Let H = {A∈GL(2,R)|(detA)2= 1}. Prove that H / GL(2,R).
In: Advanced Math
Find equations of the following. 5 2 (x − z) = 10arctan(yz), (1 + π, 1, 1)
(a) the tangent plane Incorrect: Your answer is incorrect.
(b) parametric equations of the normal line to the given surface at the specified point. (Enter your answer as a comma-separated list of equations. Let x, y, and z be in terms of t.)
In: Advanced Math
Q2. Let (E, d) be a metric space, and let x ∈ E. We say that x is isolated if the set {x} is open in E.
(a) Suppose that there exists r > 0 such that Br(x) contains only finitely many points. Prove that x is isolated.
(b) Let E be any set, and define a metric d on E by setting d(x, y) = 0 if x = y, and d(x, y) = 1 if x and y are not equal. Prove that every point x ∈ E is an isolated point. The metric d is called a discrete metric, and the space (E, d) is called a discrete metric space.
(c) Let (E, d) be a discrete metric space, as in part (b). Prove that every subset of E is both open and closed.
In: Advanced Math
In: Advanced Math
Use a computer algebra system to graph f and to find f ' and f ''. Use graphs of these derivatives to find the following. (Enter your answers using interval notation. Round your answers to two decimal places.)
f(x) =
| x3 + 5x2 + 1 |
| x4 + x3 − x2 + 2 |
The intervals where the function is increasing.
The intervals where the function is decreasing.
The local maximum values of the function. (Enter your answers as a
comma-separated list.)
The local minimum values of the function. (Enter your answers as a
comma-separated list.)
The inflection points of the function.
The intervals where the function is concave up.
The intervals where the function is concave down.
In: Advanced Math
Question 10: (1 point) We want to solve the differential equation dy/dx=[x(e^(7x+y^5)]/y4 a) This differential equation is separable and can be writen. ∫N(y)dy=∫M(x)dx , where M(x)= __________ and N(y)= __________ b) To compute the integral ∫N(y)dy, you need to use the substitution u= __________ With this substitution, we get ∫N(y)dy=∫n(u)du, where n(u)= __________ We find that ∫n(u)du= __________ +K and hence ∫N(y)dy= __________ +K, where we have added the constant of integration K for you; so you don't need to add one in your answer. c) To compute the integral ∫M(x)dx, you need to use integration by parts with f(x)= __________ and g′(x)= __________ With this choice, we get ∫M(x)dx=P(x)−∫Q(x)dx , where P(x)= __________ and Q(x)= __________ We then find that ∫M(x)dx= __________ +D where we have added the constant of integration D for you; so you don't need to add one in your answer. d)Solve ∫N(y)dy=∫M(x)dx for y to find the general solution of the given differential equation. Then, give the particular solution satisfying the initial condition y(0)=0. y(x)= __________ |
In: Advanced Math
Let ?⃗(?)=〈(?0cos?)?,−12??2+(?0sin?)?〉r→(t)=〈(v0cosθ)t,−12gt2+(v0sinθ)t〉 on the time interval [0,2?0sin??][0,2v0sinθg], where ?>0g>0; physically, this represents projectile motion with initial speed ?0v0 and angle of elevation ?θ (and ?∼9.81m/s2g∼9.81m/s2).
Find speed as a function of time ?t, and find where speed is maximized/minimized on the interval [0,2?0sin??][0,2v0sinθg].
In: Advanced Math
When using secure hash functions in an RSA signature, why do we sign the hash Sign (H (m)) instead of taking the take the hash H (Sign (m)) ?
In: Advanced Math
Stochastic Processes:
1. What does it mean for a Markov Chain to be irreducible?
2. What simple conditions imply that a Markov Chain is irreducible?
In: Advanced Math
Math of Finance
You obtain a 150,000 home loan for $25 years at 4.8% interest compounded monthly. After 8 years the rate is raised to 6.3% Compute:
a)The new payment if the term of the loan is to remain the same.
b) The term of the loan if the payment remains the same. What is the size of the concluding payment?
DO NOT USE EXCEL TO RESPOND TO THIS QUESTION. Use the appropriate formula and show work instead.
In: Advanced Math
Solve the initial value problem below using the method of Laplace transforms.
y"-4y'+13y=10e^3t y(0)=1, y'(0)=6
In: Advanced Math
Prove that the set of irrational numbers is uncountable by using the Nested Intervals Property.
In: Advanced Math
In: Advanced Math
(a) How many passwords can you make with 8 characters using
Uppercase, Lowercase,
digits using at least one Uppercase, at least one Lowercase and at
least one digit?
(b) How many numbers between 1 and 1,000,000 (inclusive) are
divisible by at least one of
3,4 and 5?
(c) How many numbers between 1 and 1,000,000 (inclusive) are
divisible by at least one of
12, 14, 15?
In: Advanced Math
1. Using domain and range transformations, solve the following recurrence relations:
a) T(1) = 1, T(n) = 2T(n/2) + 6n - 1
b) T(1) = 1, T(n) = 3T(n/2) + n^2 - n
In: Advanced Math