1) A cylindrical tank full of water measures 3 feet high and has a circular base with a radius of 1 foot. Suppose the tank is drained at a rate of ¼ cubic foot per minute. What is the rate of change in the height of the water in the tank with respect to time? 2) A spherical weather balloon is filled with helium such that its radius is growing at a constant rate of 2 feet per minute. Assuming V = volume of the balloon = 4π/3 * r^(3), where r = radius of the balloon, what is the rate of change in the volume of the balloon with respect to time at the moment it has a radius of 3 feet 3) Consider f(x) = x^(3/2)-x^(5/2) for x on [0,1] a) At what value of x does f obtain an absolute maximum value on the given interval? b) At what value of x does f obtain an absolute minimum value on the given interval? Express your answers in terms of exact values. No rounded decimal solutions. 4) Use the following function: F(x) = x * e^(x) a) On what interval(s) of x is function f increasing? b) On what interval(s) of x is function f decreasing? c) On what interval(s) of x is function f conca
ve up d) On what interval(s) of x is function f concave down Express your answers in exact values. No rounded decimals 5) Consider f(x) = (3x)/(x^(2)+3) where f’(x) = (3(3-x^(2))/((x^(2)+3)^(2)) and f’’(x) = (6x(x^(2)-9))/((x^(2)+3)^(3)). Sketch a graph of f, include in your sketch the location of key features such as intercept, extrema, inflection points, and asymptotes. 6) Consider f(x) = x^(4) – 9x^(2) Sketch a graph of f, include in your sketch the location of key features such as intercepts, extrema, inflection points, and asymptotes. 7) Find the xy coordinate on the graph of y = SQRT(x), nearest the point ( 4,0 ) in the xy coordinate plane Not L is the distance between the point (x,y) and (4,0), where L is calculated using the distance formula as follows, L = L(x,y) = SQRT(((x-4)^2)+((y-0)^(2))) 8) On any given day, the flow rate, F (Measured in cars per hour ), on a congested roadway is a function of speed of traffic on the roadway, v ( measured in miles per hour ) For any given v, F(v) = (v)/(46+0.02v^(2)) What speed will maximize the flow rate on the road?