1. Use the definition of convexity to prove that the function f(x) = x2 - 4x + 8 is convex. Is
this function strictly convex?
2. Use the definition of convexity to prove that the function f(x)= ax + b is both convex and
concave for any a•b ≠ 0.

7. Consider a two-step, serial, production process with one resource at each step. The processing time at step 1 is 10 minutes and the processing time at step 2 is 5 minutes. There is ample supply of raw materials for step 1 and ample demand.

a) What is the capacity of this process in units per hour?

b) Suppose there is variability in the processing time at step 2. Specifically, the coefficient of variation of processing time at step 2 is 2. However, there is no variability in the processing time at step 1. Now what is the total time needed for a part to go through step 2?

Hint: Total time includes processing and waiting. If step 1 is always working and processing time there is 5 minutes, how much time passes between units arriving at step 2?

c) Now what is the capacity of this process in units per hour?

Using the successor function prove distribution over addition. Do this in a detailed proof.

Let T(x1, x2) = (-x1 + 3x2, x1 - x2) be a transformation.

a) Show that T is invertible.

b)Find T inverse.

y''+4y=u_π(t)−u_3π(t) ; y(0)=0,y'(0)=0

a.Sketch the graph of the forcing function on an appropriate interval.

b.Find the solution of the given initial value problem.

c.Plot the graph of the solution.

d.Explain how the graphs of the forcing function and the solution are related.

Both parts.

a) identify Fourier series for full wave rectified sine function f(x) = | sin(x) |.

b) f(t) = cos(t) but period of 6, so t = [-3,3] (L = 6) Find the Fourier series of the resulting function.

Lydia saved $1,345,000 for retirement. The money is deposited in an account earning 3.2% compounded monthly. She is going to withdraw $5500 per month for living expenses. Create a table showing how much interest she earns each month and her monthly balance for the first 5 months of her retirement. Do this by hand with just the functions of a scientific calculator.

All necessary steps much show for these problems, please.

1. Use the Euclidean algorithm to find gcd(12345, 54321).
2. Write gcd(2420, 70) as a linear combination of 2420 and 70. The work to obtain the gcd is provided.
   2420 = 34(70) + 40

70 = 1(40) + 30

40 = 1(30) + 10

30 = 3(10) + 0

3. Determine if 1177 is prime or not. If it is not, then write 1177 as a product of primes
4. Find gcd(8370, 465) by unique factorization into products of primes.
5. Compute ?(39204) if 39204 = 2 ∙ 3 ∙ 11 .

A = { [1,2,3,2] , [-2,0,-2,-4], [0,4,4,0], [1,2,3,2]}

Can you answer the following questions regarding this matrix:

a) Find the null space of A

b) Find vectors v1,v2.... such that Null A = span {v1,v2...}

c) Is the null space of A subspace of R4

Identify the Fourier series for the half-wave rectified sine function with period 2π. This is the function that is sin(x) when sin(x) is positive, and zero when sin(x) is negative

Let y = mx where m = tan(37) is the slope. That is 37 degrees.
a. Find the Matrix that projects vectors onto this line. Find all eigenvalues
and Eigenvectors.
b. Find the Matrix that reflects vectors across this line. Find all eigenvalues
and eigenvectors.

Let f(x) = sin(x) for -pi < x < pi and be ZERO outside this interval
a. Find the Fourier Transformation. Plot on Desmos.
b. Find the Fourier Integral representation of f(x). Plot on Desmos using
reasonable limits of integration

Find the Fourier Series for the function defined over -5 < x < 5
f(x) = -2 when -5<x<0 and f(x) = 3 when 0<x<5

You can use either the real or complex form but must show work.
Plot on Desmos the first 10 terms of the series along with the original
function.

role and benefit of auditing standards in Australia