I believe this question is quite easy but lets not neglect these simple questions.
Investigate lim(x->0) (1-Cos(h))/h^2 numerically and graphically. Then prove that the limit is equal to 1/2.
1) let f(x)= 
differentiate:
f(x)+g(x), f(x)*g(x), f(x)/g(x),( f(x))^g(x) and
2) f(x)=
a) evalute the continuity at x=0
b) Is the function derivable at x=0?
Fairly easily. I can surely say we can all agree we don’t want any hard questions. (PS: I have no idea how to use laTeX, so I hope you guys can understand my fail attempt to write a math problem online.)
1) The limit below represents the derivative of f ‘(a). Find the f(x) and the a values.
Lim h->0 of ((3+h)^3) – 27)/h
f(x) = ?
a = ?
2) Find the slope of the secant line through the points of x = 2 and x=4 of the function f(x) = x^2. ( Not sure if I typed this correctly I was half asleep when I woke up: Correct me If I’m wrong)
Slope= ?
In y=mx+b form, what is the equation of a line tangent to f'(x) at x=1/2 if f(x)=e^(2x)+3x-6?
Sorry for the picture. Will need time to get used to latex.
Sample peer review Q for the test.
Is the derivative of a product the same as product of the derivatives?
i.e. it is true that
Find a counterexample:
(aka Prove the product rule).
-Izzy
Homework #5 grades are now posted in Blackboard’s grade center. Don’t freak out that the grades are low:
While it’s fresh in my mind, I think it’s worth commenting on your work. For the most part, the work was good, but Section 3.3 #64 was a minor disaster. (Again, don’t worry, I knew that’d be a harder one for you so I’d have been surprised if everyone got it.)
Please see my own hand-written solutions: Solutions5
I didn’t assign these questions because I care all that much about multiple roots, I just wanted you to be able to think about the kinds of arguments you can construct using derivatives and, in this case, the product rule. So you really had to think about two things while answering this question:
What I mean by #2 is how to construct an argument to prove something. It’s hard to think about both things at once and I’d like to focus here on #2.
In math, when you are proving Statement A is true if and only if Statement B is true, you’re really proving two things.
has a multiple root at 
and Statement B is: 
has a root at .If you haven’t worked with it before, the if and only if part of the question might throw you off. It can be particularly hard to keep track of what you’re assuming and what you’re proving. I think one reason for that is that by proving A implies B and B implies A, you’re actually saying that Statement A and Statement B are equivalent. When two things are equivalent, well, it’s hard to tell them apart! So you might accidentally assume the thing you’re trying to prove.
Compare the A if and only if B kind of result to the simpler kind A implies B. We saw an example of a result like this in class on Monday: Statement A is: 
It can also be hard to tell the difference between a proof and an assumption, or even a reason for saying something is true. I can comment more on that later, but if this kind of stuff interests you, you might like to take a class in logic.
1. If f'(x)=x^3-12x+2, at what values of x would f ‘(x) have roots?
Here are my own solutions for most of the written homework that you’ve submitted so far. Let me know if you have any questions.
Test #1 will be given in class next Wednesday, March 12. It will cover everything up to and including this Wednesday’s lecture. This includes Chapter 2 except for 2.8 (there’ll be an extra-credit question like those in 2.9 but otherwise you’re not responsible for 2.9) and Chapter 3 except for 3.4, 3.5, 3.10, and 3.11 (we’ll cover those sections after the test). Don’t forget to contribute to the class’s review sheet here.
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