Calculate the following approximations to $\int _ { 0 } ^ { 6 } x ^ { 2 } d x.$ (a) LEFT(2) (b) RIGHT(2) (c) TRAP(2) (d) MID(2)

If a matrix A is $5 \times 3$ and the product AB is $5 \times 7$, what is the size of B?

Coal gas is produced at a gasworks. Pollutants in the gas are removed by scrubbers, which become less and less efficient as time goes on. The following measurements. made at the start of each month, show the rate at which pollutants are escaping (in tons/month) in the gas:

$$\begin{array}{c|c|c|c|c|c|c|c} \hline \text{Time (months)} & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \text{Rate pollutants escape} & 5 & 7 & 8 & 10 & 13 & 16 & 20 \\ \hline \end{array}$$

Make an overestimate and anderestimate of the total quantity of pollutants that escape during the six months.

Evaluate the following integral:

$$\int _ { 0 } ^ { \pi / 2 } ( 6 + 3 \cos x ) d x$$

(a) analytically; (b) single application of the trapezoidal rule; (c) multiple-application trapezoidal rule, with n = 2 and 4; (d) single application of Simpson’s 1/3 rule; (e) multiple-application Simpson’s 1/3 rule, with n = 4; (f) single application of Simpson’s 3/8 rule; and (g) multiple-application Simpson’s rule, with n = 5. For each of the numerical estimates (b) through (g), determine the percent relative error based on (a).

(a) Find $\int \sin \theta \cos \theta d \theta.$ (b) You probably solved part (a) by making the substitution $w = \sin \theta$ or $w = \cos \theta.$ (If not, go back and do it that way.) Now find $\int \sin \theta \cos \theta d \theta$ by making the other substitution. (c) There is yet another way of finding this integral which involves the trigonometric identities

$$\begin{aligned} \sin ( 2 \theta ) & = 2 \sin \theta \cos \theta \\ \cos ( 2 \theta ) & = \cos ^ { 2 } \theta - \sin ^ { 2 } \theta. \end{aligned}$$

Find $\int \sin \theta \cos \theta d \theta$ using one of these identities and then the substitution $w = 2 \theta.$ (d) You should now have three different expressions for the indefinite integral $\int \sin \theta \cos \theta d \theta.$ Are they really different? Are they all correct? Explain.

Determine the value of k so that x+3 is a factor of x^3+4x^2-2kx+3 .

A car initially going 50 ft/sec brakes at a constant rate (constant negative acceleration), coming to a stop in 5 seconds. (a) Graph the velocity from t = 0 to t = 5. (b) How far does the car travel? (c) How far does the car travel if its initial velocity is doubled, but it brakes at the same constant rate?

A bar of metal is cooling from $1000 ^ { \circ } \mathrm { C }$ to room temperature, $20 ^ { \circ } \mathrm { C }.$ The temperature, H, of the bar t minutes after it starts cooling is given, in $^ { \circ } \mathrm { C },$ by $H = 20 + 980 e ^ { - 0 . 1 \mathrm { t } }.$ (a) Find the temperature of the bar at the end of one hour. (b) Find the average value of the temperature over the first hour. (c) Is your answer to part (b) greater or smaller than the average of the temperatures at the beginning and the end of the hour? Explain this in terms of the concavity of the graph of H.

The population of the world t years after 2010 is predicted to be $P=6.9 e^{0.012 t}$ billion. What population is predicted in 2020?

The population of the world t years after 2010 is predicted to be $P=6.9 e^{0.012 t}$ billion. What is the predicted average population between 2010 and 2020?

Let $y_1$ and $y_2$ be two solutions of $A ( x ) y ^ { \prime \prime } + B ( x ) y ^ { \prime } +C(x)y$ on an open interval $I$ where $A$, $B$, and $C$ are continuous and $A(x)$ is never zero. (a) Let $W=W \left( y _ { 1 } , y _ { 2 } \right)$. Show that

$$A ( x ) \frac { d W } { d x } = \left( y _ { 1 } \right) \left( A y _ { 2 } ^ { \prime \prime } \right) - \left( y _ { 2 } \right) \left( A y _ { 1 } ^ { \prime \prime } \right).$$

Then substitute for $A y _ { 2 } ^ { \prime \prime }$ and $A y _ { 1 } ^ { \prime \prime }$ from the original differential equation to show that

$$A ( x ) \frac { d W } { d x } = - B ( x ) W ( x )$$

(b) Solve this first-order equation to deduce Abel’s formula

$$W ( x ) = K \exp \left( - \int \frac { B ( x ) } { A ( x ) } d x \right)$$

where $K$ is a constant. (c) Why does Abel’s formula imply that the Wronskian $W \left( y _ { 1 } , y _ { 2 } \right)$ is either zero everywhere or nonzero everywhere?

The value, V, of a Tiffany lamp, worth $225 in 1975, increases at 15$V = 225 ( 1.15 ) ^ { t }.$ Find the average value of the lamp over the period 1975-2010.

Find the average value of the function over the given interval and all values of in the interval for which the function equals its average value. f(x) = 9 - x², [-3, 3]

Write the solution set of the given homogeneous system in parametric vector form.

$$\begin{array}{r}{x_{1}+3 x_{2}+x_{3}=0} \\ {-4 x_{1}-9 x_{2}+2 x_{3}=0} \\ {-3 x_{2}-6 x_{3}=0}\end{array}$$

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