Use the geometric series

$$f ( x ) = \frac { 1 } { 1 - x } = \sum _ { k = 0 } ^ { \infty } x ^ { k }$$

, for |x|<1, to find the power series representation for the following functions (centered at 0). Give the interval of convergence of the new series.

By evaluating $\mathbf{\nabla} \cdot(\mathbf{\nabla}f)$, show that the Laplacian in spherical polar coordinates is given by

$$\begin{aligned} \nabla^{2} f=& \frac{1}{r^{2}} \frac{\partial}{\partial r}\left(r^{2} \frac{\partial f}{\partial r}\right)+\frac{1}{r^{2} \sin \theta} \frac{\partial}{\partial \theta}\left(\sin \theta \frac{\partial f}{\partial \theta}\right) \\ &+\frac{1}{r^{2} \sin ^{2} \theta} \frac{\partial^{2} f}{\partial \phi^{2}} \end{aligned}$$

$$\dfrac{36}{0.12}$$

Solve each equation by finding points of intersection of two related functions.

$$-(x-3)^2+4=0$$

Suppose $V_{i}$ is the generalized eigenspace of $T$ corresponding to eigenvalue $\lambda_{i} .$ For any $k \geq 0,$ prove that the nullity of $T-\lambda_{i}$ on the subspace $\left(T-\lambda_{i}\right)^{k} V_{i}$ is the same as the nullity of $T-\lambda_{i}$ on $\left(T-\lambda_{i}\right)^{k} V$ and equals the number of Jordan blocks of $T$ having eigenvalue $\lambda_{i}$ and size greater than $k$ (so for $k=0$ this gives the number of Jordan blocks).

Shea has an annual salary of $\$ 29,640 .$ He works 40 hours a week. 52 weeks a year. How much does Shea earn per hour?

Consider a group of 20 people. If everyone shakes hands with everyone else, how many handshakes take place?

The expression a^(1/n) can be written in radical notation as _, provided that $\sqrt[n]{a}$ is a real number.

A plane takes off at St.Louis, flies straight to Denver, and then returns the same way. The plane flies at the same speed with respect to the ground during the entire flight, and there are no head winds or tail winds. Since the earth revolves around its axis once a day, you might expect that the times for the outbound trip and the return trip differ, depending on whether the plane flies against the earth's rotation or with it. ls this true, or are the two times the same?

Is the following sentence true or false? A redox reaction might produce covalent compounds.

Determine if the system has a nontrivial solution. Try to use as few row operations as possible.

$$\begin{aligned} 2 x_{1}-5 x_{2}+8 x_{3} &=0 \\-2 x_{1}-7 x_{2}+x_{3} &=0 \\ 4 x_{1}+2 x_{2}+7 x_{3} &=0 \end{aligned}$$

.

Find $f[g(x)]$ and $g[f(x)]$.

$$f(x)=-8 x+9 ; \quad g(x)=\frac{x}{5}+4$$

Consider a classical inelastic collision of the form A + B → C + D. (For example, this could be a collision such as

$$Na + Cl → Na^+ + Cl^–$$

in which two neutral atoms exchange an electron and become oppositely charged ions.) Show that the law of conservation of classical momentum is invariant under the Galiliean transformation if and only if total mass is conserved — as is certainly true in classical mechanics. (We shall find in relativity that the classical definition of momentum has to be modified and that total mass is not conserved.)

Let $f ( x ) = 2 x - 3 .$ Find $f ^ { - 1 } ( x )$