Math 231 B, C Homework assignments

Homework assignments for Math 231 B, C, Fall, 2009.

Back to Course Homepage Please refresh page to get most recent information. Section number and page references are to the Stewart text unless otherwise indicated.

Grading scheme: on each homework out of 10 points, the grade will be assigned as follows:

10 - almost perfect work
9 - about four fifths of the work is correct
8 - about three fifths of the work is correct
7 - about half of the work is correct
1-5 - less than half the work is correct (as judged by the grader)

On homeworks that are out of 20 points, just double the above guidelines.
Always look at what the grader marked wrong, and try to learn how to do that problem correctly (by yourself, from friends, or by talking to a TA at office hours).

INSTRUCTIONS.
Put a box around each answer, on every homework this semester. (This helps the graders do their job.) But don't box graphs or "proofs".
Please staple your homework - loose sheets are easily lost, and make the grader's job more difficult.
On Thursdays when you turn in homework for the preceding Friday and Monday, regard it all as one homework assignment and staple it all together. The assignment will be worth 20 points instead of 10, when it is based on two lecture days.

# Assigned Due Homework Problems

24A Fri, 11/20 Thu, 12/3 Not to turn in.
11.10: 45
To turn in.
11.10: 28, 36, 37 (comment: notice the Taylor series does not help us see the nonnegativity of sin^2), 50, 58, 72 (just go up to e⁴, not e⁶)(warning: you might prefer to write epsilon instead of e, in this problem, because "e" here is not the usual number 2.71828...)
11.11: 25, 36 (conclusion: the period of a pendulum is almost independent of the amplitude; e.g. the period for amplitude 42 degrees and the period for amplitude 10 degrees differ by only a few percent)

23 Wed, 11/18 Tue, 12/1 Not to turn in.
11.9: 1, 2, 35a
To turn in.
11.9: 4, 8, 28 (hint: see Example 8),
32 (do it two ways: 1. differentiate the series and substitute into the differential equation, 2. recognize the series as a Taylor series of some function f and substitute that function f into the differential equation),
38ab
11.10: 2, 15, 16
11.11: 33 (substitute x=d/D and find the first few terms of the Taylor series), 35ab

22B Mon, 11/16 Thu, 11/19 Not to turn in.
11.7: Read this section - it summarizes how to test series for convergence
11.8: 1, 2a, 34
To turn in.
11.6: 2, 7, 10, 20
11.8: 4, 7, 14, 21, 32a
11.8: Optional, no credit 40

22A Fri, 11/13 Thu, 11/19 Not to turn in.
11.5: 1, 3
To turn in.
11.5: 4, 8 (how will you show the terms are decreasing, that is b_n+1 < b_n?), 16, 18, 24, 30, 31

21 Wed, 11/11 Tue, 11/17 Not to turn in.
11.4: 1, 2
To turn in.
11.3: 32b, 34, 39
When using the Integral Test, remember to show that f(x) is decreasing. A graph is helpful, but not enough by itself: you should show the derivative f '(x) is negative.
11.4: 4, 6, 12, 27, 28, 29, 37
Show your reasoning, for every problem!

20B Fri, 11/6 Thu, 11/12 Not to turn in.
11.3: 1
To turn in.
11.3: 4, 6, 7, 12, 19 (hint: you need not evaluate the integral - you simply need to show it is finite, i.e. that it converges), 27 (here p>0)
In order to use the Integral Test to prove convergence of a series, in the above problems, you must show that the function f(x) is positive-valued and decreasing when x is large. Usually we consider x > 1, but you may consider x > 2 or x > 1000: we use whatever works for the function under consideration.
To show f(x) is decreasing, a graph is helpful - but it is not enough by itself. You should show the derivative f '(x) is negative.

