Math 16A HOMEWORK

Homework policies

The homework assignments are due on Fridays at 9am sharp. Late homework can be submitted until the solution set is posted; it will not be graded, but will be corrected (and a score of 1/3 will be awarded if a legitimate effort was made on the assignment).

You are not only allowed but encouraged to work together on homework. However, your submission must be your own writing (or typing) and you must indicate what help you received or what sources (e.g., websites, calc room) you used on the assignment. You may use calculators and/or computers, but you should not need to (except where indicated in the problems). Please err on the side of caution and cite all relevant sources and resources.

There will be about thirty problems a week, mostly from the textbook. I will write about two problems of my own per week, so that you get used to my problem style. They are designated by letters (Problem A, B, etc) and can found below the assignment list, under "Homegrown problems." Below that, you can find solution sets.

Your scores to HW (and quizzes and tests) will be posted on my.ucdavis.edu. You are responsible for checking that the scores online are accurate, and you have one week to notify me of any errors. If you have questions or complaints about the way the HW is graded, bring them to office hours.

Homework assignments

HW # Due Date Topics Homework assignment
HW 1 Friday April 7 coordinates, distance, lines, slope,
functions, inverses
Chapter 1: sec 1 (8,10,14,26), sec 2 (6,14,18,56), Problem A,
sec 3 (12,18,22,23,30,81,88), sec 4 (1,18,22,28,33,34,38,42,50,52),
Problem B
HW 2 Friday April 14limits, continuity, asymptotes Chapter 1: sec 5 (6,22,28,36,44,53,54)
sec 6 (2,8,12,21,24,25,30,37,38,43,46)
Chapter 3: sec 6 (7,8,12,16,17,24,26,28,33), Problems C and D
HW 3 Monday April 24
(extended)
slope of the tangent,
definition of derivative
Chapter 2: sec 1 (2,4,6,10,14,16,17,20,24)
HW 4 Friday April 28 differentiability, trig and exponential review, derivatives continued Chapter 2: sec 1 (32,38,48,50,54,66)
Chapter 8: sec 2 (1,4,12), sec 3 (6,8,59ab)
Chapter 2: sec 2 (15,16,22,24,54), Problems E and F
HW 5 Friday May 5 rules for derivatives, properties of derivatives Chapter 2: sec 3 (6,7,12,32), Chapter 2: sec 4 (8,9,19,20,24,48,56)
Chapter 2: sec 5 (20,24,28,48,59,72), Chapter 8: sec 4 (2,6,7,10,22,24,44)
Chapter 2: sec 6 (10,16,21), Problem G
HW 6 Friday May 12 implicit differentiation Problem H, Chapter 2: sec 7 (1,2,6,16,21,30)
HW 7 Friday May 19 related rates, increasing/decreasing Chapter 2: sec 7 (41,42), sec 8 (13,22,23)
Problems I, J, and K, Chapter 3: sec 1 (5,6,7,8)
HW 8 Friday May 26 critical numbers, maxes and mins, concavity Chapter 3: sec 1 (10,12,18,30,37,38), sec 2 (6,12,20,25,46)
sec 3 (23,24,25,26,28,33), Problem L
HW 9 Monday June 5
(extended)
optimization, various odds & ends Chapter 3: sec 3 (48,64,65) sec 4 (1,2,8,24), sec 5 (2,4) sec 7 (51,52),
Chapter 8: sec 4 (59,60*,62,65,66,88), Problem M
(* is extra-challenging, so that problem is optional.)

Homegrown problems

(note: a few are borrowed, where sources are cited)

