Class Summary 20120207

Dr. Brown Assigned:
7.1 8,10,26,32,42,48,57,68
7.2 2,10,20,26,38,42,56,68

[1] is Section 1
[2] is Section 2

[1A]7.1.7 to $\cos(\alpha x)$ (see Example 1). Also did $\int \cos(\beta t)dt$
[1B]7.1.10 is really important. (see Example 5). I will do $\cos^{-1}$. Basic, but uses a lot of skills. Avoid using trig substitution for the denominator because they haven't done that.
[1C][2D]I ask them to work in pairs or triplets, I had to group them manually to get them to start talking. $\int x e^x \cos x dx$. The second class didn't talk as much, but seemed to put answers down. So what I did is after they worked for about four minutes, I gave them one minute to talk about what they did with their assigned partner.
[1D]Pop question, what is $\ln(1)$?
[1E]Pop question, what is $\int \frac{1}{1+x^2} $? This one is the one most needed. But some other ones are important too. I promised I would memorize the other ones with them.
Answer with confidence!
[1F]Pop question, what is $\sin(\cos^{-1}(x))$?
[1G]Triangle. Teach the triangle to them. Also use the trigonometric identity $\cos ^2 (\theta)+\sin^2 (\theta)=1 $ to derive such identities. (For example: $\cos \sin ^{-1}(x)$
I need to ask Dr. Brown.
[I] Briefly mention that the same can be done for $\cosh \tanh^{-1}(x)$ etc.
*7.1.17 (see Example 4) Integration by parts twice.
7.1.26 changed to $y^{1/3}$ in the denominator. When integrating and taking derivatives, I personally prefer to use rational exponents. Thus I'll convert $\frac{1}{\sqrt{y}}$ to $y^{-\frac{1}{2}}$
7.1.32 Try to stick to DETAIL.
[2A]7.1.42 Mention DETAIL
7.1.48 First ask, how do you prove this? (see Example 6)
7.1.58 Instead of 7.1.57.
[2B] 7.1.57B
[2C] 7.1.68. This problem is a little more abstract, but it's a good problem to tackle. If you like this type of problem, I would encourage you try #69 and #70. A student asked this question and so I had them work in groups.

[1H]Pop question, what is $\int \tan(x) dx$ and what is $\int \sec(x)dx$
7.2.2. I determine the power of sine is odd and I save one power of sine. (pg 473)
[2E]7.2.10. Both powers are even. Did you work this problem out? I have the following solution:
$\sin ^2 (t) \cos^4(t)$
$=\frac{1}{4}\sin ^2(2t) \cos^2(t)$
$=\frac{1}{8}\left[ \sin(t)+\sin(3t) \right]^2$
From here we integrate. Oops, apparently we couldn't integrate directly from this step. Factoring and using the same summation identities would result in a long problem. One of the good ideas is to use $\sin ^2 (x)=1-\cos^2 (x)$. Then applying the half angle formulas to the result.
7.2.20 (pg 476)
7.2.26 The power of secant is even (pg 474)
7.2.38 The power of cosecant is even (pg 474 + pg 476)
7.2.42 (pg 476)
7.2.56 Straightforward.
7.2.68 If you enjoyed 68, it goes together with 67-70. And that goes together with a whole topic!