NAME:
Instructor:
+
Math 160 Calculus for Physical Scientists I
Exam 1
February 12, 2015, 5:00-6:50 pm
h
Time your class meets:
m
a
“How can it be that mathematics, being after all a product of human thought independent of experience, is
so admirably adapted to the objects of reality?”
-Albert Einstein
1. Turn off your cell phone and other devices (except your calculator).
2. Write your name on every page of the exam. Write your instructor’s name on the cover sheet.
h)
=
3. You may use a calculator on this exam. You must provide your own calculator; you may not use a
laptop computer or smart phone.
4. No notes or other references, including calculator manuals or notes stored in calculator memory,
may be used during this exam.
5. Use the back of the facing pages for scratch work and for extra space for solutions. Indicate clearly
when you wish to have work on a facing page read as part of a solution to a problem.
x→lim
a (m
(Signature)
x
+
Please do not write in this space.
HONOR PLEDGE
I have not given, received, or used any unauthorized 1-6. (18pts)
assistance on this exam. Furthermore, I agree that
7. (16pts)
I will not share any information about the
questions on this exam with any other student
8. (28pts)
before graded exams are returned.
9. (13pts)
(Date)
10. (9pts)
11. (16pts)
TOTAL
Use the graph of the function, f (x), below to answer the following multiple choice questions. Circle only
one answer for each question. These will be graded as right or wrong. No partial credit will be given.
#1-4 - 2 pts each
1. lim f (x) =
x→−∞
2. lim f (x) =
x→−4
3. lim f (x) =
x→−1
4. lim f (x) =
x→∞
(a) ∞
(a) 0
(a) 0
(a) −∞
(b) −∞
(b) 1
(b) 1
(b) ∞
(c) -2
(c) 2
(c) 2
(c) -1
(d) Does not exist
(d) Does not exist
(d) Does not exist
(d) Does not exist
(e) None of the
above.
(e) None of the
above.
(e) None of the
above.
(e) None of the
above.
5. (5pts) At x = −4, f (x) IS / IS NOT (circle one) continuous.
Using the mathematical definition of continuity, explain why you selected your answer.
6. (5pts) At x = −1, f (x) IS / IS NOT (circle one) continuous.
Using the mathematical definition of continuity, explain why you selected your answer.
7. (16pts) You are 3D printing six-sided dice for a Yahtzee tournament. After some research, you find
out that most dice have a volume of 4 cm3 each. An example of a single Yahtzee dice is illustrated
below. Note that each side length, s, has the same measure.
a
(a) In the blank below, write the function that relates the volume of a single dice, V , and the
dice’s side length, s.
V (s) =
(b) What is the perfect side length? i.e. What side length will result in a volume of 4 cm3 ? Your
answer should be written to 6 decimal places.
s0 =
(c) Due to the nature of your 3D printer, the volume of each dice will be within 0.01 cm3 of 4
cm3 . Algebraically determine the interval around s0 that will ensure that corresponding
output values are within 0.01 cm3 of 4 cm3 . If you use decimals, your answer should be
written to 6 decimal places.
< s0 <
(d) What is the maximum amount of error that can occur on either side of s0 so that the volume
values still lie within 0.01 of 4cm3 ? (i.e. what is the δ value?) If you use decimals, your answer
should be written to 6 decimal places.
Max. Error=
(e) Below is the portion of the graph of V (s) relevant for this problem. Using the graph, label the
following:
(i) V (s) = 4 cm3
(iii) s = s0
3
3
(ii) The interval of ±0.01 cm around 4cm . (iv) The interval you found in part (c).
8. (10pts) Sketch the graph of a function, F (x), that has the following properties:
• lim F (x) = 1
x→−∞
•
•
lim F (x) = ∞
x→−2−
lim F (x) = 3
x→−2+
• F (−2) = 3
• lim− F (x) = 3
x→4
• lim F (x) = −1
x→1
• lim+ F (x) = 4
x→4
• lim F (x) = 5
• F (1) = 3
x→∞
(3pts) Fill-In-The-Blanks: Using the information given above, we are guaranteed that F (x) has a
vertical asymptote at x =
because
lim F (x) =
x→
(6pts) Fill-In-The-Blanks: Using the information given above, we are guaranteed that F (x) has
horizontal asymptotes at y =
and y =
because
lim F (x) =
x→
and
lim F (x) =
x→
(9pts) Using the information given above, all of the known discontinuities of F (x) occur at
x=
(list ALL x-values, there is more than one).
For each x-value you listed in the blank above, state why F (x) is not continuous. Be sure to use
the mathematical definition of continuity.
t2 + 4
.
t→−1 |t + 2|
Below is Dave’s work, which is incorrect.
