F
Problem 1-15 16 17 18 19 20 21 22 23 24 Total Points 75 40 40 40 15 20 20 25 15 10 Score
Instructions: You will be given 150 minutes for this exam. Part A has 15 questions, Part B has 3 problems, and Part C has 6 problems. The problems are worth a total of 300 points, with each problem weighted as indicated. Your test should have 10 pages. Write your name at the top of each page. Do not detach this sheet from the test booklet. Do all your work in this booklet and circle your final answers and write them in the appropriate boxes.
In Part A, no partial credit will be assigned. You do not need to justify your work in this part of the test. Select the most appropriate answer. Keep in mind that if you use your calculator you may only get an approximate value, while the choices given in some problems may list the exact value.
In Part B, all the answers need to be exact and complete. You do not need to justify your work in this part of the test. Partial credit might be given. Always give the exact answers for the problems in Part B.
In Part C, all the answers need to be exact. Partial credit might be given, and answers without sufficient justification might receive no credit. Always give the exact answers and justify your work for the problems in Part C.
You are allowed to use a calculator without symbolic algebra capabilities for all parts of the exam. Make sure that your calculator has been approved by your instructor.
During the exam, you are not allowed to borrow or interchange calculators. You are not allowed to use any books and notes. Turn off your cell phone and any other electronic devices (but a calculator) and keep them in your pocket or bag.
Part A: Multiple Choice Answers. After you are done with Part A you must copy the letters corresponding to your answers in the space below. Your answers in the space below will be used in grading Part A of the exam; so make sure you copy them correctly.
Part A Total
Part A - Multiple Choice Examination
Select only one answer for each problem. Copy your answers to the cover page!
- Let
f (x) =
{ x + c if x < 0 ex^ if x ≥ 0.
The function f (x) is continuous at x = 0 if c =
(A) 0 (B) 1 (C) 2 (D) − 1 (E) e (F) None of the choices (A) - (E) is correct.
- Suppose that f (x) is differentiable at x = 2 and satisfies f (2) = 1 and f ′(2) = −2. Then the linearization of f (x) at x = 2 is
(A) L(x) = − 3 − 2 x (B) L(x) = 4 − 2 x (C) L(x) = −4 + x (D) L(x) = 3 + x (E) L(x) = 5 − 2 x (F) None of the choices (A) - (E) is correct.
- The function f (t) is continuous on the interval [a, b] and satisfies f (a) > 0, f (b) < 0. Which one of the following statements MUST be true?
(A)
∫ (^) b
a
f (t) dt does not exist.
(B)
∫ (^) b
a
f (t) dt = 0. (C) f ′(t) < 0 for every t in the interval (a, b). (D) f (t) is concave upward on the interval (a, b). (E) f (c) = 0 for some number c in the interval (a, b). (F) None of the choices (A) - (E) is correct.
- Suppose that f (x) is differentiable on the interval [− 1 , 2], and f (−1) = 0, f (2) = 0. Which one of the following statements MUST be true?
(A)
∫ (^2)
− 1
f (x) dx = 0. (B) f ′(x) = 0 for every x in the interval (− 1 , 2). (C) f ′(c) = 0 for some number c in the interval (− 1 , 2). (D) f (x) = 0 for every x in the interval (− 1 , 2). (E) f (c) = 0 for some number c in the interval (− 1 , 2). (F) None of the choices (A) - (E) is correct.
- The limit lim x→ 0
ex^ − cos x x
(A) − 1 (B) 1 (C) −∞ (D) ∞ (E) Does not exist (F) None of the choices (A) - (E) is correct.
- The absolute minimum value of the function f (x) = e(2x−x (^2) ) on interval [0, 2] is
(A) 1 (B) e (C) 2 (D) e^2 (E) e−^2 (F) None of the choices (A) - (E) is correct.
