Math 1210 - Calculus I - Section 2

MTWF 8:35 - 9:25 AM

Office Hours: Thursday 10:30-11:30, Friday 12:00-12:45


SYLLABUS

ANNOUNCEMENTS

Tues. April 24
The practice final solutions have been updated. There were 2 corrections:
The answer to Ch 4 #5 Part 3 is -(3x+2)^4 + 2x(3x^2 + 2)^4.
The answer to Ch 4 #7 Part 1 is 364/3.

Fri. April 20 Practice Final Solutions:
Solutions

Here is a list of things you do NOT need to study for the final exam:
Delta-Epsilon proofs of limits (1.2)
Bisection Method of finding roots (3.7)
How to find the value c in either MVT (3.6, 4.5) (Knowing that such a c exists is still important)
Secant Lines (2.1)
Derivatives/Integrals of trig functions other than sin and cos (2.4)
"Even" symmetry for integrals (4.5) (Odd symmetry is still important)
If there is any integral approximation it will be n=4. (4.6)
Horizontal Integration (5.1)
Section 5.2 (Will not be on final at all)

Everything else we learned can be on the final.

Tues. April 17 The practice final is now available: Practice Final
Solutions will be available later in the week. Please note that Problem 3.3 will be modified to y' = -xy^2, as it required things we don't know yet in its original form.
As I said in class, and previously here, the emphasis is on the final having more problems than previous tests, but the problems being shorter. For Chapters 1-3, you should also review the previous practice exams in addition to what is on the practice final.
Over the next few days I will work to fix the problems people discovered in the solutions for the previous practice exams.

Mon. April 2; The rest of the schedule for this class has been filled in and the rest of the homework assignments are available (except the bonus problems, which will be filled in soon).

The final exam will be on Monday, April 30, from 8:00-10:00 AM in the normal class room. It will be comprehensive (covers every chapter) but many of the questions will be from Chapter 4. To be able to ask more questions on it there will be some questions where I give you some of the information for free, for example a problem to find the global max/min and telling you what the critical points are. There may be some conceptual questions as well, for example asking whether the Mean Value Theorem applies or not and why (but not finding the value c). There will be examples of these on the practice final.

HOMEWORK SETS

Set 1 (Due Tuesday, Jan. 17):
Sec 1.1 # 1-4, 7, 8, 14-16, 29, 33, 43
Bonus Problem (optional) at bottom of this page

Set 2 (Due Monday, Jan. 23):
Sec 1.2 # 2, 11-20, 32
Sec 1.3 # 1-3, 9-12, 14, 16, 19, 20, 26, 28
Sec 1.4 # 1-13 odd, 8, 10
Bonus Problem (optional): 1.3 # 49
Note: The Set 1 Bonus Problem will still be accepted in Set 2.

Set 3 (Due Monday, Jan. 30)
Sec 1.5 # 1-5, 13-15, 27-32, 42
Sec 1.6 # 2-7, 13, 14, 18, 21, 26-29, 45, 48, 52
Bonus Problem (optional) at bottom of this page

Set 4 (Due Monday, Feb. 6)
Sec 2.1 # 7-10, 12-14, 23
Sec 2.2 # 1, 3, 5, 7, 8, 13-19, 38, 41, 44
Sec 2.3 # 1-9 odd, 15-22

Set 5 (Due Monday, Feb. 13)
Sec 2.3 # 25-30, 33-38, 51
Sec 2.4 # 1-7 odd, 10-14 even, 27
NOTE: On #5 and #7 you should use the quotient rule (and not Theorem B). On #10 and #12 you may use Theorem B.
Sed 2.5 # 1-14, 17-20, 33
Bonus Problem (optional) at bottom of this page

Set 6 (Due Tuesday, Feb. 21)
Sec 2.6 # 1-6, 10, 11, 23, 24 (no part e on 23, 24)
Sec 2.7 # 1-3, 6-8, 13-15, 19-22

Set 7 (Due Monday, Feb. 27)
Sec 2.8 # 1, 2, 7, 9, 11, 14, 22, 31
Sec 2.9 # 1-3, 16-19, 21, 23, 27
Bonus Problem: Sec 2.8 # 29 (hints at bottom of this page)
Note: If you want extra practice for the test, you can do 5-29 odd from the "Sample Test" in Sec. 2.10 and check your answers. Please do not turn these in though.