20A Wed, 11/4 Thu, 11/12 Not to turn in.
11.2: 1, 35
To turn in.
11.1: 65
11.2: 8, 9, 11, 25, 42 (hint: express the recurring decimal as a geometric series), 48, 60

19B Mon, 11/2 No hw to turn in. Do the following problems (mostly odd-numbered, so that answers are in the textbook).
17.3: 1 (see Example 1 on page 1126)
11.1: 7, 9, 10, 17, 19, 27, 36, 39, 58
4.8: 1, 15 (keep going until x_n and x_{n+1} agree to 6 decimal places)

19A Fri, 10/30 No hw to turn in. Do the following problems (mostly odd-numbered, so that answers are in the textbook).
9.2: 27b (similar to equation solved in class)
9.4: 7 and sketch y(t) (let t=0 be the time 8am), 8 and sketch P(t) [answer for 8a: P(t) = 10000/(1+24(11/36)^t); answer for 8b: 2.7 years]
9.5: 27 and sketch I(t) (similar to equation solved in class), 29 and sketch Q(t) and I(t)
Ch. 9 Review: 20 [answer: R=AS^k], 21

18 Wed, 10/28 Tue, 11/3 Not to turn in.
9.3: 1
To turn in.
(i) Solve dP/dt = -kt. Sketch a typical solution. (Here k > 0 is constant.)
(ii) Solve dP/dt = -kP. Sketch a typical solution. (Here k > 0 is constant.)
[Moral: we need to pay attention to whether it is the independent variable t or the dependent variable P, on the right side of the differential equation!]
9.1: 3, 13 (the training begins at t=0)
9.3:
11 (solve for y, then use the initial condition to determine the constant of integration),
35 (the last part means: what is the limit as t tends to infinity?)(answer in book is wrong: should be M-Ce^-kt),
38,
40 (hint: in part (a) you need to find a differential equation of the form dx/dt = ...? Ask yourself: if dt = 1 day, then how much is dx? i.e. how much new currency will be put into circulation that day? Notice dx will equal some fraction of 50 million.)

17B Mon, 10/26 Thu, 10/29 To turn in.
7.4: 60
7.5: 2, 6, 18, 28
Appendix G: 4 (hint: approximate the area under the curve)

17A Fri, 10/23 Thu, 10/29 To turn in.
7.4: 8, 14 (assume a does not equal b), 22, 32, 48
Hint on 7.4 # 14. You could multiply out the denominator and complete the square, and then integrate. But the denominator is already given to us as a product of linear factors, and so a simpler method is to express the integrand as C/(x+a) + D/(x+b) and figure out the constants C and D, and then integrate.
Hint on 7.4 # 22. The denominator is given to us as a product of linear factors, and so the simplest method is to express the integrand as C/s + D/s² + E/(s-1) + F/(s-1)² and figure out the constants C,D,E,F, and then integrate.
7.8:
For each of the following improper integrals, determine whether the value of the integral is finite (in which case we say the integral "converges"), or infinite (we say the integral "diverges"), or does not exist for some reason.
8, 16, 20, 28, 30, 32, 40, 47b
Hints
(i) To show the integral of a positive function is finite, you may either evaluate the integral, or else you may show it is less than or equal to some other integral that is finite.
(ii) To show the integral of a positive function is infinite, you may either evaluate the integral, or else you may show it is greater than or equal to some other integral that is infinite.
7.8:
58

16 Wed, 10/21 Tue, 10/27 To turn in.
Problem A. Show that the centroid of a triangular lamina (plate of constant density) lies at the average of the vertices. That is, write the vertices of the triangle as (a₁,b₁), (a₂,b₂), (a₃,b₃), and show that the center of mass of the lamina is ((a₁+a₂+a₃)/3,(b₁+b₂+ b₃)/3). Hint. By a translation and rotation of the lamina, you may assume it has one vertex at the origin and that the opposite side is vertical: a₁=0, b₁=0, 0 < a₂=a₃, b₂ < b₃. Sketch this triangle and find its centroid, then show the centroid equals the average of the vertices.
Problem B. Use the Theorem of Pappus to find the centroid of the right triangle with vertices at (0,0), (r,0) and (0,h). Then check your answer by using Problem A to find the centroid.
8.3: 44
8.3: 2, 5 (you can use integrals from the back of the book), 8, 10. For these problems, do not approximate with Riemann sums, and do not use the method in the book. Instead use the result from class that:
hydrostatic force perpendicular to the plate = (density of fluid)(g)(area)(depth of the centroid).
Remember only to take the centroid of the part of the plate that is under water, since that is the part that experiences pressure from the water.
The density of water at room temperature is 1000 kg/m³. (The metric system is set up rather nicely, don't you think?!)

