Math 020  / Spring 2023

Homework

As stated in the Syllabus:
You are encouraged to start working on any assignment as soon as we have covered some non-trivial part of it in class (i.e., even before it is formally assigned on MyMathLab).
Here is how to know when to stop working on an assignment whose material has not yet been fully covered:
When you see problems that stop matching any of the examples that we have covered - stop.

HW # 1  (Lec. 1: Review - 1)

(All problems except those marked in bold red must be done via MyMathLab (MML)).
Sec. 5.1:  ##  31, 33, 34, 36, 47, 45, 46, 49, 67, 69, 43, 44.

Answer for # 46:   (3/2)t^(2/3) + C
Answer for # 44:   x^3/3 + x^6/6 + C

HW # 2  (Lec. 1: Review - 2)
(All problems except those marked in bold red must be done via MML.)
Sec. 5.2:  ##  9, 23, 11, 13, 17, 25, 33, 34, 19, 65, 27, 31.

Hint for # 31:        Use the identity for  e^(m+n)  from Appendix 5.
-(1/12)/(t^3-2)^4 + C

HW # 3  (Lec. 1: Review - 3)
(All problems except those marked in bold red must be done via MML.)
Sec. 5.5:  ##  23, 25, 35, 41, 45, 47, 71, 89;
Sec. 1.6:  ##  63(c,d), 67, 68.

HW # 4  (Lec. 2: Area between curves - Theory)
(All problems except those marked in bold red must be done via MML.)
Sec. 6.1:  ##  19, 17, 23, 47, 25, 55;  49, 51.
Review Exercises for Chap. 6 (p. 431):  # 23.
Word Problem 1:  Find the area bounded by  y = 2x-4  and  y = x^2 - 3x.
Word Problem 2:  Find the difference of the areas under the given curves over
given intervals:
(a)  y1 = 1 - x^2  and  y2 = x - 1   over  0 <= x <= 2.
(b)
y1 = e^(-x)  and  y2 = x^2   over  -1 <= x <= 2, and (for (b) only)
(c)  y1 = 1/x^2  and  y2 = x^2   over  1/2 <= x <= 2.

Hint for ## 17, 23:   Here f = 0, g = given function; this can be seen from a sketch.
Hint for # 47 for Word Problem 1 (WP 1):
Sketch the two functions. Which one is  f  and which one is  g?
Hint for # 25:           Review Theorem 1 in Sec. 1.6. What is ln(1)?  ln(e)?
Note for WPs 1 and 2:    Do not forget to make sketches in each problem.
Answer for WP 1:    Integral(5x - x^2 - 4) dx from 1 to 4.
Answers for WP 2:   (a)  A1 - A2 = -2/3
(b)  A1 - A2 = -0.42
(c)  A1 - A2 = -9/8

HW # 5  (Lec. 3: Area between curves - Application - 1)
(All problems except those marked in bold red must be done via MML.)
Sec. 6.2:  ##  41, 55, 57,  45, 47, 49, 51, 53.
Word Problem 1:   Follow the steps of the class Ex. 1b and obtain the General Formula
stated right before Ex. 2.
Word Problem 2:   Reconsider Ex. 1b with  r=0.04  (i.e., half of the rate in Ex. 1).
Do you *think* that the Total Accumulated Interest (TAI) will be
equal, greater, or less than half of the TAI found in Ex. 1b for  r=0.08?
Now, use the General Formula derived in WP 1 and *compute*

Hint for ## 41, 55, 57:   Use a formula stated at the very end of topic 1.
For your own good, I recommend that you sketch  f(t)
just to make sure that it satisfies the conditions stated in the problem.
Note for # 55:               First, find the equation for f(t) as a linear function of t.
Note for # 53:               A bond is a single deposit (made at t=0), which, along with
interest, is withdrawn at the maturity date of the bond.

