Math 1151Q-001, Fall 2013
Welcome to 1151! One thing to keep in mind as you read the book is that whenever you see a definition, it's a good idea to try to think of an example that fits the definition and an example that doesn't. For instance, try to think of a piecewise function and a nonpiecewise function.
|Week ||Topic ||Assignment ||Administrivia|
|Week 1: Aug. 26-30 ||1.1-1.3, 1.5, 1.6
|Week 2: Sept. 2-6 ||2.1-2.3
||Homework #1 ||Labor Day: no class!|
|Week 3: Sept. 9-13 ||2.4-2.6
|| ||Monday: last day to drop without a "W" |
or choose the P/F option
|Week 4: Sept. 16-20 ||2.7, 2.8, 3.1
||Homework #2 |
|Week 5: Sept. 23-27 ||3.2, 3.3
||Midterm #1 |
|Week 6: Sept. 30 - Oct. 4 ||3.3-3.6
||Homework #3 |
|Week 7: Oct. 7-11 ||3.8-3.10
|Week 8: Oct. 14-18 ||3.10, 3.11, 4.1, 4.2
||Homework #4 |
|Week 9: Oct. 21-25 ||4.2, 4.3
||Midterm #2 |
|Week 10: Oct. 28 - Nov. 1||4.3-4.5
||Homework #5 ||Monday: last day to drop |
or choose to get a letter grade
|Week 11: Nov. 4-8 ||4.7-4.9
|Week 12: Nov. 11-15 ||5.1, 5.2
||Homework #6 |
|Week 13: Nov. 18-22 ||5.3, 5.4
|Week 14: Nov. 25-29 ||
|| ||Thanksgiving break: no class!|
|Week 15: Dec. 2-6 ||5.5 and review
||Homework #7 |
Tuesday, August 27
Try the "stretching and reflecting" rules with a trig function like sin(x). Compare its graph to the graphs of 2sin(x), sin(x)/2, sin(2x), and sin(x/2).
Thursday, August 29
Two things about exponential functions:
Take a minute and try to derive the other laws of logarithms from the laws for exponents!
- If you aren't sure you're right about one of the laws of exponents, try plugging in sample numbers and seeing if the right and left sides of the equation are equal.
- You should be able to tell the graphs of y=3^x, y=1^x, and y=(.2)^x apart immediately.
Thursday, September 5
I mentioned in class that I don't quite like the way our textbook defines infinite limits (Definition 5 in Section 2.2). Can you see a way that a function could satisfy that definition at a point but still not have an infinite limit there?
Keep in mind that if you are supposed to take the limit of a rational function at a point x=a and you find out that the denominator is 0 when you plug in a, you should try doing some algebra to see if you can cancel out some terms so the denominator won't be 0 anymore.
We talked about the definition of a limit of f(x) at a point x=a as playing a game today in class: if you're given an epsilon, you have to find a delta so that if you're within a distance delta of a, your f-value will be within a distance epsilon of the limit value. There are two steps to one of these problems: (1) figuring out how to calculate delta given an epsilon, and (2) showing that this delta works. Give it a try for another function like f(x)=x^3, and we'll do some more of these problems on Tuesday!
Tuesday, September 10
We did a lot of regular epsilon-delta problems today, but only one example where the limit is infinity. I suggest you try a couple more examples with infinite limits and think about what the definition would look like if the limit were negative infinity.
Thursday, September 12
Think about the hypotheses of the Intermediate Value Theorem. Why is it important to have f be continuous on a closed interval instead of an open one? Why do we need f(a) and f(b) to be different? Why do we need to choose N between f(a) and f(b)?
Today I gave two different limit definitions of the derivative. Why are they the same? Can you transform each one into the other?
Tuesday, September 17
We calculated some derivatives using the limit definition today. Remember that the key to using this definition is canceling the h in the denominator. If you can do that, then you can take these limits!
Thursday, September 19
If you can do this problem, you'll have gotten all the practical tools you need from this section: At which point on the graph of y=e^x + x^3 is the tangent line perpendicular to the line y=-x/4 + 6?
If you're up for a challenge, try proving the quotient rule on your own! (Hint: look at the method I used to prove the Product Rule in class today.)
Thursday, September 26
If you want to see the calculations of the limits I considered in class, take a look at pp. 192-93 in your book. I'll be happy to go over them with you!
You might want to practice taking derivatives of exponential functions with bases other than e. For instance, what's the derivative of g(s)=2^(tan(s))?
Tuesday, October 1
Here are some challenging implicit differentiation problems:
- Compute dy/dx when x^2 + yz = y^2.
- Find y' at the point(s) on the curve defined by -2e^x + (y^2)(cos x) = y where x=0.
Thursday, October 3
We talked in class today about taking the limit of sin(x^8)/x as x goes to 0. Try writing this function as (x^7)(sin(x^8))/(x^8). Then you can write the limit as
[lim_(x->0) x^7][lim_(x->0) sin(x^8)/(x^8)].
This second limit is the same as lim_(x->0+) sin(x)/x, which is 1, so we get 0*1 = 0.