15B Mon, 10/19 Thu, 10/22 To turn in.
8.3: 22 (and sketch the 3 points and the centroid), 24 (just find the centroid, not the moments; then sketch the 4 points and the centroid), 28, 29 (hint: why would you expect the coordinates x-bar and y-bar of the centroid to be equal?), 34, 48 (in part c, please use m=4, n=3), page 550 Discovery Project parts 1 and 2 (hint for part 2: kh=...?)

15A Fri, 10/16 Thu, 10/22 To turn in.
8.2: 28, 30 (hint: see the diagram for 6.2 #63 on page 432; check your answer in the special case R=r: you should get the same answer as the back of the book for 8.2 #33)
6.5: 6, 10, 19, 20 (hint: first express v as a function of t; then express v as a function of s)

14 Wed, 10/14 Tue, 10/20 Not to turn in.
8.1: 1
8.2: 1
To turn in.
8.1: 6 (here a>0 and b>0 are constants, called the "semi-axes" of the ellipse), 10, 12 (you may use integrals from the back of the textbook), 40a (40b is optional, for no credit)
8.2: 8, 12, 14, 25, 34 (you may assume the planes are horizontal; or vertical if you prefer)

13B Fri, 10/9 Thu, 10/15 To turn in.
7.2: 44 (see page 465)
6.4: 14, 17 (note the hint online at "Tools for Enriching Calculus"), 21 (the density of water is rho=1000 kg/m3), 29

13A Wed, 10/7 Thu, 10/15 To turn in.
7.2: 8, 12, 20 (double angle formula!), 66 (the average value of a function is defined in Sec. 6.5; in this problem, one cycle means 0 < t < 1/60)
7.3: 2 (instead of doing the indefinite integral, make it a definite integral from 0 to 3)(Correction: not from 0 to Pi/2 as earlier written), 20 (do the integral two ways: (i) u-substitution, and (ii) trig substitution), 38 (do not change the limits of integration on this problem when you substitute; instead, after substituting and integrating you can substitute back to the x-variable, and then put in the original limits of integration; note that if tan(theta)=x/b then sin(theta)=x/(x²+ b²)^1/2 by drawing the triangle...)

12B Mon, 10/5 No homework to turn in! Do the following problems (mostly odd-numbered, so that answers are in the textbook).
6.3: 1, 7, 15, 19, 25, 37 and 38 (just set the integral up: decide whether to use washers or shells), 43
7.1: 1, 5, 7, 11, 17, 21, 33, 35, 37, 41, 59
For extra practice: 7.1 # 15, 48, 57

12A Fri, 10/2 No homework to turn in! Do the following problems (all odd-numbered, so that answers are in the textbook).
6.1: 3 (hint: read p. 419), 31 (hint: split the interval into two intervals), 49
6.2: 3, 9, 11, 29, 33, 35, 41, 45, 51, 59
Hint for the solid of revolution problems: sketch the original region, sketch it rotated it about the given axis, sketch a cross-sectional "slab" or "slice" and work out its area, then obtain the volume of the solid from the integral of A(x) dx or A(y) dy. See the summary on page 427.

11 Wed, 9/30 Tue, 10/6 Required Reading
1. Learn the antiderivatives ("indefinite integrals") in the Table on page 392.
2. Learn the Integrals of Even and Odd Functions (pages 405-406, especially Figure 4; recall page 19 defines Even and Odd functions). These formulas only work when the interval is symmetric, from -a to a.
Not to turn in.
5.4: 11, 17, 55
To turn in.
5.4: 48, 50, 53
5.5: 8, 48 (do not turn in the graph), 64, 68, 70, 74
Evaluate the following integrals, using Oddness or Evenness:
(a) integral_{-pi}^{pi} sin(jx) cos(kx) dx where j and k are constants
(b) integral_{-1}^1 (x-1) (1-x^2)^{1/2} dx
(c) integral_{-infinity}^{infinity} x e^{-x^2} dx
(d) integral_{-infinity}^{infinity} |x| e^{-x^2} dx