9. (9pts) Dave computed the limit lim
+ 2)
(t + 2)(t + 2)
(t
+2)(t
t2 + 4
= lim
= lim
lim
=t+2=1
t→−1
t→−1
t→−1 |t + 2|
|t + 2|
|t +
2|
Identify at least three of his errors.
Error 1:
Error 2:
Error 3:
(4pts) Algebraically determine the correct value of the limit. Be sure to show all of your work.
t2 + 4
=
t→−1 |t + 2|
lim
10. The following questions have to do with the limit
lim cos
π x
π .
(a) (2pts) Below is a table of values for f (x) = cos
x
x→0
x
f (x)
0.1
1
0.01
1
0.001
1
0.0001
1
0.00001
1
0.000001
1
Based solely on information from this table, what would you predict as the value of
π lim cos
?
x→0
x
(b) (2pts) Below is a table of values for f (x) = cos
x
f (x)
0.15
-0.5
0.015
-0.5
0.0015
-0.5
0.00015
-0.5
π x
0.000015
-0.5
.
0.0000015
-0.5
Based solely on information from this table, what would you predict as the value of
π lim cos
?
x→0
x
π ? Explain in terms of
x
what it means for a function to have a limit (or not to have a limit) how the tables in parts (a)
and (b) support your conclusion.
(c) (5pts) From the above results, what do you conclude about lim cos
x→0
11. (16pts - 4pts each) Indicate whether each of the following statements is True or False. If the
statement is true, explain how you know it’s true. If it is false, give a counterexample and explain
why it is a counterexample. (A counterexample is an example of a function for which the “if” part
of the statement is true, but the “then” part is false.) A graph with an explanation can be used as
a counterexample.
If you use a term or phrase such as continuity, be sure to state the definition of the term or phrase
that you used.
(a) If lim+ f (x) = 51 = lim− f (x), then f (x) is continuous at x = 3.
x→3
x→3
(b) If lim− g(x) = ∞ and g(17) = 4, then g has a vertical asymptote at x = 17.
x→17
x2 − 1
and g(x) = x − 1, then f (x) is equal to g(x).
(c) If f (x) =
x+1
(d) If lim r(x) does not exist, then r(3) does not exist.
x→3
NAME:
Instructor:
Time your class meets:
Math 160 Calculus for Physical Scientists I
Exam 2
March 12, 2015, 5:00-6:50 pm
“How can it be that mathematics, being after all a product of human thought independent of experience, is
so admirably adapted to the objects of reality?”
-Albert Einstein
1. Turn off your cell phone and other devices (except your calculator).
2. Write your name on every page of the exam. Write your instructor’s name on the cover sheet.
3. You may use a calculator on this exam. You must provide your own calculator; you may not use a
laptop computer or smart phone.
e
4. No notes or other references, including calculator manuals or notes stored in calculator memory,
may be used during this exam.
ea
th
5. Use the back of the facing pages for scratch work and for extra space for solutions. Indicate clearly
when you wish to have work on a facing page read as part of a solution to a problem.
(Signature)
Br
Please do not write in this space.
HONOR PLEDGE
I have not given, received, or used any unauthorized 1-4. (18pts)
assistance on this exam. Furthermore, I agree that
5. (12pts)
I will not share any information about the
questions on this exam with any other student
6. (8pts)
before graded exams are returned.
7. (15pts)
8. (15pts)
9. (12pts)
(Date)
10. (15pts)
TOTAL
Multiple Choice: Circle only one answer for each problem unless stated otherwise. Answers will be
graded as right or wrong with no partial credit.
1. (3pts) If f 0 (7) exists and f 0 (7) = 4, then lim f (x)
x→7
(a) need not exist.
(b) is 4.
(c) is 0.
(d) is f (7).
(e) None of the above.
2. (3pts) A train travels along a straight track. The distance it has traveled after x hours is given by a
function f (x). An engineer is walking along the top of the box cars at the rate of 2.25 mi/hr in the
same direction as the train is moving. The speed of the man relative to the ground is
(a) f (x) + 2.25
(b) f (x) − 2.25
(c) f 0 (x) + 2.25
(d) f 0 (x) − 2.25
(e) None of the above.
3. (8pts) Which of the following statements are true?
(CIRCLE ALL CORRECT RESPONSES)
(a) If f (x) is differentiable at x = 42, then f (x) is also continuous at x = 42.
(b) If f (x) is continuous at x = 42, then f (x) is also differentiable at x = 42.
(c) If f 0 (x) is continuous at x = 42, then f (42) exists.
(d) If f (42) = lim f (x), then f 0 (42) exists.
x→42
f (42 + h) − f (42)
= −2, then f 0 (42) exists.
h→0
h
(e) If lim
f (42 + h) − f (42)
= −2, then f (42) exists.
h→0
h
(f) If lim
(g) If lim f (x) exists, then f (42) exists.
x→42
(h) None of the above statements are true.
