Homework 9/28

p.203 #13-15 (change the x in 15 to x^2)

Answers to p.176 #3-4

If the graphs represent f(x):
3. f(x) is increasing on (-inf, -2), (2, inf); f(x) is decreasing on (-2, 2); f(x) has critical points at -2 & 2.
4. f(x) is increasing on (-1, 0), (1, inf); f(x) is decreasing on (-inf, -1), (0,1); f(x) has critical points at -1, 0, & 1.

If the graphs represent f '(x):
3. f(x) is increasing on (-3.5, 0), (3.5, inf); f(x) is decreasing on (-inf, -3.5), (0, 3.5); f(x) has critical points at -3.5, 0, & 3.5.
4. f(x) is increasing on (-inf, -1.3), (1.3, inf); f(x) is decreasing on (-1.3, 0), (0, 1.3); f(x) has critical points at -1.3, 0, 1.3.

Homework 9/27

p.176 #43-45, 47
Solve first as if the graphs were of f(x) and then as if they were f '(x)

I won't be home until late tonight as I will be attending a lecture by Alfie Kahn. I will be online this afternoon until about 5:00.

Homework 9/26

p.160 #7-11

ARC and IRC

Average rate of change: the average rate of change of a function f(x) over the interval [a, b] is the change in f(x) over the interval divided by the change in x over the interval, in other words, the rise over the run or the average slope of the function on the interval. Since linear functions (y = mx+b) have constant slopes, the average rate of change of a linear function over any interval is m, its slope. To find the average rate of change of a non-linear function over [a, b], you must first find the points (a, f(a)) and (b, f(b)). (These are the points on f(x) when x=a and x=b, respectively.) Next you find the slope between these two points. Voila, you have found the average rate of change over the interval.

Instantaneous rate of change: the instantaneous rate of change of a function f(x) at a point x=c, is the slope of the line tangent to f(x) at x=c. To find this slope, you will need to first find the derivative of f(x) since the derivative tells you the slope of f(x) at any point. Once you have f '(x), plug in c for x to get f '(c). This value is the instantaneous rate of change of f(x) at c.

Put more simply, if you drove your car for 2 hours and covered 80 miles the average rate of change of your position (which in this example would be your average velocity) would be easy to find since you would need only divide 80 by 2 to obtain an average rate of change of 40 m/h. The instantaneous rate of change, on the other hand, would be different depending upon whether you wanted the instantaneous rate of change after 40 minutes, or after 50 minutes, or at any other time since your velocity at any given instant would be constantly changing. If you could model the distance you had traveled as a function of time, however, you could find your instantaneous speed at any given time by finding the derivative of the function.

Further reading if you are still having trouble:
http://www.mathematicshelpcentral.com/
http://www.ugrad.math.ubc.ca/

I will be on later tonight. If you have any questions, leave a comment.

The Taylor Tests are graded...

The scores varied from 50 to 100, with most of the class falling in the 70's and 80's. These are very good scores for a class's first Taylor Test, though there is certainly room for improvement. Make sure you study for tomorrow's test (which will not be multiple choice).

Practice Test Answers

1. 2 cos(2x) or 2 cos^2(x) - 2 sin^2(x)
2. 2x csc(5x) - 5x^2 csc (5x) cot (5x)
3. (2x+1)^1/2 + x(2x+1)^-1/2 or (3x+1)/(2x+1)^1/2
4. -4/(2x-1)^2
5. (1/2)x^-1/2 - (1/2)x^-3/2 or (x-1)/2x^3/2
6. -4x tan(3-x^2) sec^2(3-x^2)

1. 22
2. 19
3. 4x-3

I'll be on tonight if you have any questions & I'll be in my room in the morning around 7:30.

Homework 9/18

Finish p.150 #7-34

This (remaining) week's schedule

Wednesday (shortened period): Review/Practice Test

Thursday: Memorization Quiz, Taylor Test #1 (quiz grade)

Friday: Test

Homework 9/17

Finish p.130 #12-19, 23-25, 43-46, then do p.131 #29-30, 51-52

Extra Credit Post!

Anyone who leaves a comment on this post by 6:00 today (Saturday) receives free Extra Credit for being conscientious and thinking about calculus before Sunday night.

Homework 9/14

Complete problems 23-36 on the handout and do p.130 #11, 13, 15

I'll be online some Saturday and Sunday; leave a comment if you have a question & I'll respond whenever I see it.

Homework 9/12

p.121 #13-20, 27-34

Homework 9/10

p.87 #11-16, 23-26
p.235 #21-24
p.150 #1-2

I will be in my room in the morning until 7:45. After that, I will be in the auditorium.

I'll be online off and on for most of the evening. Leave comments if you have questions. If you're having trouble registering, try registering for a gmail account or ask a friend to post for you.

Homework 9/7

p.99 #5-8,13-16

Reminder

I forgot to mention it today, but the graphs of y = sin x and y = cos x will also be on the memorization quiz tomorrow. You will need to label 1 & -1 on the y-axis, and the four quadrantal angles (pi/2, pi, 3pi/2, 2pi) on the x-axis.

Homework 9/6

p.194 #35-42

Homework 9/5

Do problems #24-27,37-38,40-44 on the handout

Memorization quiz tomorrow (and another one friday)

Don't forget. It'll be over everything from last friday's memorization quiz as well as the special limits, continuity on an interval, and everywhere continuous.

Homework 9/4

p.84 #29-32

Homework Help Thread

I'll be back on in about 15 minutes (around 11:00). Go ahead and post any questions you have and I'll start answering them when I get on, but remember: since I don't have the worksheet with me, you need to give me specifics as to the problem you're having trouble with. Also, if someone posts a problem that you've already solved & I'm not on yet, point them in the right direction (preferably without just telling them the solution).