How do you find all solutions to: a^2+2b^2=c^2 ?

Discuss the determinants of future dividends and growth rate in dividends, and the sensitivity of the stock price to estimates of those two factors.

State and prove spectral mapping theorem

1.Suppose you have a function such that the domain of is −4≤x≤2

and the range of is −1≤y≤6.

a. What is the domain and range of the transformation

f(2(x+3))?

b. What is the domain and range of the transformation

2f(x)−3?

c. How do you know your answers are correct?

d. What can we say about how transformations affect the domain and range of a function?

2. Suppose a local vendor charges $2 per hot dog and that the number of hot dogs sold per hour x is given by x(t)=−4t^2+20t+92, where t is the number of hours since 10 AM, 0≤t≤4

a. Find an expression for the revenue per hour R as a function of x.

b. Find and simplify (R∘x)(t). What does this represent?

c. What is the revenue per hour at noon?

d. If the price were raised to $3 per hot dog with no change in the x(t) equation, which hour would produce the most revenue? Why?

e. If the price were dropped to $1 per hot dog, but that price drop caused the number of sales to increase according to the function x(t)=−9t^2+22t+138, would the vendor make more money at the original $2 price, or at the $1 price?

3. Danielle makes the claim that when the polynomial x^2−3x−10 is divided by x−5, the remainder is 0. Use what you have learned about dividing polynomials to either verify that Danielle is correct or prove that she is incorrect. What arguments would you use to support your claim? Are there any other arguments? Justify your answers.

1d. Explain the three criteria we use to evaluate solution concepts, & use them to compare Dominant Strategy Equilibrium with Nash Equilibrium?

Consider the equation
y′′ − 5y′ + 6y =e^3tcos(2t) + e^2t(3t+4)sint

(a) Determine a suitable form for yp if the method of undetermined coefficients is used

- Show steps for getting A, B, etc.

- if possible, split up the right hand side and handle each value seperately, aka y′′ − 5y′ + 6y =e^3tcos(2t),

then y′′ − 5y′ + 6y =e^2t(3t+4)sint, then adding together for final solution.

(b) Find the general solution of the equation.

R simulation:
Let X1, . . . , Xn be i.i.d. random variables from a uniform distribution on [0, 2]. Generate
and plot 10 paths of sample means from n = 1 to n = 40 in one figure for each case. Give
some comments to empirically check the Law of Large Numbers.

(a) When n is large,
X1 + · · · + Xn/n  converges to E[Xi].
(b) When n is large,
X1^2+ · · · + Xn^2/n converges to E[Xi^2 ]

T or F

1) Any N vectors spanning R^n are linearly independent

2)R5 has 7 linearly independent vectors

3) If a set of vectors with n elements is linearly dependent, then a set with n - 1 elements is also linearly dependent

4) There exists a Linear Function T:R^n -> R^n such that the range and the kernel of T are equal.

5) If a vector space has a dimension of n, then a basis for the vector space will contain n vectors

6) L: R^6 -> R^7 is one-to-one, then the range of L has a dimension of 7

7) The number of leading terms in ref(A) is equal to the dimension of the row space of matrix A

8) If a set of 4 vectors is linearly independent, then if you remove 1 vector, the set will still be linearly independent

Show that a set S has infinite elements if and only if it has a subset U such that (1) U does not equal to S and (2) U and S have the same cardinality.

i. |u| ≥ 0 and |u| = 0 iff u = 0.

ii. |au| = |a||u|.

iii. 1 |u| u (or just u/|u|) is a unit vector

Let f (x) = e^x - 4x²
a) Show that equation f (x) = 0 has three real solutions.
b) Use the Newton Method to calculate the largest of the solutions with precision.
preset of 0.01.

Given the set A = {(x, y) ∈ R² | x² + y² < 1 and y ≥ 0}. Draw sketches of cl A, int A, ∂A, (cl(A^c))^c, the limit points of A, and the isolated points of A. Try to be clear about what the sketch is describing. (The answer does not depend on whether one uses the Euclidean distance or the taxi distance on R².)