10B Mon, 9/28 Thu, 10/1 Not to turn in.
5.3: 23, 37, 51
To turn in.
5.3: 32, 34, 38, 40, 42, 46, 56 (see hint for 53), 59, 64, 70

10A Fri, 9/25 Thu, 10/1 To turn in.
5.2: 52, 53, 66 (hint: |ab|=|a| |b|)
5.3: 4 (first read the hints for #3, at Stewart's website; and print copies of the graph), 8, 10 (the variable is r; the dummy variable of integration is x), 12, 13 (hint: h(x)=g(1/x)), 16, 60
Optional (no credit) 5.3: 68

9 Wed, 9/23 Tue, 9/29 Not to turn in.
5.1: 13
5.2: 39
To turn in.
5.1: 2 (print 3 copies of the graph), 12, 14, 20, 22
5.2: 30, 34, 36, 70

8B Mon, 9/21 Thu, 9/24 To turn in.
4.7: 4, 7, 37 (hint: write h and r for the height and radius of the cone), 47, 56, 62, 72.
General hint. To find a global maximum, it is not enough just to check that the first derivative equals zero there. You also need to check the value of f at any other critical points and at the endpoints of the interval; or else use the positivity and negativity of the derivative on the left and right sides of the max point, to help you conclude that you have really found the maximum.

8A Wed, 9/16 Thu, 9/24 To turn in.
4.1: 3, 73 (remember to check the endpoints of the interval)
4.2: (Problems on Mean Value Theorem) 21a, 22, 26, 27 (do it using 26; of course, another solution is to square both sides), 29 (you can assume a < b), 36 (and sketch the graph of such an f on the same axes as the function g(x)=x; indicate the fixed point)

7B Mon, 9/14 To do, but not turn in.
3.10: 11a, 13b, 35a, 38 (check your work by estimating the error directly, using 29 degrees and 31 degrees), 39
No homework to turn in! Answers are here.

7A Fri, 9/11 To do, but not turn in.
3.5: 13, 40, 69
3.6: 40
3.9: read the instructions (a)-(e) at the bottom left of page 245; then do 12, 16, 27, 33
No homework to turn in! Answers are here.

6 Wed, 9/09 Tue, 9/15 To do, but not turn in.
3.4: 2, 27
To turn in.
3.4: 10, 22, 28 (here y=tanh(u) and you want dy/du), 32, 46, 54, 74, 84

5 Fri, 9/04 Thu, 9/10 To turn in.
4.4: # 52, 60, 66 (do not use L'Hospital's Rule on these three problems; instead use Taylor series)
Problem: Compute the fourth derivative at x=0 of f(x)=cos([1-e^sin(x)]^2). Hint: Taylor series.
2.5: # 46, 61 (if such a number exists, then find it to 2 decimal places by the Method of Bisection, from some reasonable starting values), and 65 (hint: let f(t)=monk's distance from the monastery at time t on the way up, and let g(t)=monk's distance from the monastery at time t on the way down)

4 Wed, 9/02 Tue, 9/8 Download the September 2 homework.

3B Mon, 8/31 Thu, 9/03 Evaluate the following limits by using substitution of the limiting value, or algebra, or Taylor series.
Do NOT use L'Hospital's Rule on this homework assignment.
To do, but not turn in.
2.3: 4, 11, 17
4.4: 8
To turn in.
4.4: 10, 30, 50, 75, 78
Remark on 4.4 #75: the expression equals coth(E)-1/E, where coth(x)=cosh(x)/sinh(x) is the hyperbolic cotangent.

3A Fri, 8/28 Thu, 9/3 To turn in: download the August 28 homework.

