You add a finite number of terms to a convergent series. Will the new series still converge? Explain your reasoning.

The base of a solid is enclosed by the graphs of 4 x+5 y=20, x=0, y=0. Every cross section perpendicular to the base along the x-axis is a semicircle. Find the volume of the solid.

In a hospital, the effects of high acceleration or jerk may be harmful to patients, so the acceleration and jerk need to be lower than in standard elevators. It has been determined that a 1600 -kg elevator that is installed in a hospital and that moves at an average speed of 4 m/s should have vertical motion $s(t)=4 t+0.55 t^{2}+0.1167 t^{3}$ (a) Find the acceleration of a hospital elevator. (b) Find the jerk of a hospital elevator.

How many of the carbons come from acetate? From propanoate? (i) 6-Deoxyerythronolide B$\left(\mathrm{C}_{21} \mathrm{H}_{38} \mathrm{O}_{6}\right)$

$$\begin{matrix} \text{ } & \text{Acetate} & \text{Propanoate}\\ \text{A.} & \text{0} & \text{21}\\ \text{B.} & \text{6} & \text{15}\\ \text{C.} & \text{12} & \text{9}\\ \text{D.} & \text{18} & \text{3}\\ \text{ }\\ \text{ }\\ \text{ }\\ \text{ }\\ \end{matrix}$$

Evaluate the integrals. $\int x \sec ^{-1} x d x$

Solve the heat equation $\frac{\partial u}{\partial t}=k \nabla^{2} u$ inside a circle of radius a with zero temperature around the entire boundary, if initially $u(r, \theta, 0)=f(r, \theta).$ Briefly analyze $\lim _{t \rightarrow \infty} u(r, \theta, t).$ Compare this to what you expect to occur using physical reasoning as $t \rightarrow \infty.$

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If $y=x^{a+2}+b x$, then $d y / d x=(a+2) x^{a+1}+b$.

Determine whether each of the given sets is a real linear space, if addition and multiplication by real scalars are defined in the usual way. For those that are not, tell which axioms fail to hold. The function are real-valued. All rational functions.

Find all the second partial derivatives of the given function. $f(x, y)=x^{2}+y^{2}$

Find the conformations in which the hydrogens marked in red are (a) gauche and (b) anti.

Find each indefinite integral. $\int(2+3 \cot x)^{3 / 2} \csc ^{2} x d x$

The virtual particle responsible for the color force between quarks is (a) the pion, (b) the muon, (c) the tauon, (d) the gluon.

Consider a simple metapopulation in which subpopulation A grows at a per capita rate of $r_A = 1$ and subpopulation B declines at a per capita rate of $r_B = -1.$ Suppose the per capita rate of movement between subpopulation patches is m in both directions. This gives

$$\begin{aligned} &\frac{d x_{\mathrm{A}}}{d t}=(1-m) x_{\mathrm{A}}+m x_{\mathrm{B}}\\ &\frac{d x_{\mathrm{B}}}{d t}=-(1+m) x_{\mathrm{B}}+m x_{\mathrm{A}} \end{aligned}$$

where $x_A$ and $x_B$ are the numbers of individuals in patches A and B, respectively. What is the solution to the initial-value problem if $x_{\mathrm{A}}(0)=1 \text { and } x_{\mathrm{B}}(0)=0 ?$

Calculate enough derivatives of the given function to enable you to guess the general formula for $f^{(n)}(x)$. Then verify your guess using mathematical induction. $f(x)=\cos (a x)$