Find the absolute extreme values of f(x)=312x3+16x2−12x on the interval [−4,1]. Select the correct choice: A. The absolute maximum is □ at x=□. B. There is no absolute maximum on the given interval.
Calculate the specified partial derivatives for the following functions:
(a) f(x,y)=xy3ey,fy2′′=∂y2∂2f
(b) f(p,q)=3p3q2,fp2′′=∂p2∂2f
(c) f(k,l)=5kl3,fkl′′=∂k∂l∂2f
Find the derivatives of h(x)=f(x)g(x) at x=3 and h(x)=f(x)/g(x) at x=2, given a table of f(x),f′(x),g(x),g′(x). Also, find the points where the tangent line of f(x)=(2x2+10)/(x+2) is horizontal.
Find the maximum height of a vertically moving body with s=−21gt2+v0t+s0,g>0. Which expression gives the correct max height: 2gv02+s0, gv0+s0, gv02+s0, 2gv0+s0?
Approximate the integral of f(x)=2−x2 from 0 to 1 using Riemann sums. Lisa calculates the upper sum using the maximum function value, while Theo calculates the lower sum using the minimum function value. Compare the results for 5 and 500 rectangles.
Find the rate of change of the base of a triangle with altitude 12 cm and area 97 sq cm, given the altitude is increasing at 2.5 cm/min and the area at 3.5 sq cm/min.
The exponential function f(x)=561(1.026)x models a country's population f(x) in millions, x years after 1974. Find the population in 1974, 2001, 2028, and 2055, and describe the population growth pattern.