Calculus

2) Maple syrup is being pumped into a cone-shaped vat in a factory at a rate of six cubic feet per minute. The cone has a radius of 20 feet and a height of 30 feet. How fast is the maple syrup level increasing when the syrup is 5 feet deep?

Find the centroid (x, y) of the region bounded by: y=e²x, y=0, x=0, and x = = 3. 18 IN y || ||

14. Write an ordered triple for each vector. a) AB with A(0, -3, 2) and B(0, 4,-4) b) CD with C(4, 5, 0) and D(-3, -3, 5)

(7) Suppose that a revenue function is given as R(x) = 3r³ +36r²+10r and a cost function is given as C(r) = 4r³ +15² +82x-500, where r is the number of items. (a) Find the profit function P(r) in this situation. (b) What production level a maximizes profit? (c) What is the maximum profit?

11. Determine the exact magnitude of each vector in question 10.

The masses m, are located at the points P¿. Find the moments Mr and My and center of mass of the system. 1, m2 = 2, m3 = 9. P₁ = (-2,0), P₂ = (6,4), P3 = (2,8). m1 Mx My x = y = = =

Problem 1. For the vector field F(x, y, z) = xzi+xyzj- y²k, find div F and curl F.

¸¡₁(g(x))• g'(x) dx = p) Use the Substitution Formula, f(g(x)).g'(x) dx = a g(a) 2 In x X 2 Inx -dx dx = (Type an exact answer.) f(u) du where g(x)=u, to evaluate the following integral. C...

For two rectangles, area For four rectangles, area Question 11 of 21 (Type an integer or a decimal.) (Type an integer or a decimal.) This test: 21 point(s) possible This question: 1 point(s) possible Using rectangles whose height is given by the value of the function at the midpoint of the rectangle's base (the midpoint rule), estimate the area under the graph of the following function, using two and then four rectangles. y = 9-x² between x = -3 and x = 3 Submit test

Question 2, 6.1.15 Part 1 of 4 HW Score: 16.59%, 7.13 of 43 points Points: 4.2 of 7 Save m S Consider an object moving along a line with the given velocity v. Assume t is time measured in seconds and velocities have units of Complete parts a through c. a. Determine when the motion is in the positive direction and when it is in the negative direction. b. Find the displacement over the given interval. c. Find the distance traveled over the given interval. v(t)=2t² - 16t+24; [0,7] a. When is the motion in the positive direction? Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. O A. For t-values that satisfy (Use a comma to separate answers as needed. Type your answers in interval notation.) O B. The motion is never in the positive direction.