HW # 6  (Lec. 3: Area between curves - Application - 2)
(All problems except those marked in bold red must be done via MML.)
Review Exercises for Chap. 6:   # 17;
Sec. 6.1:   ##  83, 85, 86;
Sec. 5.5:   ##  76, 75.
Word Problem:   For each of ## 76, 75, answer this additional question:
How much money will the owner of the product *lose* if they keep
operating the product for one additional year past the Useful Life (UL)?
(In both cases, you need to use an integer number of years for UL.)

Note for all problems:    Present your answer as a decimal with three decimal places.
Note for # 85:               The last sentence has a typo: "income" should be "assets".
Notes for # 75:              1) You should first *sketch* C' and R'.
2) You should do one of the integrals following Ex. 4a of Lec. 1.
Answers for # 76:          UL = 10*ln(5) = (approximately) 16;  Total profit = \$71,716
Answers for WP:           # 76:  \$118;     # 75:  -( -5/2*(e^(-9) - e^(-4)) - 1/22*(3-2) ) = \$182

HW # 7  (Lec. 4: Integration By Parts - 1 ( integrals of x*e^x ) )
(All problems except those marked in bold red must be done via MML.)
Sec. 6.3:   ##  9, 15, 41, 19, 16, 47;  75, 77, 78, 88.

Note for ## 19, 16:   Do this By Parts "in one piece," as in a class Example; do NOT expand.
Hints for # 75:          1) Use a relevant formula from the end of topic 1 of Lec. 3.
2) Follow the lines of an Example done in topic 2 of Lec. 4.
Note for # 77:          You must have already done a very similar integral in this HW.
Note for # 78:          Find a very similar integral done in class; then follow the lines of that Example.
Note for # 88:          1) Recall that the total change of any quantity equals the integral of the
rate of change of that quantity over a given time period.
2) You must have already done a more general case of this integral in this HW;
all that remains to be done is to substitute the bounds.
Answer for # 16:     -(x-1)*e^(-x) - e^(-x) + C
Answer for # 78:     4/e - 1 = 0.47
Answer for # 88:     25 - 75/e^2

HW # 8  (Lec. 4: Integration By Parts - 2 ( integrals of x*ln(x) ) )
(All problems except those marked in bold red must be done via MML.)
Sec. 6.3:   ##  11, 28, 21, 42, 43, 45, 54, 55, 57.

Hint for # 45:         Follow the method used when the integrand contains  ln(ax).
Hint for # 57:         You first need to use an identity for ln(x^m). Find it in Theorem 1 of Sec. 1.6.
Answer for # 28:    Enter  integrate ln(x)/sqrt(x) dx  into a search engine.
Answer for # 42:    Enter  integrate ln(a*x) dx  into a search engine.
Hint for # 54:         Use the search engine. The syntax for the square of the log is (ln(x))^2.

HW # 9  (Lec. 4: Integration - Choose the Right Method ) )
(All problems except those marked in bold red must be done via MML.)
The purpose of this HW is to make you decide which integration method will work in each case:
(i)   u-substitution;
(ii)  integration by parts, or
(iii) using the properties of logarithm and then using one of the integration rules R1--R3 reviewed in Lec. 1.
Properties of logarithm are found in Theorem 1 of Sec. 1.6.
Sec. 6.3:   ##  17, 39, 1043, 25, 53, 57, 59, 63.

Hint for # 10:           If unsure, first use a search engine to get an answer and then try to remember
which integration method gives answers of that form.
Note for ## 53, 57:   Remember that  ln(x^2)  is not at all the same as  (ln(x))^2.

HW # 10  (Lec. 6: Improper integrals -  Exponentials)
(All problems except those marked in bold red must be done via MML.)
Sec. 11.1:   ##  13, 23, 28, 55, 61, 63, 70.

Answer for # 28:   Diverges at -Infinity

HW # 11  (Lec. 6: Improper integrals -  Powers of x)
(All problems except those marked in bold red must be done via MML.)
Sec. 11.1:   ##  9, 11, 14, 15, 17, 19, 20, 21, 22, 57, 69, 68(a).

