MA 123 Exams Fall Semester 2013

MA 123 Exams Fall Semester 2013
Here are two midterms and the final that we gave in MA 123 in the Fall Semester of 2013. You should use them to get an idea of the format of a typical test and to see the types of questions we ask. You should not assume that the test questions this semester will be on the same topics. In fact, you are always responsible for all of the material that we cover in class as well as all of the designated material from your text. Additional information about the exams will be posted as announcements on MyMathLab.
The best way to study for our exams is to be sure that you are very comfortable with the homework assignments, the worksheets, and the examples that we present in class.
MA 123 October 24, 2013
First name (print): Last name (print):
Last five digits of ID number:
Discussion Section (circle yours):
B1: Tue 9:30–11 B2: Tue 11–12:30 B3: Tue 2–3:30 B4: Tue 3:30–5
B5: Tue 5–6:30 B6: Wed 5–6:30 B7: Thu 9:30–11 B8: Thu 11–12:30
B9: Thu 2–3:30 BA: Wed 3:30–5
Directions: Please do all of your work in this exam booklet and make sure that you cross out any work that we should ignore when we grade. Books and extra papers are not permitted. If you have a question about a problem, please ask. Remember: answers that are written logically and clearly will receive higher scores. There are 10 questions on 10 pages (not counting this cover page). Please make sure that you have all 10 pages of questions.
Calculators are not permitted on this exam.
Do not write in the following box:
PROBLEM POSSIBLE SCORE
MA 123 October 24, 2013
1. (16 points) In each part, compute dy/dx. (You do not have to simplify your answers.)
(a) y = e5x cos 2x
(b) y = x3 tan−1 x
(c) y = 105 + e3
(d) y = ln(cos(x2 + 1))
MA 123 October 24, 2013
2. (9 points) Multiple choice: Circle the correct answer. You will not receive any credit for ambiguous answers. You will receive 3 points for each right answer, 1 point if you do not answer the question, and 0 points for each wrong answer.
(a) If y = 2x+ 3
3x+ 2 , then
2
3
(b) An equation for the line that is tangent to the graph of y = 2x2 + x − 1 at the point (1, 2) is
(i) y = 5x+ 1 (ii) y = 5x+ 2 (iii) y = 5x− 3
(iv) y = 4x+ 1 (v) y = 4x+ 2 (vi) y = 4x− 2
(c) Suppose x2 + xy = 10. If x = 2, then dy
dx =
7 (vi)
MA 123 October 24, 2013
3. (6 points) More multiple choice: Circle the correct answer. You will not receive any credit for ambiguous answers. You will receive 3 points for each right answer, 1 point if you do not answer the question, and 0 points for each wrong answer.
(a) Let f be the absolute value function. That is, f(x) = |x| for all real numbers x. Which of the following three statements, (A), (B), and/or (C), are true?
(A) f is continuous at x = 0.
(B) f is differentiable at x = 0.
(C) f has an absolute minimum at x = 0.
(i) None (ii) A only (iii) B only (iv) C only
(v) A and B only (vi) A and C only (vii) B and C only
(viii) All three statements
2x− 2 . Which of the following two statements
are true?
(A) The graph has a horizontal asymptote.
(B) The graph has a vertical asymptote.
(i) Neither A nor B (ii) A only (iii) B only (iv) Both A and B
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MA 123 October 24, 2013
4. (6 points) Many choice: Circle all of the correct statements among the statements (A)–(E). You will not receive any credit for ambiguous answers.
For the function f whose graph is shown below:
(A) The number 2 is in the domain of f .
(B) lim x→2+
f(x) = lim x→2−
(E) f has a local maximum at the number 2.
2
4
MA 123 October 24, 2013
5. (9 points) For each of the following functions that are graphed in parts (a)–(c), indicate which graph (1–9) is the graph of its derivative.
(a.) graph of f
MA 123 October 24, 2013
6. (12 points) In the following figure, the graph of the function f is solid, and the graph of the function g is dashed.
-4 -2 2 4
(a) Let h(x) = g(f(x)). Calculate h′(4). Show your work.
(b) Let k(x) = f(g(x)). Calculate k′(−4). Show your work.
(c) Calculate lim x→−2−
f(x)
7. (16 points) Calculate the following limits. Show your work.
(a) lim x→∞
MA 123 October 24, 2013
8. (8 points) A lighthouse stands 300 m off of a straight shore and the focused beam of its light is revolving four times each minute. As shown in the figure below, the point P is the point on the shore closest to the lighthouse and Q is a point on the shore 400 m from P . What is the speed of the beam along the shore when it strikes the the point Q? Neglect the height of the lighthouse. You may leave your answer as a product of constants. Make sure that you specify the units of your answer and show your work.
