What is an example of a situation that you might be able to use an equation with a single unknown to help understand? What is an example of a situation that you might not be able to use an equation with a single unknown to understand? What makes an equation with a single unknown helpful in one of your examples but not the other? What patterns exist in your two examples that might be helpful in determining when to use a simple equation?

A golf ball is hit and lands 53 m away. The path of the ball took it just over a 9 m tree. Determine a function that models the path of the golf ball

How do you mathematically prove a line moves faster than the other? The problem says there are two fast food restaurants, each containing two employees working the register. Restaurant 1 has two lines to service customers, and restaurant 2 has one line for customers to wait in. How can you mathematically prove that either restaurant 1 or restaurant 2 has a better line control system?

Side note: I was thinking that this is like which function has the better growth such as an exponential line vs. a power function line, but maybe I'm wrong. Please show me the easiest way.

Give a direct proof of the following theorem, upon which case you can use it for future proofs. (Hint: note that we’ve called it a corollary as in p.81, not just a theorem.) Corollary 4.12. Every integer is even or odd.

Prove the following two statements: If the permutation α is even then α^-1 is even. If the permutation α is odd then α^-1 is odd.

Define the Hamiltonian Cycle Problem and the Travelling Salesman Problem. Give a polynomial-time transformation from the Hamiltonian Cycle Problem to the Travelling Salesman Problem to claim that if the Hamiltonian Cycle is ”Hard” (i.e., NP-Complete) then Travelling Salesman Problem must also be hard.

If a population with harvesting rate h is modeled by

dx/do=9-x^2-h

find the bifurcation point for the equation.

Which of the following sets is an abstract simplical complex? For each, if the answer is no, explain why; and if the answer is yes, give the dimension of the complex, and sketch its geometric realization, up to homeomorphism. a. {[a],[b],[a,b,c]},

b. {[a],[b],[c],[a,b,c]},

c. {[a],[b],[c],[a,b]},

d. {[a],[b],[c],[d],[a,b],[c,d]},

e. {[a],[b],[c],[d],[a,b],[b,c],[c,d],[a,d],[a,c],[a,b,c]}.

A:=<<0,-1,1>|<4,0,-2>|<2,-1,0>|<2,1,1>>;
Matrix(3, 4, [[0, 4, 2, 2], [-1, 0, -1, 1], [1, -2, 0, 1]])

(a) Use the concept of matrix Rank to argue, without performing ANY calculation, why the columns of this matrix canNOT be linerly independent.

(b) Use Gauss-Jordan elimination method (you can use ReducedRowEchelonForm command) to identify a set B of linearly independent column vectors of A that span the column space of A. Express the column vectors of A that are not included in the set B as a linear combination of the vectors in the set B.

(c) Do the columns of matrix A span the entire Euclidean space
"real^3"
? Explain why yes or why not.

Solve the following initial value problems by the method of undetermined coefficients:

A.) y′′ + y = 4 sin t − cos t.

B.) y′′ = 5t4 − 2t.

A spring is stretched by 7in by a mass weighing 15lb. The mass is attached to a dashpot mechanism that has a damping constant of 0.1lb⋅s/ft and is acted on by an external force of 5cos(9t)lb. Determine the steady state response of this system. Use 32ft/s2 as the acceleration due to gravity. Pay close attention to the units.

Solve the following differential equation:

y''+e^{-y}=0 where y is only a function of x.