1. Find the general solution of the given system using the characteristic equation

dxdt=6x-y
dydt=5x+2y

1. Find the general solution of the given system using the characteristic equation

dxdt=6x-y
dydt=5x+2y

1. Test the series below for convergence using the Root Test.
∞∑n=1 (2n/7n+5)^n
The limit of the root test simplifies to lim n→∞ |f(n)| where
f(n)=   
The limit is:    
Based on this, the series

• Diverges
• Converges

2. Multiple choice question.  We want to use the Alternating Series Test to determine if the series:

∞∑k=4 (−1)^k+2 k^2/√k5+3

converges or diverges.
We can conclude that:

• The Alternating Series Test does not apply because the terms of the series do not alternate.
• The Alternating Series Test does not apply because the absolute value of the terms do not approach 0, and the series diverges for the same reason.
• The series converges by the Alternating Series Test.
• The series diverges by the Alternating Series Test.
• The Alternating Series Test does not apply because the absolute value of the terms are not decreasing.

Find the equation of a circle that passes through (1,7). (6,2) and (4,5) using a matrix.

Use two functions below for parts a and b.
 ?(?)=??−?
 ?(?)=ln(?)+ln(1−?)+3
a) Find the stationary points, if any, of the following functions and label them accordingly (local or global minima/maxima or inflection point).
b) Characterize the above functions as convex, concave or neither convex nor concave

Determine the eigenvalues and the corresponding normalized eigenfunctions of the following Sturm–Liouville problem: y''(x) + λy(x) = 0, x ∈ [0;L], y(0) = 0, y(L) = 0,

True/False

If SS is a sphere and FF is a constant vector field, then ∬SF⋅dS=0.

Part 1. Describe the boundaries of the triangle with vertices (0, 0), (2, 0), and (2, 6). (a) Describe the boundary with the top function, bottom function, left point, and right point. (b) Describe the boundary with the left function, right function, bottom point, and top point.

Part 2. Consider the triangle with vertices (0, 0), (3, 0) and (6, 6). This triangle can be described using only one of the two perspectives presented above: top-bottom or left-right. Explain which perspective can be used and describe the region using that perspective. Write and label the boundary functions and points. If you want to use the other perspective, then you’ll have to split the shape into two different parts, each of which can be described using that perspective.

Part 3. Split the triangle in the previous exercise into two triangles. Describe each triangle as a region using the perspective you didn’t use in the previous exercise.

Solve the initial value problem y’cosx = a + y where y(π/3)=a and 0

please explain how you do everything

Find the area of the surface correct to four decimal places by expressing the area in terms of a single integral and using your calculator to estimate the integral.
The part of the surface z = e^{-x^2 - y^2} that lies above the disk
x2 + y2 ≤ 49

Please write clearly and show work. I am having trouble the the rdr integral.

Find dy/dx for a & b

a) sin x+cos y=1

b) cos x^2 = xe^y

c)Let f(x) = 5 /2 x^2 − e^x . Find the value of x for which the second derivative f'' (x) equals zero.

d) For what value of the constant c is the function f continuous on (−∞,∞)?

f(x) = {cx^2 + 2x, x < 2 ,

  2x + 4, x ≥ 2}

Find the integral that represents the volume of the following solids:

1. below the surface z=1+xy and over the triangle with vertices (1,1), (4,1) and (3,2).

2. enclosed by the planes y=0, z=0, y=x and 6x+2y+3z=6

Set up the triple integral of an arbitrary continuous function f(x, y, z) in spherical coordinates over the solid shown. (Assume a = 4 and b = 8. ) f(x, y, z) dV E = 0 π/2 f  , , dρ dθ dφ 4

Show that the equation has exactly one real root.

3x + cos(x) = 0

Solve this differential equation:

9x - 7i > 3(3x - 7u)

Please provide detailed solution

Multivariable Calculus

[A] Consider the region R in the first quadrant that is outside the circle r = 1 and inside the four-leaved rose r = 2 sin 2θ).

(A.1) Draw a sketch of the circle and the four-leaved rose (include the entire graph) and shade the region R. Feel free to use your graphing calculator.

(A.2) Write the following double integral as an iterated integral in polar coordinates. Do not evaluate the integral in this part. Be sure to use appropriate notation. (In order to find the interval for theta, you will have to find the TWO values of theta for which the circle and four-leaved rose intersect (in the first quadrant). Set the two functions equal to each other and solve the resulting equation; it should be a simple trig equation. Also note that the function that you are integrating, cos 2θ, is already written in polar form and thus will not need to be converted. Do not use decimal approximations for your angles; they should include a factor of π if you have found them correctly.)

∫_R ∫ cos 2θ dA

(A.3) Evaluate the integral in (A.2). Show all work!!! (After evaluating the inner integral, the outer integral should only require a U-substitution. Do not give a decimal approximation to the integral and do not use a computer program to calculate your antiderivatives and/or integrals.)