91Ó°ÊÓ

If \(g(x)=\frac{1}{32} x^{4}-5 x^{2},\) find \(g^{\prime}(4)\) (A) \(\quad-72\) (B) \(\quad-32\) (C) \(\quad 24\) (D) \(\quad 32\)

Sea grass grows on a lake. The rate of growth of the grass is \(\frac{d G}{d t}=k G\) where \(k\) is a constant. (a) Find an expression for \(G\) , the amount of grass in the lake (in tons), in terms of \(t\) , the number of years, if the amount of grass is 100 tons initially and 120 tons after one year. (b) In how many years will the amount of grass available be 300 tons? (c) If fish are now introduced into the lake and consume a consistent 80 tons/year of sea grass, how long will it take for the lake to be completely free of sea grass?

Water is being poured into a hemispherical bowl of radius 6 inches at the rate of 4 in \(^{3} / \mathrm{sec} .\) (a) Given that the volume of the water in the spherical segment shown above is \(V=\pi h^{2}\left(R-\frac{h}{3}\right)\) , where \(R\) is the radius of the sphere, find the rate that the water level is rising when the water is 2 inches deep. (b) Find an expression for \(r\) , the radius of the surface of the spherical segment of water, in terms of \(h\) . (c) How fast is the circular area of the surface of the spherical segment of water growing (in in'/3ec) when the water is 2 inches deep?

An object moves with velocity \(v(t)=t^{2}-8 t+7\) (a) Write a polynomial expression for the position of the particle at any time \(t>0 .\) (b) At what time(s) is the particle changing direction? (c) Find the total distance traveled by the particle from time \(t=0\) to \(t=4\) .

Find a positive value \(c,\) for \(x,\) that satisfies the conclusion of the Mean Value Theorem for Derivatives for \(f(x)=3 x^{2}-5 x+1\) on the interval \([2,5]\) . (A) 1 (B) \(\frac{11}{6}\) (C) \(\frac{23}{6}\) (D) \(\frac{7}{2}\)

Find the slope of the normal line to \(y=x+\cos x y\) at \((0,1)\) (A) 1 (B) \(\quad-1\) (C) 0 (D) \(\quad\) Undefined