Determine whether the statement is true and give an explanation or counterexample. The sequence of partial sums associated with the series $\sum_{k=1}^{\infty} \frac{1}{k^2+1}$ converges.

In the last investigation, you studied intersection, union, and set difference. Another useful set operation is set to complement. The complement of a set A is everything in the "universe" that is not in A. This is written A$^{\prime}$ and is read " A compliment" or "not A." In terms of set difference, you can think of A$^{\prime}$ as "the universe" -A. The "universe" is defined by the universal set, which is the set of all objects that are under consideration. That is, the universal set defines the overall context.

a. Consider the universal set U = {5, 10, 15, 20, 25, 30, 35}. Suppose A = {10, 20, 30} and B = {5, 10, 15, 20}.

i. Find A$^{\prime}$.

ii. Find (A $\cap$ B)$^{\prime}$.

b. Give a general description of A$^{\prime}$, for any set A, using set-builder notation.

c. Consider the universal set Z = {x | x is an integer }. Find the complement of the set E = {x | x is an even integer}.

d. You can illustrate the universal set in a Venn diagram by drawing a large rectangle around the circles that represent the sets on which you are operating. Draw Venn diagrams that illustrate the situations in Parts a and c.

At certain states, the p-v-T data of a gas can be expressed as $Z=1-A p / T^{4},$ where Z is the compressibility factor and A is a constant. a. Obtain an expression for $(\partial p / \partial T)_{v}$ in terms of p, T, A, and the gas constant R. b. Obtain an expression for the change in specific entropy, $\left[s\left(p_{2}, T\right)-s\left(p_{1}, T\right)\right].$ c. Obtain an expression for the change in specific enthalpy, $\left[h\left(p_{2}, T\right)-h\left(p_{1}, T\right)\right].$

What is the area, to the nearest tenth, of a figure that is formed using a rectangle 10 feet long and 7 feet wide and a semicircle with a diameter of 7 feet?

Kumar rides his bicycle such that the back wheel rotates 10 times in 5 s. Determine the angular velocity of the wheel in a) degrees per second b) radians per second

Use the model for projectile motion, assuming there is no air resistance and g=32 feet per second per second. Determine the maximum height and range of a projectile fired at a height of 6 feet above the ground with an initial speed of 400 feet per second and an angle of $60^{\circ}$ above the horizontal.

A room is 15 feet long and 12 feet wide. What is the area of the room is square yards?

Generalize If point $B$ lies on minor arc $A C$, and $A B$ and $B C$ are non-overlapping adjacent arcs, what is an expression for the measure of the major arc associated with $A C$ in terms of $A B$ and $B C$ ?

In the following problem, rewrite the given expression in terms of cos $\theta$. Then simplify.

$$2 \sin \theta \cos \theta \cot \theta$$

Verify each identity.

$$\cos ^ { 2 } \frac { \theta } { 2 } = \frac { \sin \theta + \tan \theta } { 2 \tan \theta }$$

An object with negligible air resistance is dropped from a plane. During the first second of fall, the object falls 16 feet; during the second second, it falls 48 feet; during the third second, it falls 80 feet; and during the fourth second, it falls 112 feet. Assume this pattern continues. How many feet will the object fall in 8 seconds?

Solve the rational equation. If an equation has no solution, so state.

$$\frac{6}{x}-\frac{x}{3}=1$$

Rewrite the equation, using rational exponents. Consider the equation

$$\sqrt[3]{4x-4}=\sqrt{x+1}$$

In this exercise, write an equation that expresses each relationship. Then solve the equation for $y$.

$x$ varies directly as $z$ and inversely as the difference between $y$ and $w$.