Set 8 (Due Monday, Mar. 5)
Sec 3.1 # 1, 2, 5-12, 23, 29, 30
Sec 3.2 # 3-5, 11-13, 19-22, 26
Sec 3.3 # 1-3, 11-14, 17, 18, 21-23, 39-41
Bonus: Bottom of Page

Set 9 (Due Monday, Mar. 19)
Sec 3.4 # 4, 6, 7, 9, 10, 13, 19, 23, 33, 44
Sec 3.5 # 2-5, 9, 11, 16, 17, 24, 25, 58
Bonus: Bottom of Page

Set 10 (Due Monday, Mar. 26)
Sec 3.6 # 3-9, 14, 19 (do not have to sketch graphs)
Sec 3.7 # 1, 2, 6, 9, 11, 15
Sec 3.8 # 1-9, 17, 19, 20, 29, 30, 34
Bonus: Bottom of Page

Set 11 (This homework will not be collected!)
Sec 3.9 # 1, 3, 5, 6, 10, 11, 12, 21, 28

Set 12 (Due Monday, Apr. 9)
Sec 4.1 # 1-3, 9, 11, 19-24, 38, 49-51, 53, 54, 57

Set 13 (Due Monday, Apr. 16)
Sec 4.2 # 1, 3, 5, 7, 8, 11, 12, 15, 24, 25
Sec 4.3 # 1, 4, 9-12, 17-22
Bonus: Bottom of Page

Set 14 (Due Monday, Apr. 23)
Sec 4.4 # 1-5, 11, 15, 18, 35, 41, 42, 45-49, 55
Sec 4.5 # 1, 4, 8, 11, 15-18, 36, 41
Sec 4.6 # 1, 5, 8, 9
Bonus Problems

Set 15 (Not Collected!)
Sec 5.1 # 1, 3, 6, 7, 12-18, 21, 23
Sec 5.2 # 1-3, 6, 7, 12, 18, 23, 26

Practice Final
Solutions

CLASS SCHEDULE

Day Sections Covered Extra announcements
Week 1
January 9 Review (0.4-0.7)
January 10 Review (0.4-0.7), 1.1 Syllabus handed out
January 11 1.1, 1.2
January 13 1.2
Week 2
January 16 No Class!
January 17 Begin 1.3 HW Set #1 Due
January 18 Finish 1.3, 1.4
January 20 Finish 1.4, Begin 1.5 HW Set #1 returned
Week 3
January 23 Finish 1.5 HW Set #2 Due
January 24 Begin 1.6
January 25 Finish 1.6
January 27 Exam Review Day ( Practice Exam , Solutions ), HW #2 returned
Week 4
January 30 Exam 1, Chapter 1 HW Set #3 Due
January 31 Sec 2.1
February 1 Sec 2.2
February 3 Begin Sec 2.3 HW Set #3 returned
Week 5
February 6 Finish Sec 2.3 HW Set #4 Due, Exam 1 Returned
February 7 Sec 2.4
February 8 Begin Sec 2.5
February 10 Finish Sec 2.5 HW #4 returned
Week 6
February 13 Sec 2.6 HW Set #5 Due
February 14 Sick Day
February 15 Sec. 2.7
February 17 Begin Sec 2.8 HW #5 Returned
Week 7
February 20 No Class! Presidents
February 21 Finish Sec 2.8 HW #6 Due
February 22 Sec 2.9
February 24 Test Review Exam 2 ( Practice Exam 2 , Solutions )
Week 8
February 27 Exam 2, Chapter 2 HW #7 Due
February 28 Section 3.1
February 29 Section 3.2
March 2 Section 3.3
Week 9
March 5 Section 3.4 HW #8 Due, Test + HW#7 Returned?
March 6 Section 3.4/3.5
March 7 Section 3.5
March 9 Bonus Day - Interesting applications HW #8 Returned
March 12-16 Spring Break
Week 10
March 19 Section 3.6 HW #9 Due
March 20 Section 3.7/3.8
March 21 Section 3.8
March 23 Begin Section 3.9 HW #9 Returned
Week 11
March 26 Section 3.9 HW #10 Due
March 27 Review Day ( Practice Exam 3 )( Solutions )
March 28 Exam 3 Ch. 3
March 30 Exam 3 No HW This week.
Week 12
April 2 Section 4.1 HW #10 Returned
April 3 Section 4.1
April 4 Section 4.2
April 6 No Class
Week 13
April 9 Section 4.2 HW #12 Due, Test Returned
April 10 Section 4.3
April 11 Section 4.3
April 13 Section 4.4 HW #12 Returned
Week 14
April 16 Section 4.4 HW #13 Due
April 17 Section 4.5
April 18 Section 4.6
April 20 Section 5.1 HW #13 Returned
Week 15
April 23 Section 5.2 HW #14 Due
April 24 Section 5.2
April 25 Final Exam Review
April 27 No Class - Finals week
Week 16
April 30 Final Exam 8:00-10:00am, same room ( Practice Final )( Solutions )

BONUS PROBLEMS

Remember that on these problems, you must show all work and complete all parts of the problem to receive credit. The grading will be harder on these than on normal homework!