Hint/Answer for # 14:   See posted page 6-9. The results stated there are valid also if the
lower integration bound equals -Infinity.
Hint/Answer for # 20:   See posted page 6-9.
Hint for # 57:               First, do a u-substitution; then do an improper integral with this  u.
Answer for # 68(a):      10 mln barrels

HW # 12  (Lec. 7: Probability density function - 1)
(All problems except those marked in bold red must be done via MML.)
Sec. 11.2:   ##  11, 12, 13, 14, 25, 41, 43, 47.

Answer for # 14:   no  (plot the PDF if you are unsure of this answer)
Note for # 25:       "The first two conditions of the PDF" referred to in the problem statement
are those listed on p. 630. In the Notes, they are Properties 1 and 3 on p. 7-7.
Hint for ## 41, 43:   (k*f) must satisfy one of the Properties of the PDF.
Note for # 47:        Make sure that you record your answers in your notebook where you do HW.
These answers will be important when we cover sec. 11.4.

HW # 13  (Lec. 7: Probability density function - 2)
(All problems except those marked in bold red must be done via MML.)
Sec. 11.2:   ##  15, 17, 27, 55, 57, 61;  45, 49, 19, 21, 23, 29, 31, 33, 37, 59, 63.

Note for ## 45, 49:      Review the properties of CDiF.
Note for ## 21, 31:      You need to do these problems using the CDiF.
Hint for # 31(B):         P( X <= x) + P(X >= x) = 1.
Clarification for # 33: Graph both the PDF and CDiF.
Hint for # 59:               To integrate the PDF, use a u-substitution.
Hint for ## 59, 63:       Part (C) in these problems requires you to first find the CDiF.
Review Ex. 4 in the posted Lecture 7.

HW # 14  (Lec. 8: Mean, Variance, Median - 1)
(All problems except those marked in bold red must be done via MML.)
Sec. 11.3:   ## 5, 17, 19, 13, 14, 49, 51, 35.

Hint for # 13:          For the "prediction" part, think of your intuitive definition of "average".
You may also consult the Geometric interpretation of mean on p. 8-9.
Hint for # 14:          Variance and Standard deviation characterize the same property of a
random variable (Standard deviation is just the square root of variance).
So, what property is that? Find the answer on pp. 8-3,4,
Hint for # 49:          We computed a similar integral near the end of Lec. 7.
Note for # 35:
This problem will become important when we study Sec. 11.4.
Hint for # 35:          How do we integrate a product of two functions? Review topic 1 of Lec. 1
(i.e., posted pages 1-1 through 1-4).

HW # 15  (Lec. 8: Mean, Variance, Median - 2)
(All problems except those marked in bold red must be done via MML.)
Sec. 11.3:   ## 7, 11, 27, 29, 23, 45, 47, 53; 37, 38, 39.

Note for # 23:        What are  ln(1)  and  ln(e)? See Theorem 1 in Sec. 1.6 if you forgot.
Note for # 47:        You have already done a similar problem in this assignment.
Note for # 53:        See the Note for # 49 in the previous HW.
Note for ## 37-39:  The concept of quartiles is somewhat similar to the concept of median,
and quartiles are used in many applications. To clarify:
the probability that X is between  min(x)  and  x1  is 0.25;
the probability that X is between  x1  and  x2  is 0.25;
the probability that X is between  x2  and  x3  is 0.25;
the probability that X is between  x3  and  max(x)  is 0.25.
So, what is the probability that X is below x1? below x2? below x3?

HW # 16  (Lec. 9: Exponential and Normal PDFs - 1 (Exponential and one other))
(All problems except those marked in bold red must be done via MML.)
Sec. 11.4:   ## 3, 39, 71, 73, 79, 65, 66.

Notes for ## 65, 66:  1) Make sure to read a Preamble found just before these problems.
Answer for # 66:      Sqrt(p/(p-2))/(p-1); so must have p>2.

HW # 17  (Lec. 9: Exponential and Normal PDFs - 2 (Normal, Numeric problems))
(All problems except those marked in bold red must be done via MML.)
Sec. 11.4:   ## 9, 17, 19, 27, 29, 31, 47, 49, 51, 53, 55--58.

