Suppose that you want to find the Fourier series of $f(x)=x+x^{2}$. Explain why to find $b_k$ you would need only to integrate $x \sin \left(\frac{k \pi x}{l}\right)$ and to find $a_k$ you would need only to integrate $x^{2} \cos \left(\frac{k \pi x}{l}\right)$.

Use ZPP. Find all the solutions to each equation. a. $(x-2)(x+1) = 0$ b. $(2x-3)(x+5) = 0$ c. $3x(x+8) = 0$ d. $x^2-4x = 0$

Factor each polynomial. $3 x^{2}+4 x+1$

What is inertia, and why is it important in the laws of motion?

Solve the linear system using elimination.

$$\begin{array} { l } { - x + \frac { 1 } { 2 } y = - 19 } \\ { x - y = 12 } \end{array}$$

Can the Doppler effect be observed with longitudinal waves, with transverse waves, or with both?

Describe how the focal length of a convex lens changes as the lens becomes more curved.

The product of two consecutive even integers is 224. Find the integers.

Prove that if p ones and q zeros are placed around a circle in an arbitrary manner, where p, q, and k are positive integers satisfying p$\geq$kq, the arrangement must contain at least k consecutive ones.

Suppose that electricity is draining from a capacitor at a rate that is proportional to the voltage V across its terminals and that, if t is measured in seconds, dV/dt = -V/40. Solve this equation for V, using V_0 to denote the value of V when t = 0. How long will it take the voltage to drop to 10% of its original value?

Write the decimal as a fraction or a mixed number.

$$- 1 . \overline { 63 }$$

A function f has derivative

$$f ^ { \prime } ( x ) = \frac { 1 } { x ^ { 4 } + 1 }.$$

Where on the interval [1, 4] does f take on its maximum value?

State the instructions of the function in words. R(r) = 3(2r + 5)

Solve equation $s-5=3$ and check your answer.