2 Wed, 8/26 Tue, 9/1 To turn in: download the August 26 homework.

1 Mon, 8/24 Thu, 8/27 To do, but not turn in.
Diagnostic Tests (pages xxiv-xxviii): A 6b, 9abcde; B 5b; C 4c, 7a; D 9
1.6: 59, 60
To turn in.
1.5: 18, 20, 26 (you can do part d using logarithms)
1.6: 20 (and graph m as a function of v, for 0< v < c), 50b, 52b, 58 (and graph Q as a function of t > 0, and graph t as a function of Q)
Hyperbolic trig functions: recall the definitions of the "hyperbolic cosine" cosh(x)=(e^x+e^-x)/2 and "hyperbolic sine" sinh(x)=(e^x-e^-x)/2.
(a) Prove the hyperbolic Pythagorean theorem: cosh²(x)-sinh²(x)=1.
(b) Guess and prove hyperbolic double angle identities: sinh(2x)=...? cosh(2x)=...?

#	Assigned	Due	Homework Problems
24A	Fri, 11/20	Thu, 12/3	Not to turn in. 11.10: 45 To turn in. 11.10: 28, 36, 37 (comment: notice the Taylor series does not help us see the nonnegativity of sin^2), 50, 58, 72 (just go up to e⁴, not e⁶)(warning: you might prefer to write epsilon instead of e, in this problem, because "e" here is not the usual number 2.71828...) 11.11: 25, 36 (conclusion: the period of a pendulum is almost independent of the amplitude; e.g. the period for amplitude 42 degrees and the period for amplitude 10 degrees differ by only a few percent)
23	Wed, 11/18	Tue, 12/1	Not to turn in. 11.9: 1, 2, 35a To turn in. 11.9: 4, 8, 28 (hint: see Example 8), 32 (do it two ways: 1. differentiate the series and substitute into the differential equation, 2. recognize the series as a Taylor series of some function f and substitute that function f into the differential equation), 38ab 11.10: 2, 15, 16 11.11: 33 (substitute x=d/D and find the first few terms of the Taylor series), 35ab
22B	Mon, 11/16	Thu, 11/19	Not to turn in. 11.7: Read this section - it summarizes how to test series for convergence 11.8: 1, 2a, 34 To turn in. 11.6: 2, 7, 10, 20 11.8: 4, 7, 14, 21, 32a 11.8: Optional, no credit 40
22A	Fri, 11/13	Thu, 11/19	Not to turn in. 11.5: 1, 3 To turn in. 11.5: 4, 8 (how will you show the terms are decreasing, that is b_n+1 < b_n?), 16, 18, 24, 30, 31
21	Wed, 11/11	Tue, 11/17	Not to turn in. 11.4: 1, 2 To turn in. 11.3: 32b, 34, 39 When using the Integral Test, remember to show that f(x) is decreasing. A graph is helpful, but not enough by itself: you should show the derivative f '(x) is negative. 11.4: 4, 6, 12, 27, 28, 29, 37 Show your reasoning, for every problem!
20B	Fri, 11/6	Thu, 11/12	Not to turn in. 11.3: 1 To turn in. 11.3: 4, 6, 7, 12, 19 (hint: you need not evaluate the integral - you simply need to show it is finite, i.e. that it converges), 27 (here p>0) In order to use the Integral Test to prove convergence of a series, in the above problems, you must show that the function f(x) is positive-valued and decreasing when x is large. Usually we consider x > 1, but you may consider x > 2 or x > 1000: we use whatever works for the function under consideration. To show f(x) is decreasing, a graph is helpful - but it is not enough by itself. You should show the derivative f '(x) is negative.
20A	Wed, 11/4	Thu, 11/12	Not to turn in. 11.2: 1, 35 To turn in. 11.1: 65 11.2: 8, 9, 11, 25, 42 (hint: express the recurring decimal as a geometric series), 48, 60
19B	Mon, 11/2		No hw to turn in. Do the following problems (mostly odd-numbered, so that answers are in the textbook). 17.3: 1 (see Example 1 on page 1126) 11.1: 7, 9, 10, 17, 19, 27, 36, 39, 58 4.8: 1, 15 (keep going until x_n and x_{n+1} agree to 6 decimal places)
