Class 11 - Review (9/28/2017, Thursday)

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

• 8.2 due 9/28 Thur
• 8.4 due 10/2 Mon

Team HW 2

• Assignment 2 link
• Due 9/29 Fri

Here are hints for Team HW 2.

1a. To determine overestimate / underestimate, you need to know whether the function $f(x)$ increases or decreases in the interval $[1, 4]$. This can be done by taking derivative:

$f'(x) = \dfrac{(x^{2} + 2x) - 2x(x + 1)}{(x^{2} + 2x)^{2}}.$

1b. Note that $x^{2} + 2x = x(x + 2)$. Apply the partial fraction trick.

1c. Your computation will involve the natural log $\ln$.

1d. You should notice that you have some problem at $x = 0$.

2a. Note that each interval has equal length of $0.5$.

2b. Sum $2^{-2} + 2^{-1} + 2^{0} + 2^{1} + 2^{2}$.

2c. Suppose that the number $c = 1/0$ exists. Then $0 = 0 \cdot c = 0 \cdot (1/0) = 1$.

2d. You don't need any hint for this.

3a. You are asked to compute $f(3) - f(0)$.

3b. Think about $g(t)$ separately for the intervals $[0, 0.5], [0.5, 2],$ and $[2, 3]$. Recall that the line with slope $m$ passing through $(a, b)$ is given by $y = m(x - a) + b$. You should think about the rest of the problem on your own.

3c. Recall that average value of $F(t)$ for $a \leq t \leq b$ is equal to
$$\frac{1}{b - a}\int_{a}^{b}F(t)dt.$$
Think about the rest of the problem on your own.

3d. Possibly except at $t = 0.5, 2$, you are given $H(t) = g'(t) - f'(t)$. For ii, the distance between Car 1 and Car 2 at time $t = x$ is equal to
$$\left|\int_{0}^{x}g(t) - f(t)dt\right|.$$
Now, think about how to get rid of the absolute value by breaking the interval $[0, 3]$ into pieces.

4a. You should start by writing
$$P(x) = -2 + \int_{1}^{x}p(t)dt.$$

4b. Write out $p(x)$ explicitly by breaking up the interval $[-3, 3]$. Then use the hint for 4a to explicitly compute $P(x)$.