, \ rqht hous.e" L"{E
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9. (8 points) Let f(x) = 1√ x+ 2
.
Using only the definition of the derivative and the limit theorems discussed so far this semester, calculate f ′(1). You will not receive any credit if you do not start with the definition of the derivative or if you use limit theorems not discussed so far this semester.
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10. (8 points) Consider the function
f(x) = 4x1/3 − x4/3.
(a) Calculate the critical points of f . Show your work.
(b) Find the absolute maximum and absolute minimum values of f over the interval −1 ≤ x ≤ 8. Show your work.
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Discussion Section (circle yours):
B1: Tue 9:30–11 B2: Tue 11–12:30 B3: Tue 2–3:30 B4: Tue 3:30–5
B5: Tue 5–6:30 B6: Wed 5–6:30 B7: Thu 9:30–11 B8: Thu 11–12:30
B9: Thu 2–3:30 BA: Wed 3:30–5
Calculators are not permitted on this exam. Do not write in the following box:
MA 123 November 21, 2013
(a)
∫ sin 2x dx =
(i) 1 2
cos 2x+ C (ii) −2 cos 2x+ C (iii) − sin2 x+ C
(iv) cos2 x+ C (v) −1 2
cos 2x+ C
(b) Given a function f such that f ′′(x) = 2, f ′(0) = 1, and f(0) = 3/2, then f(1) =
(i) 10 3
(ii) 17 3
(iii) 7 2
(iv) 3 10
(v) 17 2
(vi) 13 3
(vii) 23 3
(viii) 21 2
(iv) 1 4
(viii) 4
(a) When the region bounded by the graph of f(x) = 8 − 2x and the x-axis on the interval [0, 4] is approximated by left Riemann sums, the approximations
(i) always underestimate the area of the region.
(ii) always overestimate the area of the region.
(iii) are always equal to the area of the region.
(iv) none of (i)–(iii).
Note that left Riemann sums are sometimes called Riemann sums with left endpoints.
(b) If the interval [3, 5] is divided into n = 4 subintervals of equal length, the grid points xk including the endpoints of [3, 5] are
(i) xk = 3 + k for k = 1, 2, 3, 4 (ii) xk = 3 + k for k = 0, 1, 2, 3
(iii) xk = 3 + k for k = 0, 1, 2, 3, 4 (iv) xk = 3 + k/2 for k = 1, 2, 3, 4
(v) xk = 3 + k/2 for k = 0, 1, 2, 3 (vi) xk = 3 + k/2 for k = 0, 1, 2, 3, 4
(c) The sum (22 + 42 + 62) · 2 is a
(i) left Riemann sum for f(x) = x2 on [1, 7] with n = 3.
(ii) right Riemann sum for f(x) = x2 on [1, 7] with n = 3.
(iii) midpoint Riemann sum for f(x) = x2 on [1, 7] with n = 3.
(iv) none of (i)–(iii).
3. (6 points) Even more multiple choice: Circle the correct answer. You will not receive any credit for ambiguous answers. You will receive 3 points for each right answer, 1 point if you do not answer the question, and 0 points for each wrong answer.
(a) Suppose that f1(x) and f2(x) are two functions that tend to infinity as x → ∞. Recall that the notation f1 << f2 means that f2 grows faster than f1 as x→∞. Consider the four functions:
f1(x) = 10x f2(x) = x12 f3(x) = ln(2x) f4(x) = ex
Which of the following statements is true?
(i) f1 << f2 << f3 << f4 (ii) f2 << f3 << f4 << f1
(iii) f3 << f4 << f1 << f2 (iv) f4 << f1 << f3 << f2
(v) f3 << f2 << f1 << f4 (vi) f4 << f2 << f1 << f3
(vii) f3 << f2 << f4 << f1 (viii) f2 << f1 << f4 << f3
(ix) f1 << f2 << f4 << f3 (x) f2 << f1 << f3 << f4
(b) If the second derivative of a function f is f ′′(x) = x(x+1)(x−2)2, then the graph of f has inflection points if x =
(i) −1 only (ii) 0 only (iii) 2 only
(iv) −1 and 0 only (v) −1 and 2 only (vi) 0 and 2 only
(vii) −1, 0, and 2
3
4. (9 points) Let f(x) = √ x.
(a) Find the linear approximation L(x) to f(x) at x = 25. Show your work.
(b) Using your answer in part (a), approximate √
28. Express your answer as a fraction.
(c) Is your approximation smaller or larger than √
28? Justify your answer with a sentence and/or a labeled diagram.