19A	Fri, 10/30		No hw to turn in. Do the following problems (mostly odd-numbered, so that answers are in the textbook). 9.2: 27b (similar to equation solved in class) 9.4: 7 and sketch y(t) (let t=0 be the time 8am), 8 and sketch P(t) [answer for 8a: P(t) = 10000/(1+24(11/36)^t); answer for 8b: 2.7 years] 9.5: 27 and sketch I(t) (similar to equation solved in class), 29 and sketch Q(t) and I(t) Ch. 9 Review: 20 [answer: R=AS^k], 21
18	Wed, 10/28	Tue, 11/3	Not to turn in. 9.3: 1 To turn in. (i) Solve dP/dt = -kt. Sketch a typical solution. (Here k > 0 is constant.) (ii) Solve dP/dt = -kP. Sketch a typical solution. (Here k > 0 is constant.) [Moral: we need to pay attention to whether it is the independent variable t or the dependent variable P, on the right side of the differential equation!] 9.1: 3, 13 (the training begins at t=0) 9.3: 11 (solve for y, then use the initial condition to determine the constant of integration), 35 (the last part means: what is the limit as t tends to infinity?)(answer in book is wrong: should be M-Ce^-kt), 38, 40 (hint: in part (a) you need to find a differential equation of the form dx/dt = ...? Ask yourself: if dt = 1 day, then how much is dx? i.e. how much new currency will be put into circulation that day? Notice dx will equal some fraction of 50 million.)
17B	Mon, 10/26	Thu, 10/29	To turn in. 7.4: 60 7.5: 2, 6, 18, 28 Appendix G: 4 (hint: approximate the area under the curve)
17A	Fri, 10/23	Thu, 10/29	To turn in. 7.4: 8, 14 (assume a does not equal b), 22, 32, 48 Hint on 7.4 # 14. You could multiply out the denominator and complete the square, and then integrate. But the denominator is already given to us as a product of linear factors, and so a simpler method is to express the integrand as C/(x+a) + D/(x+b) and figure out the constants C and D, and then integrate. Hint on 7.4 # 22. The denominator is given to us as a product of linear factors, and so the simplest method is to express the integrand as C/s + D/s² + E/(s-1) + F/(s-1)² and figure out the constants C,D,E,F, and then integrate. 7.8: For each of the following improper integrals, determine whether the value of the integral is finite (in which case we say the integral "converges"), or infinite (we say the integral "diverges"), or does not exist for some reason. 8, 16, 20, 28, 30, 32, 40, 47b Hints (i) To show the integral of a positive function is finite, you may either evaluate the integral, or else you may show it is less than or equal to some other integral that is finite. (ii) To show the integral of a positive function is infinite, you may either evaluate the integral, or else you may show it is greater than or equal to some other integral that is infinite. 7.8: 58
16	Wed, 10/21	Tue, 10/27	To turn in. Problem A. Show that the centroid of a triangular lamina (plate of constant density) lies at the average of the vertices. That is, write the vertices of the triangle as (a₁,b₁), (a₂,b₂), (a₃,b₃), and show that the center of mass of the lamina is ((a₁+a₂+a₃)/3,(b₁+b₂+ b₃)/3). Hint. By a translation and rotation of the lamina, you may assume it has one vertex at the origin and that the opposite side is vertical: a₁=0, b₁=0, 0 < a₂=a₃, b₂ < b₃. Sketch this triangle and find its centroid, then show the centroid equals the average of the vertices. Problem B. Use the Theorem of Pappus to find the centroid of the right triangle with vertices at (0,0), (r,0) and (0,h). Then check your answer by using Problem A to find the centroid. 8.3: 44 8.3: 2, 5 (you can use integrals from the back of the book), 8, 10. For these problems, do not approximate with Riemann sums, and do not use the method in the book. Instead use the result from class that: hydrostatic force perpendicular to the plate = (density of fluid)(g)(area)(depth of the centroid). Remember only to take the centroid of the part of the plate that is under water, since that is the part that experiences pressure from the water. The density of water at room temperature is 1000 kg/m³. (The metric system is set up rather nicely, don't you think?!)
