Class 10 - Quiz 2 (9/26/2017, Tuesday)

2nd Gateway

The second Gateway will start on Sept 26, Tuesday and you will earn extra 10% quiz points if you pass it by Sept 29, Friday. (Of course, 20% total if you passed 1st Gateway early.)

Webwork

• 8.1 due 9/27 Wed
• 8.2 due 9/28 Thur

Team HW 2

• Assignment 2 link
• Due 9/29 Fri

Quiz 2

Here are the problems you need to study for Quiz 2.

Class 6

Exercise*. Solve Part (b) of Number 5 in Exam 1, Fall 2015. (Hint: On the interval $[0, 1]$, note that $g(x)$ is negative and decreasing. Since $-1 \leq g(x) \leq 0$ on $[0, 1]$, we must have $1 \leq g(x) + 2 \leq 2$ on $[0, 1]$. Thus, we see $g(x) + 2$ is positive and decreasing on $[0, 1]$. What about $1/(g(x) + 2)$? Answer this and apply the discussion prior to this exercise.)

Exercise*. Solve Part (a) of Number 5 in Exam 1, Fall 2015.

Exercise*. Solve Part (c) of Number 7 in Exam 1, Winter 2017.

Exercise*. Solve Part (d) of Number 7 in Exam 1, Winter 2017.

Exercise*. Solve Part (c) of Number 5 in Exam 1, Fall 2015. (Hint: if $f'' > 0$, then $f$ is concave up, and if $f'' < 0$, then $f$ is concave down. )

Exercise*. Compute the integral

\[\int_{e^{2}}^{e^{\pi}} \dfrac{3x + 1}{x^{2} + x} dx.\]

(Hint: first, note that $\frac{3x + 1}{x^{2} + x} = \frac{3x + 3}{x^{2} + x} - \frac{2}{x^{2} + x}$. Integrate the first term by substitution method by choosing $u = x^{2} + x$, and integrate the second term using partial fraction decomposition.)

Exercise*. Compute the integral

\[\int_{e^{2}}^{e^{\pi}} \dfrac{2 - x}{x^{2} + 3x + 2} dx.\]

(Hint: first, note that $\frac{2 - x}{x^{2} + 3x + 2} = \frac{7/2}{x^{2} + 3x + 2} - \frac{x + 3/2}{x^{2} + 3x + 2}$. Integrate the second term by substitution method by choosing $u = x^{2} + 3x + 2$, and integrate the first term using partial fraction decomposition realizing that $x^{2} + 3x + 2 = (x + 1)(x + 2)$.)

Class 7

Exercise*. Compute

\[\int \frac{3x + 11}{x^{2} - x - 6} dx.\]

Exercise* Solve 6b in Exam 1, 2017 Winter. (Hint: You need to draw this first. You will need to draw the circle of radius $\sqrt{2}$ somewhere, which you need to find, first and draw the vertical line $x = 3$. Then do the rotation as they say. You will get a doughnut shaped figure. Slice horizontally. Your thickness will be "$dy$", and you will have to compute volumes of two figures and subtract one from another.)

Class 8

Exercise* Solve 6b in Exam 1, 2017 Winter. (Hint: the volume of each shell must be $2\pi (3 - x)f(x) \Delta x.$ Figure out what $f(x)$ should be and where $x$ varies.)

Exercise* Solve 6a in Exam 1, 2017 Winter. (Hint: Draw a picture and figure out what the thickness should be!)

Exercise*. Solve 7b in Exam 1, 2016 Winter.

Exercise* Solve 3a in Exam 1, 2017 Winter.

Class 9

Exercise*. Solve 3a and 3b of Exam 1, Winter 2017.

Exercise*. Solve 9a, b, and c of Exam 1, Winter 2017.

Exercise*. Solve 9 of Exam 1, Winter 2016.

Exercise*. Solve 9 of Exam 1, Fall 2015.