Problem A If your IQ starts out at 112 and it rises by 3 points for every day you attend class, write an expression for your IQ in terms of the number of days you attend. Marilyn vos Savant used to be in the Guinness Book of World Records with a listed IQ of 228. Until what date would we have to continue the term before you surpassed her?
Problem B Write a function y=f(x) which is linear (its graph is a line). Find the slope of the line. Find the inverse function-- it should be a line, if f has an inverse-- and its slope. Are they parallel? Perpendicular? Neither? What is the relationship of the slopes, in general, for linear functions that are inverses of each other?
Problem C (i) Come up with an example of a linear function f(x) and a second-degree polynomial g(x) such that the limit as x goes to 1 is equal to 5 in both cases.
(ii) Are the compositions f(g(x)) and g(f(x)) continuous functions? Explain how you know.
Problem D Suppose you are trying to learn a new skill, and your "learning curve" is given by a function L(t), so that L(0) is how much you know at the beginning and L(t) is how much you know at time t. The domain is all numbers greater than or equal to zero, and L(t) is always at least zero as well (since you can't have negative knowledge). Is it possible for L(t) to have a horizontal asymptote? If possible, what would it mean?-- would a horizontal asymptote be good news or bad news?
Problem E Let f(x)=sin x, g(x)=cos x, h(x)=tan x, j(x)=ex, k(x)=ln x.
Give the domain and range of each.
Problem F Draw the graph of f(x)=sin x and draw several tangent lines at various points. Find a value c such that the tangent line to sin x at x=c never intersects the curve in any point except (c, sin c). For that value of c, what is f '(c)? Explain how you know.
Problem G For each function f(x) below,
(a) neatly graph f and f ' for the domain [-2,3] ....... (the use of a computer or graphing calculator will be helpful);
(b) on the same axes, draw the vertical lines x= -1, x=1, and x=2 with dotted lines or in a different color;
(c) mark the points where those lines intersect the graph of f and the graph of f ' ;
(d) for each marked point on the graph of f(x), write the slope at that point;
(e) for each marked point on the graph of f '(x), write the y-value at that point.
The functions are f(x)=x2-x and f(x)=(x+1)*sin(x+1)+3. ....... (The " * " symbol means multiplication.)
Problem H The relationship between Fahrenheit and Celsius temperature is given by F=(9/5)C+32.
(a) Suppose the temperature is changing over time. (This means that both F and C depend on t.) When F=212, what is C?
(b) Take d/dt of both sides of the equation relating F and C. What is dC/dt in terms of dF/dt?
(c) Say the temperature is rising at 5 degrees Fahrenheit per hour. What is the rate of change in degrees Celsius per hour?
Problem I
(Math 106, JHU)
Often, the population levels of different species in the same ecosystem are related. Suppose that you are studying a region where the wolf population is related to the bear population in a way that is modeled by the formula

W + B3 = constant.

At the time of your study, there are 6 bears and 37 wolves in the region. At that time, you estimate that the bear population is increasing at the rate of one bear per year. What is happening to the wolf population?
Problem J
(Math 102, UBC)
One problem often studied by ecologists is the spread of "invading" organisms, applied to everything from squirrels to oak trees. In a classic paper from 1951, J.G. Skellam studied the dispersal of muskrats throughout Europe after their accidental release into the wild by a farmer. (See mentions here and here.)

Skellam modeled the spread of the population by adopting the assumption that the square root of the area occupied by the muskrats was increasing at a constant rate, k. Assume that the area A is circular, and that the center of the circle is the site where the muskrats were released. Let B equal the square root of the area and let r be the radius of the circle. Write down a formula for the relationship between A and B and another for the relationship between A and r.

Find dr/dt, and simplify your answer until it only depends on k and (the number pi).

Problem K
(Math 106, JHU)
There is a relationship between brain function and alcohol level; one way to model it is by considering the number of neurons at a given time, n, and the alcohol level in the blood at a given time, a. Suppose the are related by the formula

na2 + n2ea = constant.

The rate of increase of the number of neurons is given by dn/dt, and the relative rate of increase is given by (1/n)(dn/dt).

Find the relative rate of increase of the number of neurons in your brain if you start out at a=0 but da/dt=2. Once you have your answer, explain in words how to interpret it.

Problem L (a) Is f(x)=cos x increasing, decreasing, or "stable" when x=0?
(b) Give all the intervals on which cos x is decreasing.
(c) Give all inflection points of cos x.
(d) What are all critical points of g(x)= ex? Explain.
Problem M The concentration of a drug in the body varies after the drug is administered: there is an early surge and then the level drops slowly back down towards zero. This is usually modeled by a drug concentration curve of the form C = a t e-bt for some constants a and b that depend on the particular drug.

Suppose that you want to administer a drug called optimazol to a patient who is suffering from a dread disease (like, say, lack of interest in calculus). For this drug, the concentration curve is

C = 14.6 t e- 0.2 t,

where the concentration is measured in nanograms-per-milliliter.

Time is measured in hours. How long after the drug is injected will it reach peak concentration in the patient's blood?

Homework solutions

Most solution sets written by our TA, Jim Matthews.

Solution Set 1 (note: for Problem A, the date works out to June 30 by my calculation)

Solution Set 2 (note: there's a typo in #1.6.30. In the final answer, the discontinuity should be at 4, not 0. The rest is right.)

Solution Set 3

Solution Set 4

Solution Set 5

Solution Set 6

Solution Set 7

Solution Set 8

Solution Set 9


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