15B	Mon, 10/19	Thu, 10/22	To turn in. 8.3: 22 (and sketch the 3 points and the centroid), 24 (just find the centroid, not the moments; then sketch the 4 points and the centroid), 28, 29 (hint: why would you expect the coordinates x-bar and y-bar of the centroid to be equal?), 34, 48 (in part c, please use m=4, n=3), page 550 Discovery Project parts 1 and 2 (hint for part 2: kh=...?)
15A	Fri, 10/16	Thu, 10/22	To turn in. 8.2: 28, 30 (hint: see the diagram for 6.2 #63 on page 432; check your answer in the special case R=r: you should get the same answer as the back of the book for 8.2 #33) 6.5: 6, 10, 19, 20 (hint: first express v as a function of t; then express v as a function of s)
14	Wed, 10/14	Tue, 10/20	Not to turn in. 8.1: 1 8.2: 1 To turn in. 8.1: 6 (here a>0 and b>0 are constants, called the "semi-axes" of the ellipse), 10, 12 (you may use integrals from the back of the textbook), 40a (40b is optional, for no credit) 8.2: 8, 12, 14, 25, 34 (you may assume the planes are horizontal; or vertical if you prefer)
13B	Fri, 10/9	Thu, 10/15	To turn in. 7.2: 44 (see page 465) 6.4: 14, 17 (note the hint online at "Tools for Enriching Calculus"), 21 (the density of water is rho=1000 kg/m3), 29
13A	Wed, 10/7	Thu, 10/15	To turn in. 7.2: 8, 12, 20 (double angle formula!), 66 (the average value of a function is defined in Sec. 6.5; in this problem, one cycle means 0 < t < 1/60) 7.3: 2 (instead of doing the indefinite integral, make it a definite integral from 0 to 3)(Correction: not from 0 to Pi/2 as earlier written), 20 (do the integral two ways: (i) u-substitution, and (ii) trig substitution), 38 (do not change the limits of integration on this problem when you substitute; instead, after substituting and integrating you can substitute back to the x-variable, and then put in the original limits of integration; note that if tan(theta)=x/b then sin(theta)=x/(x²+ b²)^1/2 by drawing the triangle...)
12B	Mon, 10/5		No homework to turn in! Do the following problems (mostly odd-numbered, so that answers are in the textbook). 6.3: 1, 7, 15, 19, 25, 37 and 38 (just set the integral up: decide whether to use washers or shells), 43 7.1: 1, 5, 7, 11, 17, 21, 33, 35, 37, 41, 59 For extra practice: 7.1 # 15, 48, 57
12A	Fri, 10/2		No homework to turn in! Do the following problems (all odd-numbered, so that answers are in the textbook). 6.1: 3 (hint: read p. 419), 31 (hint: split the interval into two intervals), 49 6.2: 3, 9, 11, 29, 33, 35, 41, 45, 51, 59 Hint for the solid of revolution problems: sketch the original region, sketch it rotated it about the given axis, sketch a cross-sectional "slab" or "slice" and work out its area, then obtain the volume of the solid from the integral of A(x) dx or A(y) dy. See the summary on page 427.
11	Wed, 9/30	Tue, 10/6	Required Reading 1. Learn the antiderivatives ("indefinite integrals") in the Table on page 392. 2. Learn the Integrals of Even and Odd Functions (pages 405-406, especially Figure 4; recall page 19 defines Even and Odd functions). These formulas only work when the interval is symmetric, from -a to a. Not to turn in. 5.4: 11, 17, 55 To turn in. 5.4: 48, 50, 53 5.5: 8, 48 (do not turn in the graph), 64, 68, 70, 74 Evaluate the following integrals, using Oddness or Evenness: (a) integral_{-pi}^{pi} sin(jx) cos(kx) dx where j and k are constants (b) integral_{-1}^1 (x-1) (1-x^2)^{1/2} dx (c) integral_{-infinity}^{infinity} x e^{-x^2} dx (d) integral_{-infinity}^{infinity} \|x\| e^{-x^2} dx
