what are some differences and similarities between the models in Euclidean, Spherical, and hyperbolic geometries?

We have 600 m 2 of material to make a rectangular shaped crate with

square sides and an open top. Find the maximum possible volume of the crate.

For each problem, find the valuesc that satisfy Rolle's Theorem

5. y=x^2+4x+5 [-3,-1]

6. y=x^3-2x^2-x-1 [-1,2]

7. -x^3+2x^2+x-6 [-1,2]

8. x^3-4x-x+7 [-1,4]

Evaluate the improper integral from 0 to infinity xe^-x

Find the area of the indicated region. We suggest you graph the curves to check whether one is above the other or whether they cross, and that you use technology to check your answer. Between y = 2x^2 + 6x − 3 and y = −x^2 + 3x + 3 for x in [−2, 2]

Find a sequence {an} whose first five terms are -2, 8/2, -26/6, 80/24, -242/120 …. and find if it was converge or diverge

(1 point) Use Euler's method to solve

dBdt=0.04B

with initial value B=1000 when t=0 .

A.Δt=1 and 1 step: B(1)≈

B. Δt=0.5 and 2 steps: B(1)≈

C. Δt=0.25 and 4 steps: B(1)≈

D. Suppose B is the balance in a bank account earning interest. Be sure that you can explain why the result of your calculation in part (a) is equivalent to compounding the interest once a year instead of continuously. Then interpret the result of your calculations in parts (b) and (c) in terms of compound interest.

Find the integrals of the following:

1) 6sin²xcos³x dx

2) 9xcos²x dx

3) 3cos²xtan³x dx

4) 7cot⁵xsin⁴x dx

5) 3tan²xsecx dx

Find the area of the region bounded by the parabolas x = y^2 - 4 and x = 2 - y^2

the answer is 8 sqrt(3)

The curves of the quadratic and cubic functions are f(x)=2x-x^2 and g(x)= ax^3 +bx^2+cx+d. where a,b,c,d ER, intersect at 2 points P and Q. These points are also two points of tangency for the two tangent lines drawn from point A(2,9) upon the parobala. The graph of the cubic function has a y-intercept at (0,-1) and an x intercept at (-4,0). What is the value of the coefficient "b" in the equation of the given cubic function.

2. Let f(x)=2x^2−4x+7/5x^2+5x−9, evaluate f '(x) at x=3 rounded to 2 decimal places.

f '(3)=

3. Let f(x)=(x^3+4x+2)(160−5x) find f ′(x).
f '(x)=

4. Find the derivative of the function f(x)=√x−5/x^4

f '(x)=

5. Find the derivative of the function f(x)=2x−5/3x−3

f '(x)=

6. Find the derivative of the function g(x)=(x^4−5x^2+5x+4)(x^3−4x^2−1). You do not have to simplify your answer.

g '(x)=

7. Let f(x)=(−x^2+x+3)^5

a. Find the derivative. f '(x)=

b. Find f '(3)=

8.

Let f(x)=(x^2−x+4)^3

a. Find the derivative. f '(x)=

b. Find f' ′(-2)=

9. Let f(x)=2x^2−4x+3/3x^2+2x+11
Evaluate f '(x)=4 rounded to 2 decimal places.
f ′(4)=   

10. Let f(x)=(3x−5x^3)/(3+√x)

find f '(x)

f '(x)=

11. Find the derivative of the function f(x)=√7x+4

f '(x)=

The function P(x)=−1.5x^2+ 900x−4500 gives the profit when x units of a certain product are sold. Find
a) the profit when 75 units are sold  dollars
b) the average profit per unit when 75 units are sold dollars per unit
c) the rate that profit is changing when exactly 75 units are sold dollars per unit
d) the rate that profit changes on average when the number of units sold rises from 75 to 150. dollars per unit
e) The number of units sold when profit stops increasing and starts decreasing. (Round to the nearest whole number if necessary.)
    units

. Find the general solution to the ODE:

x^2 y" + 5xy' + 3y = x^2

1. The altitude of a triangle is increasing at a rate of 33 centimeters/minute while the area of the triangle is increasing at a rate of 4.54.5 square centimeters/minute. At what rate is the base of the triangle changing when the altitude is 11.511.5 centimeters and the area is 8686 square centimeters?

2. Gravel is being dumped from a conveyor belt at a rate of 30 cubic feet per minute. It forms a pile in the shape of a right circular cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 19 feet high? Recall that the volume of a right circular cone with height h and radius of the base r is given by using the volume formula?

3. A rotating light is located 14 feet from a wall. The light completes one rotation every 4 seconds. Find the rate at which the light projected onto the wall is moving along the wall when the light's angle is 5 degrees from perpendicular to the wall.

4. At noon, ship A is 40 nautical miles due west of ship B. Ship A is sailing west at 20 knots and ship B is sailing north at 20 knots. How fast (in knots) is the distance between the ships changing at 5 PM? (Note: 1 knot is a speed of 1 nautical mile per hour.

5.A street light is at the top of a 12 ft tall pole. A woman 6 ft tall walks away from the pole with a speed of 5 ft/sec along a straight path. How fast is the tip of her shadow moving when she is 30 ft from the base of the pole?

6.An inverted pyramid is being filled with water at a constant rate of 45 cubic centimeters per second. The pyramid, at the top, has the shape of a square with sides of length 3 cm, and the height is 5 cm.
Find the rate at which the water level is rising when the water level is 3 cm.

7.A circle is inside a square.

The radius of the circle is decreasing at a rate of 2 meters per day and the sides of the square are increasing at a rate of 2 meters per day.
When the radius is 3 meters, and the sides are 22 meters, then how fast is the AREA outside the circle but inside the square changing?
The rate of change of the area enclosed between the circle and the square is ____________ square meters per day.

8. A police car is located 40 feet to the side of a straight road.
A red car is driving along the road in the direction of the police car and is 200 feet up the road from the location of the police car. The police radar reads that the distance between the police car and the red car is decreasing at a rate of 95 feet per second. How fast is the red car actually traveling along the road?
The actual speed (along the road) of the red car is _________ feet per second

The curves of the quadratic and cubic functions are f(x)=2x-x^2 and g(x)= ax^3 +bx^2+cx+d. where a,b,c,d ER, intersect at 2 points P and Q. These points are also two points of tangency for the two tangent lines drawn from point A(2,9) upon the parobala. The graph of the cubic function has a y-intercept at (0,-1) and an x intercept at (-4,0). What is the standard equation of the tangent line AP.