Construct a truth table for the statement. p∧~p

(a) Use a slope field to represent the solution of each differential equation. (b) Use the slope field representation to draw some graphs of the family of solutions to the differential equation.

Construct a truth table for the compound statement. Determine the truth value of each compound statement if the given statements are true. (~p ˅ q) ˄ r; q, r

Describe all ways in which the tetrahedral group T can operate on a set of two elements.

For small, slowly falling objects, the assumption made in Eq. (i) of Problem 8 that the drag force is proportional to the velocity is a good one. For larger, more rapidly falling objects, it is more accurate to assume that the drag force is proportional to the square of the velocity. (a) Write a differential equation for the velocity of a falling object of mass m if the drag force is proportional to the square of the velocity. Assume that the upward direction is positive. (b) Determine the limiting velocity after a long time. (c) If m = 0.025 kilograms (kg), find the drag coefficient so that the limiting velocity is −35 m/s.

Find f(2), f(3), f(4), and f(5) if f is defined recursively by f(0) = −1, f(1) = 2, and for n = 1, 2, . . . a) f(n + 1) = f(n) + 3f(n − 1). b) f(n + 1) = f(n)² f(n − 1). c) f(n + 1) = 3f(n)² − 4f(n − 1)². d) f(n + 1) = f(n − 1)/f(n).

Simplify.

$$\sqrt { 196 c ^ { 6 } d ^ { 4 } }$$

A spherical raindrop evaporates at a rate proportional to its surface area. Write a differential equation for the volume of the raindrop as a function of time.

Use Separation of Variables to find the general solution. yy' = x

Use the separation of variables technique to derive the solution $(7)$ to the differential equation $(6)$.

The half-life of a radioactive material is the time required for an amount of this material to decay to one-half its original value. Show that for any radioactive material that decays according to the equation Q'=−rQ,the half-lifeτand the decay rate r satisfy the equation rτ=ln 2.

A radio active material,such as the isotope thorium-234,disintegrates at a rate proportional to the amount currently present. If Q(t)is the amount present at time t , then dQ/dt=−rQ, where r>0 is the decay rate.(a) If 100 mg of thorium-234 decays to 82.04 mg in 1 week, determine the decay rater.(b) Find an expression for the amount of thorium-234 present at any time t.(c) Find the time required for the thorium-234 to decay to one-half its original amount.

Does the problem involve permutations or combinations? Explain your answer. (It is not necessary to solve the problem.) How many different 4-card hands can be dealt from a 52-card deck?

The components of vector $\vec{A}$ are $A_{x}$ and $A_{y}$ (both positive), and the angle that it makes with respect to the positive x axis is $\theta$. Find the angle $\theta$ if the components of the displacement vector $\vec{A}$ are (a) $A_{x}=12 \mathrm{m}$ and $A_{y}=12 \mathrm{m}$, (b) $A_{x}=17 \mathrm{m}$ and $A_{y}=12 \mathrm{m}$, and (c) $A_{x}=12 \mathrm{m}$ and $A_{y}=17 \mathrm{m}$.