- Suppose that f (x) is differentiable. If f (0) = 1, f ′(0) > 0, and f ′(x) is decreasing on the interval (− 1 , 1), circle one graph that best represents the curve of f (x).
y
1 x
y
x
y
x
y
x x
y
(A) (B) (C) (D) (E)
(F) None of the choices (A) - (E) is correct.
- A particle is moved along the x-axis by a force that measures f (x) = cos x pounds at a point x feet from the origin. Then the work done in moving the particle from x = 0 to x = π/2 feet is
(A) 0 lb-ft (B) 1 lb-ft (C) 1.5 lb-ft (D) 2 lb-ft (E) 2.5 lb-ft (F) None of the choices (A) - (E) is correct.
- Suppose that f (x) is continuous on the interval [0, 2]. If
∫ (^1)
0
f (x) dx = 5 and
∫ (^2)
1
f (x) dx = −6,
then
∫ (^2)
0
(2x + f (x)) dx =
(A) − 3 (B) − 1 (C) 0 (D) 1 (E) 3 (F) None of the choices (A) - (E) is correct.
- The length of the parametric curve given by x = 2 cos t, y = sin t with 0 ≤ t ≤ 1 is
(A)
∫ (^1)
0
(cos t − 2 sin t) dt (B)
∫ (^1)
0
√ 3 + cos^2 t dt (C)
∫ (^1)
0
√ 3 + sin^2 t dt
(D)
∫ (^1)
0
√ 1 + 3 sin^2 t dt (E)
∫ (^1)
0
√ 1 + 3 cos^2 t dt (F) None of the choices (A) - (E) is correct.
Part B -Fill in the blanks.
- Compute the indicated derivatives.
(A) The table shows several values of f and f ′. Let g(x) = f (ex). Evaluate g′(0). x^ f^ (x)^ f^
′(x) − 1 2 4 0 − 3 2 1 5 − 2
Answer:
g′(0) =
(B) Let h(x) = [f (x)]^2. Use the same data for f and f ′^ given in Part (A) to evaluate h′(−1).
Answer:
h′(−1) =
(C) Consider the equation ey^ + x^2 y = 0. Find dy dx in terms of x and y.
Answer:
dy dx
(D) Let k(x) =
∫ √x
5
cos(t^2 + 1) dt. Find k′(x).
Answer:
k′(x) =
- Let F (x) =
∫ (^) x
0
f (t) dt, where f is the function consists of straight lines and a semicircle shown
in the following figure. For each question write your answer in the box, and explain in the blank space.
y
0 1 2 3 4 5 6 7 8 9 10
2 1
3
! 2
! 1
f(t)
t
(A) Determine the values of f (9), F (9), and F ′(9).
Answer:
f (9) = F (9) = F ′(9) =
(B) List all the x value(s) at which F (x) attains a local maximum value.
Answer:
x =
(C) List all the interval (s) on which F (x) is decreasing.
Answer:
(D) List all the interval(s) on which F (x) is concave upward.
Answer:
Part C – Essay Questions - SHOW YOUR WORK!!
- In three hours, the velocity of a car at each half hour was recorded as follows:
Time (hours) 0 .5 1 1.5 2 2.5 3 Velocity (mi/h) 25 45 50 55 40 45 37
Estimate of the average velocity (in miles/hour) of the car over these three hours using Simpson’s Rule for n = 6.
Average velocity =
- A cylindrical can is to be made with top and bottom made of copper costing 2 cents per square inch and curved side made of aluminum costing 1 cent per square inch. The total cost is to be no more than 300π cents total. Find the dimensions of the can that will maximize its volume.
height = radius =
- Find the area of the region bounded by the curve y = cos x and the lines y = 2, x = 0, and x = π 2
1 2 x
2
1
y
0
y = cos x
y=
!/
Area =
- Let R be the region enclosed by the curves y =
x and y = x^2. Find the volume of the solid obtained by rotating R about the x-axis.
x
y
R
y = (^) x
y = x 2
Volume =
