2. Use the Comparison Test or Limit Comparison Test to determine whether each series is convergent or divergent. Carefully explain your justification for using each test.

(a) Draw and shade in the rectangles needed to represent a left sum approximation of the distance traveled on the interval [0,30] with n = 6 rectangles. (b) Use the graph to estimate the distance traveled on the interval [0,30] with a left Riemann sum with n = 6 rectangles. Note: 1 second is equal to 1/3600 hours. (c) Does this give an overestimate or an underestimate of the distance traveled? (d) Interpret the Riemann Sum in the context of the situation. Include units in your answer.

A 4 (a) Determine all the elements of Z35. (b) Show that U = {1, 8, 22, 29} a subgroup of Z35 is, so (U, O) < (Z85, 0). (c) Find a subgroup of Z35 that contains 11 as an element. (1+2+1 Points) Please justify all answers. Solutions without a solution path will not be evaluated.

(2 + 2 Points) A 3 Let Pk: Z10 -> Z10 be an illustration with the rule Pk (n) = (k n) mod 10, k N. The figure yr is an encryption of Z10 if yr is injective. (a) Alice would like to send her friend Bob her phone number 0152-347896. For security reasons, she would like to encrypt the number and add a Find encryption of the form pr. First, she tries to do this with the- educations y2 and p3. What is your phone number in each case, after you click on every digit that has applied figures y2 and p3 respectively? (b) Show that v3 is an encryption. For which k € Z10 N N is the decoding (i.e. reverse graphing) of y3? (c) Why is 22 not an encryption? (d) For which k € N is yr an encryption and for which k € N is yk no encryption? Justify your answer. 4k (1+2+1 +2 Points) For the summer semester beginners: A figure f: X-> Y is called injective if there are no two x values that are mapped to the same y value: for all x1 # x2 € x, f(x1) # f(x2) apply. An equivalent formulation is: if f(x1) = f(x2) for two values x1, x2 € X, then x1 must be = x2 apply.

2. For the functions f and g defined by, f(x) = cosh (3x), g(x) = log(x) find the composite functions fog and go f stating clearly the intended domain and range.

Problem 3 i) Write a program in MATLAB/Octave/Scilab or Python (any other language please contact me) that randomly generates a positive definite matrix. ii) In order to check your previous code, write a function that implements Sylvester's criterion and use this function to check every matrix created in i). Problem

Determine whether each series is convergent or divergent. For each alternating series that is convergent, determine whether it is conditionally or absolutely convergent.

(a) Use the table to determine a midpoint Riemann Sum with n = (b) Interpret your answer in the context of that situation. What are you estimating with this Riemann Sum?

(3) OpenStax Section 1.1 435. Let r; denote the total rainfall in Portland on the jth day of the year in 2009. Interpret the following in the context of the situation.

Let A 1(a) Be (G,-) a group. For which x € G does x2 = x apply? (b) Specify three different o € S4 for which 03 = o applies. How many o € S4 are there any that satisfy this equation? (1 + 3 Points)