A car is traveling at 80 feet per second (approximately 55 miles per hour) and slows down as it passes through a busy inter section. The car's velocity is shown in the following table.

$$\begin{array}{c|c|c|c|c|c|} \hline \text { Time since brakes applied (seconds) } & {0} & {3} & {6} & {9} & {12} \\ \hline \text { velocity (ft/sec) } & {80} & {62} & {47} & {32} & {22} \\ \hline \end{array}$$

Use a left sum to approximate the total distance the car has traveled over the 12 second interval.

Express in the form $z=x+y i$ the complex number z whose modulus and argument are given. $|z|=1, \quad \arg (z)=\frac{3 \pi}{4}$

Determine whether each improper integral converges or diverges. If it converges, find its value. $\int_{1}^{\infty} \frac{4}{x} d x$

Find all equilibrium values of the given system of differential equations.

$$\begin{aligned} &\frac{d x}{d t}=x y^{2}-x\\ &\frac{d y}{d t}=x \sin \pi y \end{aligned}$$

Determine whether each integral is improper. For those that are improper, state the reason. $\int_{2}^{3} \frac{d x}{x-1}$

Show that $\frac{2 j+1}{j^{2}(j+1)^{2}}=\frac{1}{j^{2}}-\frac{1}{(j+1)^{2}}$; hence evaluate $\sum_{j=1}^{n} \frac{2 j+1}{j^{2}(j+1)^{2}}$.

A baseball thrown directly upward at 96 ft/sec has velocity v(t) = 96 - 32t ft/sec at time t seconds. (a) Graph the velocity from t = 0 to t = 6. (b) When does the baseball reach the peak of its flight? How high does it go? (c) How high is the baseball at time t = 5?

A function $T: V_{2} \rightarrow V_{2}$ is defined by the formula given for T(x, y), where (x, y) is an arbitrary point in $V_2$. In each case determine whether T is one-to-one on $V_2.$ If it is, describe its range $T(V_2);$ for each point $(u, v)$ in $T( V_2)$, let $(\mathrm{x}, y)=T^{-1}(u, v)$ and give formulas for determining x and y in terms of u and v. $T(x, y)=\left(e^{x}, e^{y}\right).$

Find the solution of each of the following initial-value problems. $y^{\prime \prime}+y^{\prime}+y=H_{\pi}(t)-H_{2 \pi}(t) ; \quad y(0)=1, y^{\prime}(0)=0$

Find the lengths of the given curves. $y=\frac{x^{3}}{3}+\frac{1}{4 x}$ from x = 1 to x = 4

Calculate the given iterated integrals. $\int_{0}^{1} \int_{0}^{y}\left(x y+y^{2}\right) d x d y$

In the real linear space $C(0,2 \pi)$ with inner product $(f, g)=\int_{0}^{2 \pi} f(x) g(x) d x$, let $f(x) = x$. In the subspace spanned by $\mathrm{u},,(\mathrm{x})=1, u_{1}(x)=\cos \mathrm{x}, u_{2}(x)=\sin \mathrm{x}$, find the trigonometric polynomial nearest to f.

Let $v(r, t)=t^{n} e^{-r^{2} /(4 l)}$. Find a value of the constant n such that v satisfies the following equation $\frac{\partial v}{\partial t}=\frac{1}{r^{2}} \frac{\partial}{\partial r}\left(r^{2} \frac{\partial v}{\partial r}\right)$.

Consider a function f(x) that is odd around x = L/2. Show that the even coefficients (n even) of the Fourier cosine series of f(x) on $0 \leq x \leq L$ are zero.