1.Determine which amounts of postage can be formed using just 3-cent and 10-cent stamps.

2.Prove your answer to (a) using the principle of mathematical induction. Be sure to state explicitly your inductive hypothesis in the inductive step.

3.Prove your answer to (a) using strong induction. How does the inductive hypothesis in this proof differ from that in the inductive hypothesis for a proof using mathematical induction?

q = a(b+c(de+f)), t = !(q+a), m = aq, where the inputs are a, b, c, d, e, f and q; and outputs are t and m.

Please provide the truth table.

Please write the Sum of Products for the output t.

Implement the Sum of Products of t with AND2, OR2 and NOT gates.

Convert the previous part by using ONLY NAND2 gates.

Provide MIPS code for the last part.

Give examples of practical and theoretical math from 1700s,1800s,1900s. What was the interaction between the theoretical and the applied. What extent practical considerations at times get ahead of theoretical.

S(x) is a cubic spline for the function f(x) = sin(pi x/2) + cos(pi x/2) at the nodes x_{0 ⁼} 0 , x₁ = 1 , x₂ = 2

and satisfies the clamped boundary conditions. Determine the coefficient of x³ in S(x) on [0,1] ans. pi/2 -3/2

Consider rolling two 6-sided dice. What is the probability that

1. at least two of the rolls have a sum that exceeds 6?
2. at least 7 of the rolls have a sum that is even?
3. exactly three rolls have a sum that equals 5?

Give an example of a continuous function that is not uniformly continuous. Be specific about the domain of the function.

prove that if f is a univalent function in D then w=f(z) is conformal mapping in every point in D

Please solve the following equation by using the frobenius method.

xy′ = (x + 1)y

My apologies, the original image did not upload. Thank you!

Use the graph to find the limit L (if it exists). If the limit does not exist, explain why. (If an answer does not exist, enter DNE.)

h(x) = -x/2 + x²

(a)

lim x→2 h(x)

L =

(Select One)

The limit does not exist at x = 2 because the function is not continuous at any x value.

The limit does not exist at x = 2 because the function approaches different values from the left and right side of 2.    

The limit does not exist at x = 2 because the function value is undefined at x = 2.

The limit does not exist at x = 2 because the function does not approach f(2) as x approaches 2.The limit exists at x = 2.

(b)

lim x→1 h(x)

L =

(Select One)

The limit does not exist at x = 1 because the function does not approach f(1) as x approaches 1.

The limit does not exist at x = 1 because the function approaches different values from the left and right side of 1.    

The limit does not exist at x = 1 because the function is not continuous at any x value.

The limit does not exist at x = 1 because the function value is undefined at x = 1.The limit exists at x = 1.

Find the solution of the given initial value problem:

3y′′′+27y′−810y=0

y(0)=11, y′(0)=39, y′′(0)=−261

Let E be a projective conic through the quadrilateral ABCD, and let the tangents to E at A and C meet at P on the Line BD. Show that the tangents to E at B and D meet at a Point Q on AC.

Hint: Consider poles and polars.

Solve this system using LU-factorization.

3x_1-2x_2+ x_3 =-5

x_2- x_3= -2

-3x_1+ x_2 + 2x_3 = 13

Theorem (Three Tangents): Let a non-degenerate plane conic touch the sides BC, CA, and AB of a triangle ABC in R² at the points P, Q, and R respectively. Then AP, BQ, and CR are concurrent.

Please provide a proof of the Three Tangents Theorem without reference to Ceva's Theorem.

Hint: Consider the Three Point Theorem

a) Using the method of undetermined coefficients, find the general solution of yʺ + 4yʹ − 5y = e^−4x

b) Solve xy'=(x+1)y^2

c) Solve the initial value problem : (x−1)yʹ+3y= 1/ (x-1)^2 + sinx/(x-1)^2 , y(0)=3

What is the dominant color with these x and y values? x=7/15, y= 7/15 (z= 1/15 of course)

Select one:
a. Reddish green
b. Greenish blue
c. Violet
d. Yellow