Consider the following function. f(x) = x(x−10)3 Find the first and second derivatives. f′(x) = f′′(x) = Find the points of inflection of the graph of the function. (If an answer does not exist, enter DNE.) smaller x-value (x, y) = ( ) larger x-value (x, y) = ( ) Determine the open intervals on which the graph of the function is concave upward or concave downward. (Enter your answers using interval notation. If an answer does not exist, enter DNE.) concave upward concave downward
Consider the following equations. f(x) = x+2 g(x) = 12 x+2 Sketch the region bounded by the graphs of the functions. Find the area of the region.
Set up a definite integral that yields the area of the region. (Do not evaluate the integral.) f(x) = 4 = 2x dx
Match each solid with the integral that represents its volume, and give the dimensions of each solid. (a) right circular cylinder π∫0 h(rxh)2 dx π∫0 hr2 dx π∫−rr(r2−x2)2 dx π∫−bb(a1−x2 b2)2 dx π∫−rr[(R+r2−x2)2−(R−r2−x2)2]dx Give the dimensions of the solid. A right circular cylinder with radius and height (b) ellipsoid π∫0 h(rxh)2 dx π∫0 hr2 dx π∫−rr(r2−x2)2 dx π∫−bb(a1−x2 b2)2 dx (b) ellipsoid π∫0 h(rxh)2 dx π∫0 hr2 dx π∫−rr(r2−x2)2 dx π∫−bb(a1−x2 b2)2 dx π∫−rr[(R+r2−x2)2−(R−r2−x2)2]dx Give the dimensions of the solid. An ellipsoid with axes . (c) sphere π∫0 h(rxh)2 dx π∫0 hr2 dx π∫−rr(r2−x2)2 dx π∫−bb(a1−x2 b2)2 dx π∫−rr[(R+r2−x2)2−(R−r2−x2)2]dx Give the dimensions of the solid. A sphere with radius . (d) right circular cone π∫0 h(rxh)2 dx π∫0 hr2 dx π∫−rr(r2−x2)2 dx π∫−bb(a1−x2 b2)2 dx π∫−rr[(R+r2−x2)2−(R−r2−x2)2]dx Give the dimensions of the solid. A right circular cone with radius of the base and height (e) torus π∫0 h(rxh)2 dx π∫0 hr2 dx π∫−rr(r2−x2)2 dx π∫−bb(a1−x2 b2)2 dx π∫−rr[(R+r2−x2)2−(R−r2−x2)2]dx Give the dimensions of the solid. A right circular cone with radius of the base and height . (e) torus π∫0 h(rxh)2 dx π∫0 hr2 dx π∫−rr(r2−x2)2 dx π∫−bb(a1−x2 b2)2 dx π∫−rr[(R+r2−x2)2−(R−r2−x2)2]dx Give the dimensions of the solid. A torus with the radius of its circular cross section as and the distance from the axis of the torus to the center of its cross section as