(1) z=ln(x^2+y^2), y=e^x. find ∂z/∂x and dz/dx.

(2) f(x1, x2, x3) = x1^2*x2+3sqrt(x3), x1 = sqrt(x3), x2 = lnx3. find ∂f/∂x3, and df/dx3.

Determine the centroid, C(x̅, y̅, z̅), of the solid formed in the first octant bounded by z + y − 16 = 0 and 2x² − 32 + 2y = 0.

Solve y'''-4y'=32xe^2x-24x² , y(0)=0 y'(0)=0 y''(0)=-1

Determine the volume of the solid of revolution generated by rotating around the axis; x=9 the region bounded by the curves: y^2=9-x; y=3-x

Consider a car traveling on a highway. If the car travels 100 miles in 2 hours, which theorem guarantees that the car must have been traveling at 50 mph at some point in those two hours? (You may assume position and velocity are continuous and differentiable)

An equation of an ellipse is given. 2x^2 + 64y^2 = 128 (a) Find the vertices, foci, and eccentricity of the ellipse. vertex(x, y)= (smaller x-value) vertex(x, y)= (larger x-value) focus(x, y)= (smaller x value) focu (x, y)= (larger x-value) eccentricity (b) Determine the length of the major axis. Determine the length of the minor axis. (c) Sketch a graph of the ellipse.

Find all horizontal and vertical asymptotes (if any). (If an answer does not exist, enter DNE. Enter your answers as a comma-separated list of equations.)

r(x) =

The displacement (in centimeters) of a particle moving back and forth along a straight line is given by the equation of motion

s = 3 sin(πt) + 4 cos(πt), where t is measured in seconds. (Round your answers to two decimal places.)

(a) Find the average velocity during each time period.

(i)    [1, 2]
_______cm/s

(ii)    [1, 1.1]
_______ cm/s

(iii)    [1, 1.01]
______ cm/s

(iv)    [1, 1.001]

_______cm/s

B) Estimate the instantaneous velocity of the particle when t=1.

1. "Give an example of a function that is defined on the set of integers that is not a one-to-one function."
Keep in mind that the above domain must be the set of integers. Identify what your codomain is, too.

2. "Give an example of a function that is defined on the set of rational numbers that is not an onto function."
The above domain must be the set of rational numbers. Identify what your codomain is, too. This is especially important when explaining why a function is not onto.

3. "Give an example of a function defined on the real numbers that is a one-to-one correspondence."
The above domain must be the set of real numbers. Identify what your codomain is, too.

With the area between the graphs:
x+y=14
x+6=y^2

Find the area two different ways

First way:
[a,b]integral(f(x))dx + [b,c]integral(g(x))dx =

whats is a,b,c, f(x), and g(x):

Second way:
[a, b]integral(h(y))dy

what is a, b, and h(y)

A parallelogram has consecutive sides with lengths 9 and 7 and diagonals of integral length How long are these diagonals?

(6) A diagonal of a rectangle in neutral geometry divides the rectangle into triangle I and triangle II. Can the angle-sum of triangle I be less than 180°? Why or why not?

Use spherical coordinates to find the volume of solid within the sphere x^2 + y^2 + z^2 = 16 and above the cone 3z^2 = x^2 + y^2 and lying in the 1st octant.

When Ashley was asked to round 345 to the nearest 100, she rounded 345 to 350 and then rounded 350 to 400. Billie claimed the answer was 300. How would you respond to these students?

A shop sells two competing brands of socks, Levis and Gap. Each pair of socks is obtained at a cost of 33 dollars per pair. The manager estimates that if he sells the Levis socks for xx dollars per pair and the Gap socks for yy dollars per pair, then consumers will buy 10−3x+y10−3x+y pairs of Levis socks and 1+3x−2y1+3x−2y pairs of Gap socks. How should the manager set the prices so that the profit will be maximized? Round your answers to the nearest cent.