4. (4pts) The graph of f (x) = (x + 1)2/3 below has a cusp at x = −1. Which of of the following
mathematical statements are consistent with the behavior of the function at x = −1?
(CIRCLE ALL CORRECT RESPONSES)
(a) lim−
f (−1 + h) − f (−1)
= −1
h
(b) lim+
f (−1 + h) − f (−1)
=1
h
(c) lim−
f (−1 + h) − f (−1)
= −∞
h
(d) lim+
f (−1 + h) − f (−1)
=∞
h
h→0
h→0
h→0
h→0
f (−1 + h) − f (−1)
h→0
h
(e) lim
(f) f 0 (−1) does not exist.
(g) f 0 (−1) does exist.
(h) None of the above.
does not exist.
5. (12pts) Suppose that f (x) denotes a function defined for all real numbers. The statement below is
true sometimes. Give an example of a function for which it holds true and an example of a function
for which it does not hold true. Explain your reasoning. Provide your answers by filling in the table
below:
A function that is not continuous has an absolute maximum.
Example of True
Example of False
Why is the statement true for your example?
Why is the statement false for your example?
6. (8pts) The function of f (x) is represented by the graph below.
Explain in complete sentences in terms of the graph what the equation f 0 (a) = lim
h→0
means. (Be sure to talk about secant lines and the tangent line for the function).
f (a + h) − f (a)
h
7. (15pts - 5pts each) Use the information provided in the table to compute the following derivatives.
Be sure to show all of your work. An answer with no supporting work will receive no credit.
x
f(x)
f ’(x)
g(x)
g’(x)
0
1
2
3
4
2
1
2π
7
51
-1
h(x) = f (x) · sin(πx)
k(x) = g(tan(x))
(a) h0 (2)
(b) k 0 (0)
d
(c)
dx
x
g(x)
x=0
8. (15pts) The graph below is defined by the equation y 2 = 2 sin (πx) + xy
(a) Algebraically verify that the point (−1, −1) lies on the graph above.
(b) In the graph above, draw the line tangent to the curve at the point (−1, −1).
(c) Using calculus, determine
dy
.
dx
(d) Find the equation of the line tangent to the curve above at the point (−1, −1). If you choose
to use decimals, round all values to at least 6 decimal places. Exact form is also acceptable.
9. (a) (12pts) The graphs of a function and its derivatives (f (x), f 0 (x), and f 00 (x)) are shown below.
A fourth, a random graph, is also included. Identify each graph as f (x), f 0 (x), f 00 (x) or
“Random” by filling in the blanks with the correct label.
(1)
(2)
(3)
(4)
Graph 1 =
Graph 2 =
Graph 3 =
Graph 4 =
(b) In 2-4 sentences, explain why the graph you chose as “Random” does not relate to the other
graphs (i.e. was not chosen to be f (x), f 0 (x) or f 00 (x)). Your explanation should include a
discussion of slopes.
10. (15pts) Below is a complete statement of the Mean Value Theorem from Thomas’ Calculus (CSU
edition):
Suppose that y = f (x) is continuous over a closed interval [a, b] and differentiable on the interval’s
interior (a, b). Then there is at least one point c in (a, b) at which
f (b) − f (a)
= f 0 (c)
b−a
Now consider the following problem.
Problem Statement: It took 14 seconds for a mercury thermometer to rise from −19◦ C to 100◦ C
when it was taken from a freezer and placed in boiling water. Show that at some point in time the
mercury was rising at a rate of 8.5◦ C/sec.
(a) In 2-4 sentences, explain why the Mean Value Theorem applies in this context.
(b) Evaluate the following expression in terms of the problem statement:
f (b) − f (a)
b−a
(c) What does the Mean Value Theorem conclude in the context of the problem statement?
NAME:
Instructor:
Time your class meets:
its
Math 160 Calculus for Physical Scientists I
Exam 3
April 16, 2015, 5:00-6:50 pm
im
“How can it be that mathematics, being after all a product of human thought independent of
experience, is so admirably adapted to the objects of reality?”
-Albert Einstein
1. Turn off your cell phone and other devices (except your calculator).
ts
l
2. Write your name on every page of the exam. Write your instructor’s name on the cover sheet.
3. You may use a calculator on this exam. You must provide your own calculator; you may not use
a laptop computer or smart phone.
as
i
4. No notes or other references, including calculator manuals or notes stored in calculator memory,
may be used during this exam.
5. Use the back of the facing pages for scratch work and for extra space for solutions. Indicate
clearly when you wish to have work on a facing page read as part of a solution to a problem.
