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.)

1. horizontal versus oblique asymptote

a. under what circumstances does a rational function have a horizontal asymptote of y=0

b. under what circumstances does a rational function have a horizontal asymptote that is not y=0

c. under what circumstances does a rational function have an oblique asymptote

d. find horizontal and /or oblique asymptote for each of the following

i. f(x)=x^2-3x+8/x+1

ii. g(x)=4/x+1

iii. h(x)=3x+4/x+1

e. what are the vertical asymptote for each of the functions f, g, and h in problem 1d?

A student asks if the precision with which manufacturers must calibrate their tools is at all related to statistics. How do you respond?

If three resistors with resistance of R1, R2, R3 are wired in parralle, their combined combined resitance R is given by the following formula:

1/R= 1/R1 + 1/R2 +1/R3

The design of a voltmeter requires that the resitance R2 be 80 ohms greater than the restance of R1, that the restance of R3 be 5 ohms greater than R1, and that their combined restance be 10 ohms. Find the value of each resistance.

R1=________ohms

R2=________ohms

R3=________ohms

3: Find the volume of the torus shown using:

a. The Shell Method.

b. The Washer Method.

Show that if the opposite angles of a quadrilateral add up to two right angles, then the vertices are concyclic

A tank contains 400 gallons of brine (salt in solution) in which 125 pounds of salt has been dissolved. Freshwater (with no salt added) runs into the tank at a rate of 4 gallons per minute, and the stirred mixture is drained from the tank at the same rate. (1) Find the amount of salt in the tank after an hour. (2) How long does it take to reduce the amount of salt in the tank to 10 pounds?

Find all pairs of x and y for both of the following two equalities hold:

x + y = 10

xy - 4

A tank contains 90 kg of salt and 2000 L of water: Pure water enters a tank at the rate 8 L/min. The solution is mixed and drains from the tank at the rate 4 L/min. What is the amount of salt in the tank initially? Find the amount f salt in the tank after 4.5 hours. Find the concentration of salt in the solution in the tank as the time approaches infinity. (Assume your tank is large enough to hold the solution.)