4
5. (9 points) Let R be the region bounded by the graph of f(x) = 1/x and the x-axis between x = 1 and x = 3. Approximate the area of R by the left Riemann sum with n = 4 subintervals. (The left Riemann sum is sometimes called the Riemann sum with left endpoints.) Write your approximation as a sum of 4 terms. DO NOT SIMPLIFY AND DO NOT USE SIGMA NOTATION.
6. (9 points) Determine
7. (9 points) A cyclist traveling at 40 ft/s starts to slow down at a constant rate of 4 ft/s2. How many feet does she travel before she comes to a complete stop? Show your work.
8. (9 points) Calculate lim x→0
(3− 2ex)1/x. Show your work.
6
9. (4 points) For function f that is graphed immediately below, indicate which graph or graphs (1–6) is/are the graph(s) of its antiderivatives. (Recall that a function has more than one antiderivative.)
Graph of f
10. (12 points) Consider the graph of the first derivative f ′(x) of a function f(x) defined on the interval 0 ≤ x ≤ 9:
1 2 3 4 5 6 7 8 9
-2
-1
1
2
3
4 Graph of the FIRST DERIVATIVE of f
(a) On what intervals is f(x) increasing? Justify your answer with a sentence.
(b) At what values of x does f(x) have a local maximum? Justify your answer with a sentence.
(c) On what intervals is the graph of f(x) concave up? Justify your answer with a sentence.
(d) What are the x-coordinates of the inflection points for f(x)? Justify your answer with a sentence.
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11. (12 points) Consider all rectangles that are inscribed in the ellipse x2 + 4y2 = 4 as shown in the following figure.
-2 -1 1 2
-1
1
(a) The area of the rectangle is the objective function. Express it as a function of one variable. Make sure that you indicate what the variable represents.
(b) Find the dimensions of the rectangle of maximum area. Show your work.
(c) Using the First Derivative Test, show why the dimensions that you produced in part (b) maximize the area.
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MA 123 Final December 17, 2013
Calculators are not permitted on this exam.
Do not write in the following box:
Name, etc. 1
1. (12 points) In both parts, compute dy/dx. (You do not have to simplify your answers.)
(a) y = ex 3
x2 − 5
2. (6 points) A runner accelerates from the starting line at the rate of 7 + 2t ft/s2 until she reaches a speed of 30 ft/s. How far has she run from the starting line? Show your work.
1
3. (9 points) Let f and g be two functions that are differentiable for all real numbers. Their values and the values of their derivatives at x = 1, 2, 3, 4, 5 are given in the table:
x = 1 2 3 4 5
f(x) = 0 3 5 1 0
f ′(x) = 5 2 −5 −8 −10
g(x) = 4 5 1 3 2
g′(x) = 2 10 20 15 22
Compute the following derivatives. Show your work and express your answers as integers or fractions.
(a) h′1(2) where h1(x) = xf(x)
(b) h′2(4) where h2(x) = f(x)g(x)
(c) h′3(3) where h3(x) = f(g(x))
2
4. (12 points) Calculate the following integrals:
(a)
3
5. (8 points) Consider the function f(t) graphed below. (The curved arc is 1/4 of a circle.)
1 2 3 4 5 6
-1
0
1
Let
A(x) =
∫ x
0
∫ x
2
(a)
∫ 4
0
√ u du
(b) If the line tangent to the graph of the function f at the point (1, 7) passes through the point (−2,−2), then f ′(1) is
(i) −5 (ii) 1 (iii) 3 (iv) 7 (v) undefined
5
(a) Suppose that a particle moving along a line has velocity v(t) = t3− 3t2 + 12t+ 4. Then its maximum acceleration over the interval 0 ≤ t ≤ 3 is
(i) 9 (ii) 12 (iii) 14 (iv) 21 (v) 40
(b) How many inflection points does f(x) = 2x6 − 6x5 + 5x4 have?
(i) 0 (ii) 1 (iii) 2 (iv) 3 (v) 4
6
8. (12 points) Suppose that the functions f and f ′ are continuous on an open interval I that contains the numbers a and b. Simplify the following expressions by eliminating the integral sign.
(a) d
9. (12 points) Calculate the following limits. Show your work.
(a) lim x→0−
10. (12 points) An inverted conical water tank with a height of 12 ft and a radius of 6 ft is drained through a hole in the vertex at rate of 2 ft3/s. What is the rate of change of the depth of the water when the depth is 3 ft? Show your work.
Note: The volume V of a cone with base radius r and height h is V = 1 3 πr2h.
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MA 123 Exams Fall Semester 2013