Consider the following functions.
f1(x) = cos(2x), f2(x) = 1, f3(x) = cos2(x)
g(x) = c1f1(x) + c2f2(x) + c3f3(x)
Solve for c1, c2, and c3 so that g(x) = 0 on the interval (−∞, ∞). If a nontrivial solution exists, state it. (If only the trivial solution exists, enter the trivial solution {0, 0, 0}.)
{c1, c2, c3} =?

Determine whether f1, f2, f3 are linearly independent on the interval (−∞, ∞).
linearly dependent or linearly independent?  

Let V and W be Banach spaces and suppose T : V → W is a linear map. Suppose that for every f ∈ W∗ the corresponding linear map f ◦ T on V is in V ∗ . Prove that T is bounded.

A mass that weight 15lb15lb stretches a spring 8in8in. The system is acted on by an external force 9sin(43–√t)lb9sin⁡(43t)lb.If the mass is pulled down 3in3in and then released, determine the position of the mass at any time tt. Use 32ft/s232ft/s2 as the acceleration due to gravity. Pay close attention to the units. Answer must be in inches

5. Provide a counterexample to a false statement. (The statement might be a “for all,” “there
exists,” or P ==> Q.)
6. Compute the product of two sets.
7. Given a relation (as ordered pairs or as a diagram), determine the domain, range, and target
of the relation.
8. Given a relation (as ordered pairs or as a diagram), determine if a pair is in the relation.
9. Convert a relation from a list of ordered pairs to a mapping diagram.
10. Convert a relation from a mapping diagram to a list of ordered pairs.

1. Write the following complex numbers in polar form.

a. 6 + ?2

b. 10 + ?0

c. -1/j

d. 1/6 (18 + ?24)

e. sqrt(-j)

1. Locate a root of sin(x)=x2 where x is in radians. Use a graphical technique and bisection with
the initial interval from 0.5 to 1. Perform the computation until ea is less than es=2%. Also
perform an error check by substituting your final answer into the original equation.
2. Determine the positive real root of ln(x
2
)=0.7 using three iterations of the bisection method,
with initial interval of [0.5:2].
3. Determine the lowest positive root of f(x)=7sin(x)e
-x
-1 using (a) the Newton-Raphson
method (three iterations, xi=0.3). (b) the secant method (five iterations, xi-1=0.5 and xi=0.4). (c)
the modified secant method (three iterations, xi=0.3, δ=0.01).
4. Use the Newton-Raphson method to find the root of f(x)=e
-0.5x
(4-x)-2. Employ initial guesses
of (a) 2, (b) 6, and (c) 8. Explain your results.

2.48. Is it true that {ax + by + cz|x, y, z ∈ Z} = {n · gcd(a, b, c)|n ∈ Z}?

2.49. What are all the integer values of e for which the Diophantine equation 18x + 14y + 63z = e has an integer solution. Find a solution for each such e.

2.50. For integers a, b and k > 0, is it true that a | b iff a^k | b^k ?

Question 5.

Use MATLAB to solve for and plot the response of the following models for 0≤t ≤1.5, where the input is f (t) =5t and the initial conditions are zero

a. 3¨ x +21˙ x +30x = f (t) b. 5¨

A (Turn in the MATLAB script and answers from MATLAB, .m file, screen shots if needed)

B (Turn in the MATLAB plot with t being time in SI units)

C Comment on the response the analytical solution compared with the MATLAB and the plots. (Do the calculations and MATLAB agree ? Why and Why not.? Do the plots make sense?)

Question 1. Find the general solution of the following ODEs

a) y′′ − 2y′ − 3y = e−x + 2e3x
b) y′′ + y′ + y = x2 + 4 cos x.

class : Analysis Real

Ques : give 2 example periodic function and show the periodic function is uniform contiunity

Calculate the relative (sub-space) topology with respect to the usual (metric) topology in R (the set of real numbers), for the following sub-sets of R:

X = Z, where Z represents the set of integers

Y = {0} U {1 / n | n is an integer such that n> 0}

Calculate (establish who are) the closed (relative) sets for the X and Y sub-spaces defined above.

Is {0} open relative to X?

Is {0} open relative to Y?

A child swinging on a swingset with chains of length l wants to swing and jump off the swing to get as far as possible. At what angle θ_jump should the child let go of the swing to maximize the landing distance? What is the distance jumped as a function of θ_jump?

Results should be functions of θ_jump and variables/parameters which should be introduced as required.

Model swinging using simple pendulum and jumping using projectile motion. Note, the child is not being pushed.

At 1:43pm, a can of soda is placed in a fridge where the temperature is 34◦F.
At 1:46pm, the temperature of the can had dropped 15◦F. At 1:49pm, the temperature
of the can had dropped another 6◦F. What was the temperature of the can when it was
put in the fridge?

For a tetrahedron

-describe the type of groups of a rectangle

-describe the orders of the groups

-describe the structure of the groups

-describe the elements of the groups (make sure to name all the elements and describe them as a group of permutations on the vertices)

- describe each group as subgroups of permutation groups

-describe all possible orders, types and generators for each subgroup of the group

-are any of these groups cyclic and or abelian?

-are any of these subgroups cyclic and or abelian?

-are there subgroups of every possible order?

-which subgroups are isomorphic and how do you know?