(a) Prove that there exist unique scalars $c_0,c_1,\dots,c_n$ such that

$$f(-1)+f(-2)=c_0f(0)+c_1f(1)+\dots+c_nf(n)$$

for every polynomial $f(x)$ in $\mathcal{P}_n$. (b) Find the scalars $c_0,c_1,c_2,c_3,$ and $c_4$ such that

$$f(-1)+f(-2)=c_0f(0)+c_1f(1)+c_2f(2)+c_3f(3)+c_4f(4)$$

for every polynomial $f(x)$ in $\mathcal{P}_4$.

Juanita runs every third day and swims every fifth day. If Juanita runs and swims today, in how many days will she run and swim again on the same day?

Fill in the blanks. Given a relation in x and y, if to each value of x in the domain there corresponds exactly one value of in the range, y is said to be a _____ of x. We call x the independent _____ and y the _____ variable.

Simplify: 8 1/3 · 1 4/5

Find the: a. smallest multiple of 6 that is greater than 200 b. greatest multiple of 11 that is less than 500.

We now have $5,000 in assets and are given a choice between investment 1 and investment 2. With investment 1, 80% of the time weincrease our asset position by$295,000, and 20% of the time we increase our asset position by $95,000. With investment 2, 50% of the time we increase our asset position by$595,000, and 50% of the time we increase our asset position by $5,000. Our utility function for final asset position x is u(x). We are given the following values for u(x): u(0) = 0, u(640,000) = .80, u(810,000) = .90, u(0) = 0, u(90,000) = .30, u(1,000,000) = 1, u(490,000) = .7. a. Are we risk-averse, risk-seeking, or risk-neutral? Explain. b. Will we prefer investment 1 or investment 2?

Convert each rectangular equation to a polar equation that expresses f in terms of

$$\theta.$$

$$2 x + 3 y = 8$$

Simplify the function. List any restrictions on the domain.

$$f ( x ) = \frac { x ^ { 2 } + 6 x - 16 } { x ^ { 2 } - 4 }$$

The region below the curve y=1/sqrt(4+x^2) from x=0 to x=2 is revolved avbout the x-axis. Find the volume of the resulting solid

Evaluate the expression when a=16, b=8, and c=7. 3a+2.1(4)

Let the domain of the function

$$f(x) = \frac { 1 } { 5 } x - 12$$

be {-5, 0, 10}. What is the range? F. {-5, 0, 10} G. {0, 12, 13} H. {-13, -12, -11} I. {-13, -12, -10}

A car goes into a skid and gradually comes to a stop, accelerating at a constant rate. At the midpoint of the skid, how much of its kinetic energy has it lost?

Find an equation of the curve that satisfies the given conditions. At each point (x, y) on the curve the slope equals the square of the distance between the point and the y-axis; the point (−1, 2) is on the curve.

Simplify

$$\frac { x ^ { 16 } - 1 } { x - 1 }$$

by long division and by factoring. Which method do you prefer? Explain your answer.