10B	Mon, 9/28	Thu, 10/1	Not to turn in. 5.3: 23, 37, 51 To turn in. 5.3: 32, 34, 38, 40, 42, 46, 56 (see hint for 53), 59, 64, 70
10A	Fri, 9/25	Thu, 10/1	To turn in. 5.2: 52, 53, 66 (hint: \|ab\|=\|a\| \|b\|) 5.3: 4 (first read the hints for #3, at Stewart's website; and print copies of the graph), 8, 10 (the variable is r; the dummy variable of integration is x), 12, 13 (hint: h(x)=g(1/x)), 16, 60 Optional (no credit) 5.3: 68
9	Wed, 9/23	Tue, 9/29	Not to turn in. 5.1: 13 5.2: 39 To turn in. 5.1: 2 (print 3 copies of the graph), 12, 14, 20, 22 5.2: 30, 34, 36, 70
8B	Mon, 9/21	Thu, 9/24	To turn in. 4.7: 4, 7, 37 (hint: write h and r for the height and radius of the cone), 47, 56, 62, 72. General hint. To find a global maximum, it is not enough just to check that the first derivative equals zero there. You also need to check the value of f at any other critical points and at the endpoints of the interval; or else use the positivity and negativity of the derivative on the left and right sides of the max point, to help you conclude that you have really found the maximum.
8A	Wed, 9/16	Thu, 9/24	To turn in. 4.1: 3, 73 (remember to check the endpoints of the interval) 4.2: (Problems on Mean Value Theorem) 21a, 22, 26, 27 (do it using 26; of course, another solution is to square both sides), 29 (you can assume a < b), 36 (and sketch the graph of such an f on the same axes as the function g(x)=x; indicate the fixed point)
7B	Mon, 9/14		To do, but not turn in. 3.10: 11a, 13b, 35a, 38 (check your work by estimating the error directly, using 29 degrees and 31 degrees), 39 No homework to turn in! Answers are here.
7A	Fri, 9/11		To do, but not turn in. 3.5: 13, 40, 69 3.6: 40 3.9: read the instructions (a)-(e) at the bottom left of page 245; then do 12, 16, 27, 33 No homework to turn in! Answers are here.
6	Wed, 9/09	Tue, 9/15	To do, but not turn in. 3.4: 2, 27 To turn in. 3.4: 10, 22, 28 (here y=tanh(u) and you want dy/du), 32, 46, 54, 74, 84
5	Fri, 9/04	Thu, 9/10	To turn in. 4.4: # 52, 60, 66 (do not use L'Hospital's Rule on these three problems; instead use Taylor series) Problem: Compute the fourth derivative at x=0 of f(x)=cos([1-e^sin(x)]^2). Hint: Taylor series. 2.5: # 46, 61 (if such a number exists, then find it to 2 decimal places by the Method of Bisection, from some reasonable starting values), and 65 (hint: let f(t)=monk's distance from the monastery at time t on the way up, and let g(t)=monk's distance from the monastery at time t on the way down)
4	Wed, 9/02	Tue, 9/8	Download the September 2 homework.
3B	Mon, 8/31	Thu, 9/03	Evaluate the following limits by using substitution of the limiting value, or algebra, or Taylor series. Do NOT use L'Hospital's Rule on this homework assignment. To do, but not turn in. 2.3: 4, 11, 17 4.4: 8 To turn in. 4.4: 10, 30, 50, 75, 78 Remark on 4.4 #75: the expression equals coth(E)-1/E, where coth(x)=cosh(x)/sinh(x) is the hyperbolic cotangent.
3A	Fri, 8/28	Thu, 9/3	To turn in: download the August 28 homework.
2	Wed, 8/26	Tue, 9/1	To turn in: download the August 26 homework.
1	Mon, 8/24	Thu, 8/27	To do, but not turn in. Diagnostic Tests (pages xxiv-xxviii): A 6b, 9abcde; B 5b; C 4c, 7a; D 9 1.6: 59, 60 To turn in. 1.5: 18, 20, 26 (you can do part d using logarithms) 1.6: 20 (and graph m as a function of v, for 0< v < c), 50b, 52b, 58 (and graph Q as a function of t > 0, and graph t as a function of Q) Hyperbolic trig functions: recall the definitions of the "hyperbolic cosine" cosh(x)=(e^x+e^-x)/2 and "hyperbolic sine" sinh(x)=(e^x-e^-x)/2. (a) Prove the hyperbolic Pythagorean theorem: cosh²(x)-sinh²(x)=1. (b) Guess and prove hyperbolic double angle identities: sinh(2x)=...? cosh(2x)=...?