Ca
(Date)
lcu
(Signature)
lu
sh
HONOR PLEDGE
I have not given, received, or used any
unauthorized assistance on this exam.
Furthermore, I agree that I will not share any
information about the questions on this exam
with any other student before graded exams are
returned.
Please do not write in this space.
1-3. (12pts)
4. (12pts)
5. (12pts)
6. (12pts)
7. (20pts)
8. (20pts)
9. (12pts)
TOTAL
Multiple Choice (12pts - 4pts each): Problems 1 - 3 are multiple choice. Circle only one answer
for each problem unless stated otherwise. Answers will be graded as right or wrong with no partial credit.
1. You want to estimate the area underneath the graph of a positive function by using four
rectangles of equal width. The rectangles that will always give the best estimate of this area are
those with height obtained from the:
(a) Left endpoints
(c) Right endpoints
(b) Midpoints
(d) Not enough information
2. Let f (x) be a polynomial with critical points at x = −3 and x = 4, and let k be a nonzero
constant.
What are the critical points of g(x) = kf (x)?
(a) x = −3 and x = 4
3
4
(b) x = − and x =
k
k
(c) x = 3k and x = 4k
(d) Cannot be determined
Z 3
Z
3. Given that
D(x) dx = 5, what is the value of
1
(a)
(b)
(c)
(d)
(e)
1
(2 − D(x)) dx?
3
-1
1
3
7
Cannot be determined.
4. (12pts) Find the function that satisfies the following properties:
• the second derivative is f 00 (x) = 6x
• the graph of f (x) passes through the point (1, 1) and has a horizontal tangent line at (1, 1)
f (x) =
5. (12pts) The table below provides data about the velocity, v(t), for a new race car recorded by
an automotive testing device during an acceleration test, in which the race car accelerates for 3
seconds.
Time (sec.)
0 0.5 1 1.5 2
2.5
3
Velocity (m/s) 0 3 6.6 9.8 13 16.1 19.1
(a) Using 6 subintervals of equal length, compute an overestimate of the distance covered by
the race car during the acceleration test. Be sure to write out all terms used to do the
computation.
(b) What is the definite integral that would give the exact distance covered by the race car
during the acceleration test? Fill in the blanks below.
Z
(
)d
6. (12pts) True/False: Indicate whether each of the following statements is True or False. If the
statement is true, explain how you know it’s true. If it is false, give a counterexample. (A
counterexample is an example that shows the statement is false.)
(a) If f 0 (7) = 0, the f (x) must have a local maximum or local minimum at x = 7.
(b) If f (x) and g(x) are both decreasing functions, then H(x) = f (x) + g(x) is also a
decreasing function.
7. (20pts) Suppose you own an apartment complex with 100 apartments. Every month, you make
a total profit given by:
P = rq − 80(100 − q) − 18600
where r is the price you charge for rent, and q is the number of rooms rented.
r
(a) If the demand for rooms is given by q = 100 − , use algebra to show that we can simplify
10
the total profit to a function of one variable:
P =−
1 2
r + 92r − 18600
10
(b) You do not want a negative profit (i.e. you want P ≥ 0). Thus, the value of r for
maximizing your profit should lie in the following interval (fill in the blanks):
≤r≤
(c) Find the rent price and quantity of rooms rented that maximize your monthly profit.
Justify/show that this monthly profit is the absolute maximum using calculus.
8. (20pts) Suppose that we have a function, f (t).
Below is the graph of the derivative of f (t) on the interval [0, 5].
(c) Identify the interval(s) on which f (t) is
decreasing:
(d) Identify the interval(s) on which f (t) is
concave up:
(e) Identify the interval(s) on which f (t) is
concave down:
(f) Identify where f (t) has local maxima, if
applicable:
(a) Identify the critical point(s) of f (t):
t=
(b) Identify the interval(s) on which f (t) is
increasing:
t=
(g) Identify where f (t) has local minima, if
applicable:
t=
Suppose that f (t) represents a person’s distance from the largest tree in a jungle. Write a brief
story describing the motion of the person starting at time t = 0 and ending at time t = 5.
A full-credit answer will indicate that you clearly understand the relationship between a
function and an antiderivative.
9. (12pts) Rick Deckard claims that
Z
1
x
+C
dx =
2
(1 − x)
1−x
Indiana Jones claims that
Z
1
x
dx =
+C
2
(1 − x)
1 − x2
Han Solo claims that
Z
1
1
+C
dx =
2
(1 − x)
1−x
Who is correct? (CIRCLE ALL CORRECT STATEMENTS)
I. Rick Deckard is correct
II. Indiana Jones is correct
III. Han Solo is correct
For each statement, explain/show clearly (using calculus) how you know it is correct or
incorrect.