Using the Maclaurin series, show that if f(z) is even, its integral (with a suitable constant of integration) is odd.

Identify the constant of variation. Then write the equation of variation.

w varies inversely with z; w = -8 when z = 3.

Prove $\frac{d}{d x} \ln |a x|=\frac{d}{d x} \ln |x|$ for any constant a.

Describe how air pressure changes with an increase in altitude. Does it change at a constant rate?

Consider the following gaseous sample in a cylinder fitted with a movable piston. Initially, there are $n$ moles of the gas at temperature $T$, pressure $P$, and the volume $V$. Choose the cylinder that correctly represents the gas after each of the following changes. (1) The pressure on the piston is tripled at constant $n$ and $T$. (2) The temperature is doubled at constant $n$ and $P$. (3) $n$ moles of another gas are added at constant $T$ and $P$. (4) $T$ is halved and pressure on the piston is reduced to a quarter of its original value.

In the position shown the thin rod moves at a constant speed $u=3\ \mathrm{in} . / \mathrm{s}$ out of the tube $B C$. At the same time tube $B C$ rotates at the constant rate $\omega_2=1.5\ \mathrm{rad} / \mathrm{s}$ with respect to arm $C D$. Knowing that the entire assembly rotates about the $X$ axis at the constant rate $\omega_1=1.2\ \mathrm{rad} / \mathrm{s}$, find the velocity and acceleration of end $A$ of the rod.

Use a computer to graph the parametric surface. Get a printout and indicate on it which grid curves have u constant and which have v constant. x=sin v, y = cos u sin 4v, z = sin 2u sin 4v, 0≤u≤2π, -π/2≤v≤π/2

Consider the function r(u, v) = 2u cos vi + 2u sin vj + vk. (a) Sketch a graph of the function where is held constant at u = 1. Identify the graph. (b) Sketch a graph of the function where is held constant at v = 2π/3. Identify the graph. (c) Assume that a surface is represented by the vector-valued function r = r(u, v). What generalization can you make about the graph of the function when one of the parameters is held constant?

Use mathematical induction to prove statement.

For constant positive integers $m$ and $n$, show that $\left(x^m\right)^n=x^{m n}$.

Determine the equations of the elastic curve using the $x_1$ and $x_2$ coordinates. $E I$ is constant.

What mathematical constant does $\lim _ { x \rightarrow 0 } ( 1 + x ) ^ { 1 / x }$ appear to equal?

In which one of the following situations can the equations of kinematics not be used? (a) When the velocity changes from moment to moment, (b) when the velocity remains constant, (c) when the acceleration changes from moment to moment, (d) when the acceleration remains constant.

Which of the objects below is accelerating?(a) Object moving at constant speed along a straight line (b) Object moving at constant speed in a circle (c) Object slowing down while moving in a straight line (d) Both b and c describe accelerating objects.

The value of the equilibrium constant for the reaction below is 40.0 at a specified temperature. What would be the value of that constant for the reverse reaction under the same conditions?

$$\mathrm { H } _ { 2 } ( g ) + \mathrm { I } _ { 2 } ( g ) \Longleftrightarrow 2 \mathrm { HI } ( g )$$