Note for ## 19, 31, 51, 53:   Review Ex. 4 in the Notes.
Note for ## 55--57:              Find the answers conspicously displayed in the Notes.
Answer for # 58:                  No; again, find the explanation in the same place in the Notes.

HW # 18  (Lec. 9: Exponential and Normal PDFs - 3 (Normal, Word problems))
(All problems except those marked in bold red must be done via MML.)
Sec. 11.4:   ## 75, 77, 78, 81, 82, and the reworded # 85 below.
Reworded # 85:
The instructor grades a test on a curve while assuming that the grades are normally distributed.
The instructor further determines that the average grade for the test is 70 and the standard deviation
is 8. The instructor wants to set grade brackets in such a way that (approximately) 10% of the grades
and A's, 20% are B's, 40% are C's, 20% are D's, and 10% are F's. What should these grade brackets
be? Round the answers to the nearest integer.
(I think I have made it somewhat clearer than in the book, but feel free to read
the book's version as well. Numbers in both versions are the same.)

Note for most of problems:    Review Ex. 4 in the Notes.
Hint for # 85:                        In all previous problems, you have first found  z  and then, G(z)
from the Table in Appendix C. Here, you will work backwards.
That is:
1) Determine the probability that you want. For example, for the
threshold between C and either B or D, you will take 0.2. (Why?)
And for the threshold between B and A you will take 0.4. (Why?)
2) Find the closest decimal for that probability in the Table; e.g.,
for P = 0.2, it appears to be 0.1985. Then determine which  z
it corresponds to.
3) Using the equation  (s - mu)/sigma = z,  find the score  s.

HW # 19  (Lec. 10: Differential equations - 1 (Verifying solution, finding "C"))
(All problems except those marked in bold red must be done via MML.)
Sec. 9.1:   ## 12, 13, 17, 19, 20,  29, 31, 32, 33.

Answer for # 32:       C = 0

HW # 20  (Lec. 10: Differential equations - 2 (Setting up a DE; Graphing a solution))
(All problems except those marked in bold red must be done via MML.)
Sec. 9.1:   ##
51, 52, 53,  47, 49.

Answer for # 52:    dP/dt = -k*P, where k is some positive constant (why is it positive?)
Note for # 53:        Think whether your proportionality constant should be positive or negative.
For that, recall that the pizza is hotter than the room and decide whether
the solution to your DE should increase or decrease. Review the bullet point
right before topic 3 on p. 10-5.
Note for ## 47,49:  You must review the graphs of e^{ax} for a>0 and a<0 and know them cold!
See Sec. 1.5 or the first page of Lecture 6, for example. Or just find them online.

HW # 21  (Lec. 10: Differential equations - 2 (Setting up a DE; Graphing a solution))
(All problems except those marked in bold red must be done via MML.)
Sec. 9.1:   ## 47, 49, 65, 63, 64.

Note for all problems:   You must do each of them twice:
The first time by the method illustrated in
in Ex. 6(a) and 7(a) in the Notes, and
the other time, by the method illustrated in Ex. 6(b) and 7(b).
And yes, you did ## 47, 49 (by the former method) in HW 20,
but practice makes perfect.
Partial answer for # 64(B):     The respective C values are -1000 and 1000.

HW # 22  (Lec. 11: Separable Differential equations - 1 (Method of solution))
(All problems except those marked in bold red must be done via MML.)
Sec. 9.2:   ## 9, 10, 13, 16;  17, 18, 19, 20;  25, 26, 27, 29;  31, 38, 37.

Note for ## 9, 10, 13, 16:   Review topic 4 of Lecture 10.
Note for ## 17 -- 20:          Review topic 0 of Lecture 11.
Note for # 37:                    First write this as an explicitly separable DE.
Answer for # 10:                dS/dt = k*S,  k>0
Answer for # 16:                The amount of drug is decreasing proportionally to
the amount of it left, and initially there is 5 ml of drug.
Answer for # 18:                y' = x*(2-y)
Answer for # 20:                y' = (2-x)/y
Answer for # 26:                y = 10 + C*e^t, C=5
Answer for # 38:                y = 3 + C*e^{x^2+4x}, C = -1

HW # 23  (Lec. 11: Separable Differential equations - 2 (Word problems))
(All problems except those marked in bold red must be done via MML.)
Sec. 9.2:   ## 11, 15;  59, 75, 63, 65;  71, 72, 73, 74, 68, 82.