Set 1 Bonus (Due Tuesday, Jan. 17): Define a function f(x) on the real numbers as follows: f(x) = x if x is rational, and f(x) = -x if x is irrational. Sketch a graph of f(x) the best you can. Is there anywhere that the limit of f(x) exists? Prove it using a delta-epsilon proof!

Set 2 Bonus (Due Monday, Jan. 23): Section 1.3 Prob #49.

Set 3 Bonus (Due Monday, Jan. 30): Assume that the temperature on the Earth is a continuous function (in other words, there are no sudden jumps in temperature). Convince me that at every moment, there are two points exactly opposite each other somewhere on the planet at which the temperature is the same!
Hint: You only need look at the equator. Think of the equator cut-and-pasted onto the number line from 0 to 1. Now the temperature on the equator is a function defined from 0 to 1. Two points are exactly opposite if they are a distance 1/2 apart in this model. Very importantly, the function must have the same values at 0 and 1. You will need to use the Intermediate Value Theorem and some cleverness to prove the claim above.

Set 5 Bonus (Due Monday, Feb. 13): : In Chapter 4 we will learn about anti-derivatives, which represent the reverse process of taking a derivative. In many cases it is easy to find an antiderivative by imagining what one must have done to get to that derivative. For example, an antiderivative of 5x^4 is x^5 ("reverse" power rule), and an antiderivative of 6(2x+1)^2 is (2x+1)^3 ("reverse" chain rule).
1) Find an antiderivative for the following (in other words, find a function whose derivative is the given expression):
a) 3x^2
b) 2x*cos(x^2)
c) 1/x^3
d) 6(4x+1)*(2x^2 + x + 3)^5
2) Why have I been saying "an" antiderivative instead of "the" antiderivative? Why might there be more than one?

Set 7 Bonus (Due Monday, Feb. 27): Sec. 2.8 # 29.
Hint: The first sentence is the most difficult one to understand. For quantity A to be proportional to quantity B means that A = k * B for some constant k. The problem tells you that the RATE of snow melt (the amount of volume being lost) is proportional to surface area. Note that k will be negative here! For part (a) you will want to show that dr/dt is constant.

Set 8 Bonus (Due Monday, Mar. 5): A function f can be written as a composition f(t) = g(r(t)).
1) Show that if r(t) has a critical point at t=0 then f(t) has a critical point at t=0.
2) Show that if r(t) has an inflection point at t=0 and g(t) has an inflection point at t = r(0) then f(t) has an inflection point at t=0.
Hint: Chain Rule.

Set 9 Bonus (Due Monday, Mar. 19): A polynomial g(x) satisfies the following properties:
1) g(x)-g''(x) is greater than or equal to 0 for every value of x.
2) The highest power of x in g(x) has an even exponent and a positive coefficient.
Prove that g(x) is never negative, that is, g(x) is greater than or equal to 0 for every value of x.
Hint: Property (2) tells you what the function is doing as x becomes very large positive or negative. Supposing the function is negative somewhere, show that it achieves a global minimum with a negative value. What does the second derivative look like there?

Set 10 Bonus (Due Monday, Mar. 26): I mentioned in class that the integral of 1/x cannot be found using the power rule of integration (you would get 1/0 x^0 if you tried to).
Suppose there is a function F(x) such that F'(x) = 1/x and F(1) = 0. Using what you know about graphing functions from Section 3.5, sketch a graph of what F(x) might look like. (You will learn in Calculus 2 what the function F(x) is.)

Set 13 Bonus (Due Monday, Apr. 16): Let s(x) be the following function:

s(x) = integral from x to x^2 of (sin x)(cos t) dt

Find d/dx (s(x)).
Hint: This is like #22 from the 4.3 homework that I briefly went over in class. You will need both the chain rule and the product rule while working this probem. Also read the hint on #25.

Set 14 Bonus (Due Monday, Apr. 23):
Bonus Problems