Let D be the region bounded by the paraboloids z = 8 - x² - y² and z = x² + y². Write six different triple iterated integrals for the volume of D. Evaluate one of the integrals.

15. State the period,amplitude, and midline of the following functions.

(a) y=3sin(2x)+5

(b)5cos((pi/3)x)+2

(c) 2y = cos(10t-60)+2

The temperature at time t hours is

T(t) = −0.6t² + 2t + 70

(for 0 ≤ t ≤ 12). Find the average temperature between time 0 and time 10.

Solve the follwing game theory matrix,

Find the optimal strategies for both players

For the function g(x) = x^2-3 and the point P(2,1)

Use either LIMIT definition of the derivative to find the slope of the tangent line to the graph of g at P AND determine an equation of the tangent line at P.

This is an ODE related question,

A 1500 L tank initally contains 600 L of water with 50 kg of salt in it. Water enters the tank at a rate of 9 L/hour and the water entering the tank has a salt concentration 0.1 kg/L. A well mixed solution leaves the tank at a rate of 6L/hour.

a. The integrating factor that can be used to solved the differential equation for q(t) is?

b. The amount of salt in the tank when the tank overflows is?

Consider the non-right triangle shown below that has side lengths of 1.4, 1.829 and 2.6 cm and interior angle measures of 0.74, α , and β rad. 0.74 radαβ1.4 cm1.829 cm2.6 cm What is the value of α ?

α =

β =

Consider the non-right triangle shown below that has side lengths of 1.5, 1.817, and 2.6 cm and interior angle measures of 0.75, α, and β degrees.

What is the value of α =

What is the value of  β =

Find the volume of the solid obtained by rotating the region bounded by x = 4- (y-1) ^ 2; x + y = 4 on the X axis, you must graph the region

Find an appropriate model for the data. Round values to the nearest hundredth.

Find a set of parametric equations for the tangent line to the curve of intersection of the surfaces at the given point. (Enter your answers as a comma-separated list of equations.)

z = x² + y²,    z = 16 − y,    (4, −1, 17)

Use the first derivative test to determine the location of each local extremum and the value of the function at this extremum.

f(x)=4xe^-4x

1. The function has a local maximum of __at x=___.

Optimization problem for calculus 3.

Possible use of Lagrange Multiple?

Question: What are the largest and smallest outputs of the function x ²+2y² on the ellipse x²+y²+xy=3

8. (Section 1.3 #10) Let p be an integer other than 0, ±1. Prove p is prime if and only if for all a 2 Z
either (a, p) = 1 OR p | a.
9. (Section 1.3 #16) Prove that (a, b) = 1 if and only if there is no prime p such that p | a AND p | b.

There are 5 paltonic solids. Now the question is, what are the inscribed solids? What shape is contained within each platonic solid with a vertex at the center of each side?

1. Tetrahedron (4) faces made of Triangles

2. Hexahedron (6)Faces made of squares

3. Icosahedron (20) Faces made of Triangles

4. Octahedron (8) Faces made of Triangles

5. Dedecaheron (12) faces made of pentagon shapes