Notes for # 65:                      1) Set t=0 in 2010.
2) The rate that the last question of the problem asks for is,
of course, given by the r.h.s. of your differential equation.
Note for ## 71 -- 82 and 68:   All these problems are on the Limited growth/decay model.
So review the corresponding topics in the Lecture.
Hint for # 71:                        What temperature should you include in the r.h.s. of your DE:
500F or 800F, so that it would be the counterpart of the 72F
in the DE of Example 6?
Notes for # 72:                      1) Similarly, here, what temperature should be the counterpart
of the 72F from Example 6?
2) Once you decide on the above question, set up the model
taking into account the fact that the bar is now cooling down
(as opposed to heating).
Notes for # 68:                       1) Set t=0 at the "two years ago" mark.
2) Disregard the question about the rate and its interpretation,
found towards the end of the problem.
Answer for # 72:                    k = 0.050, t = 43.7 min
Answer for # 68:                    S(3) = 5.32M,  9.0 years
Answer for # 82:                    (A) 61 words;  (B) 48 hours.

HW # 24  (Lec. 12: Linear differential equations, with application to Mixing)
(All problems except those marked in bold red must be done via MML.)
Sec. 9.3:   ## 23, 25, 26, 27, 29, 35;   69, 71, 72.
Word Problems:
1.   Mortgage repayment problems
(A) Repeat steps of the solution of Ex. 7 in Lecture 11 for the case where the  APR = r = 4.5%
(i.e., 25% below that in Ex 7). Find the total amount you will pay to the bank.
Denote M[r] the total amount you pay to the bank with the APR = r.
Compare  (M[4.5%] - P0)  with  (M[6%] - P0),  where P0 is the principal (same as in Ex. 7).
Is the former 25% smaller than the latter? Or more than that? Or less than that?
(B) Repeat the assignment of part (A) for r = 7.5% (i.e., 25% higher than that in Ex. 7).
Answer the same questions as in part (A).
2.   Pollution remediation
(A) Repeat # 69 assuming that the incoming water is fresh (has no pollutants). How many pounds of
pollutants are in the tank after 2 hours?
(B) Repeat # 71
assuming that the incoming water is fresh (has no pollutants). How many pounds of
pollutants are in the tank after 2 hours?

Notes for all problems in Sec. 9.3:    Review relevant Rules from Theorem 1 in Appendix A.5.

Answer for # 26:                              e^{3x} + C e^{2x}, C=1
Notes for ## 69, 71, 72 and WP2:    1) The capacity of the tank (1000 gal) is not used in these problems.
2) Recall that the phrase "rate (in pounds per gallon)" at the end
concentration, not rate, of pullutants.
3) It may look like asking you to do two word problems, WP2,
has significantly increased the amount of work that you need to do.
Well, it has not. You should realize it soon after you begin doing
WP2(A).
Hint for WPs 1:                               Find a formula in Ex. 7 that gives  m  as a function of  r.
Answer for # 72:                              m = 300(t+2) - 200/Sqrt[1+(t/2)];  concentr = m/(200+100t)
Answer for WP2(A):                        m = 400/(1 + (t/8))^2;
concentr = m/(200+25t)
Answer for WP2(B):                        m = 400*e^{-t/4};   concentr = m/200

HW # 25  (Lec. 13: Functions of several variables, and Visualization of functions of 2 variables)
(All problems except those marked in bold red must be done via MML.)
Sec. 7.1:   ## 41;  74, 72(A,B), 66, 67, 68;  57, 58;  33, 34, 62(A,B).