First of all, WebAssign is not playing nicely right now with the newest update of Google Chrome. You may not be able to see all the content, so either use a different browser or make sure that your third-party cookies are enabled.
Here is the problem I promised you:
And finally, here is a quokka.
- Suppose you have 1/5 of a radioactive substance left after 50 years. What is its half-life (that is, how long will it take for only half the original amount to remain)? Note that you don't have an actual amount (like 100 g) in this problem, but you don't need it!
Tuesday, October 8
Here's a general question about related rates: Suppose I gave you a two-part problem like you saw today. For instance, on the baseball problem, I could have asked "How fast is he moving away from third base when he's 2/3 of the way to first base, and how fast is he moving away when he's halfway to first base?" How many of the steps we did would have been useful for answering either question?
Here are some more related rate questions:
- Suppose, as in class today, that a baseball player is running to first base from home plate at a rate or 15 ft/s. How is his distance from second base changing when he is 10 feet from first base?
- Quite frankly, any problem in Section 3.9 would be good. I recommend #35 for something marginally more complicated and anything between #11 and #34 for a general, all-purpose problem.
Thursday, October 10
We've seen that the linearization of a function at a tends to become less accurate as values farther away from a are chosen. Think about f(x)=e^x, g(x)=x^2+2x+1, and h(x)=sqrt(x). Linearize these functions at x=1. Which of these linearizations is the most accurate? Why do you think that is?
Confirm that the derivative of cosh x is sinh x and vice versa!
Tuesday, October 15
I suggested that you think about why the converse of Fermat's Theorem isn't true. (The converse of an "if...then..." statement is the sentence you get when you swap the sentences that appear in the "if" and "then" clauses. For example, the converse of "If the sky is blue, then pigs are purple" is "If pigs are purple, then the sky is blue.")
Thursday, October 17
Here's an application of the Mean Value Theorem: Suppose I drive north on I-91 after Springfield. The Massachusetts and Vermont state police collaborate to find out that I travel for 3 hours and drive 198 miles in that time. Can they deduce from that information that I was speeding?
Now we get to start thinking more about the geometrical meanings of the first and second derivatives. Remember that the first derivative represents the slope and the second derivative represents the concavity, and you'll be set. (Also, remember that when a graph is concave up, it looks like a smile.)
Thursday, October 24
Try finding the inflection points, critical numbers, and local extrema for more functions (you could start with g(t)=t^(4/3)+9).
Think about the differences between the First and Second Derivative tests. Which one gives you more information? When would each be easier to use?
Tuesday, October 29
Remember that L'Hospital's Rule doesn't apply unless the original limit has an indeterminate form.
Thursday, October 31
Today we sketched a curve with great attention to detail. Try doing the same for a polynomial, a rational function, and y=e^(-x^2).
Here are some more optimization problems for you to try!
- You want to build a rectangular enclosure bordering a neighbor's stone wall using the 50 feet of barbed wire you already have so the area in the enclosure is maximized. What are the dimensions of the enclosure, and what is its area?
- Find two numbers whose difference is 81 and whose product is as large as possible.
Tuesday, November 5
One last tip about optimization problems: sometimes it's easier to maximize or minimize the square of a function than the function itself. For instance, if you're asked to maximize f(x)=(x^2+3x+7)^(1/2) on an interval I, you could maximize f^2(x) = x^2+3x+7 instead.
Thursday, November 7
We now know two approximation methods: linearization and Newton's Method. When would each one be most useful?
We approximated the area under the curve y=x^2 from x=0 to x=4 using 4 subintervals with right endpoints, left endpoints, and midpoints. How much better does each of these approximations get if you use 8 subintervals?
Tuesday, November 12
Can you calculate the area under the curve y=x+4 from x=1 to x=3 using the limit method we used in class today? (The sum 1+2+3+...+n is equal to n(n+1)/2.) Note that you can check your work geometrically.
Thursday, November 14
I said in class that we can use arbitrary sample points for each subinterval in a Riemann sum instead of always using the left endpoint, right endpoint, or midpoint. Could you write a few sentences explaining carefully why this is true?
We've done several applications of the first part of the Fundamental Theorem of Calculus. Do you know what the conditions are for the application of this theorem?
Tuesday, November 19
Go through and make a list of all the things we learned in this class that we put to use in the proof of the Fundamental Theorem of Calculus!
Thursday, November 20
Take a look at the list of applications of the Net Change Theorem in your book. Can you think of others?
Just think of u-substitution as reversing the Chain Rule. If you can figure out what the f and g are, you'll be ready to go!
Tuesday, December 3
Remember that when you're doing a definite integral using u-substitution, you need to either (1) change the bounds on the integral but not convert the function of u back to a function of x or (2) keep the original bounds on the integral but convert the function of u back to a function of x before plugging them in.
Thursday, December 5
Here's the related rates problem I was going to give you. It's got a really nice twist!
Suppose a ladder is sliding down a wall at a rate of 1.5 m/s. When the base of the ladder is 3 meters away from the wall, the base is moving away from the wall at a rate of 2 m/s. How long is the ladder?
Good luck studying for your exams!