Answer for # 74:     8k,  2500k
Answer for # 66:     5*9*8,  5*4*27
Notes for # 68:        1) The relation between Revenue, Cost and Profit was considered
in one of the class examples.
2) Unlike # 67 and a similar Example considered in class, here all quantities
are expressed not in terms of prices, but in terms of the demands of bicycles
(e.g., x=12 means that 12 ten-speed bicycles have been "demanded," i.e., sold).
Answer for # 68:      p=155, q=80, R=2750, P=1300
Answer for # 58:      B=(2,0,0), H=(0,4,3)
Answer for # 34:      25-x-10+12x = 15+11x
Answer for # 62:      (A)  f(x) = const + 10x - x^2, these are downward-facing parabolas
(B)  f(y) = const + 25y - 5y^2
, these are also downward-facing parabolas

HW # 26  (Lec. 14: Partial derivatives)
(All problems except those marked in bold red must be done via MML.)
Sec. 7.2:   ## 19, 20, 21, 22, 25, 27, 29;  85, 87 (plus additional assignment), 88, 96, 98
39, 40, 41, 42, 43, 44, 71, 73.
Find the Revenue and compute R_p(10,10) and R_q(10,10) and interpret the results.

Note for # 88:         JFYI, unlike in # 87 and class Examples, here the variables are not the prices of,
but demands for, the two products.
Note for ## 96, 98:  Review the Rule of thumb on p. 14-7. In view of it, you will not need to substitute

Answer for # 20:     2x - 14y
Answer for # 22:     4x^2 - 10xy
R_p(10,10) = 160 =>
increasing the price of brand A by from \$10/lb to \$11/lb while keeping the price
of brand B fixed at \$10/lb will increase the revenue by \$160;
R_q(10,10) = 280 =>

increasing the price of brand B by from \$10/lb to \$11/lb while keeping the price
of brand A fixed at \$10/lb will increase the revenue by \$280
.
Answer for # 88:     R_x(10,5) = 60 =>

an increase in the demand of 10-speed bikes from 10 to 11 per day while
the demand of 3-speed bikes stays at 5 per day will increase the revenue by \$60;
P_x(10,5) = -20 =>
the above action will decrease the profit by \$20.
Answer for # 96:      Find it in a class Example.
Answer for # 98:      L_w = k*v^2 =>
increasing the weight of the car by 1% will increase the skid length by 1%;

L_v = 2k*w*v =>

increasing the speed of the car by 1% will increase the skid length by 2%.

HW # 27  (Lec. 15: Optimization problems (a.k.a. maxima and minima of functions of 2 variables))
(All problems except those marked in bold red must be done via MML.)
Sec. 7.3:   ## 41, 44(B), Matched Problem (p. 459) for Example 4 (p. 458), # 47.
Word problems:
1.   In # 87 of Sec. 7.2 (see HW 26), find the prices of brands A and B that maximize the revenue.
2.   In a laboratory test, antibiotics A and B are used together to eliminate certain disease-causing bacteria.
The rate of elimination of the targeted bacteria  f(x,y) = xy - 2x^2 - y^2 + 110x + 60y, where
x  and  are amounts of antibiotics A and B in milligrams. Find the amounts of the two medicines
that will maximize the antibiotic effect (i.e., maximize the rate  f(x,y)  of bacteria elimination).
3.   Subjects in a psychology experiment who practice for their Calculus exam for  x  hours and then
rest for  y  hours are found to achive an exam score of
f(x,y) = 2xy - 4x^2 - y^2 + 22x - 4y + 65
(on average). Find the numbers of hours of practice and rest that would maximize the expected
exam score.

General note for all problems:
You must perform the 2nd-derivative test to verify that the solution you found is indeed a maximum
or a minimum, as required in each problem.

Answer for # 44(B):   p = 40,  q = 50,  max P = \$1200
Answer for the Matched problem for Example 4 is found on p. 461
Answer for WP1:      p = 850/11 = 77.27,  q = 1050/11 = 95.45
Answer for WP2:      x = 40, y = 50
Answer for WP3:      3 hours of practice and 1 hour of rest will give a maximum score of 96 points