Expert: Alon Mandes - 12/18/2008

Question
1. f(x) = x^3 - 6x^2; find where is the function decreasing
2. f(3) = 2, f"(3)= 1/3, g(3) = 6, g'(3) = 1/2. If h(x) = f(x)/g(x), find h' (3)
3. Write the equation of teh tangeant line to the curve y = the cubed root of x^2 -1 @x=3
4. write the equation of teh tangeant line to the curve x^2+xy+2y^2=4 @(1,1)
5.  If s(t) = 1-t/1+t what is the acceleration at t=3
6. Radius of sphere increasing at 2in./sec. How fast is the volume increasing when the radius is 4 in.? 11 in.?
7. Find all values concave down f(x)= 3/x-2
8. f ' (x-2)(x+3)^2(x+1)^3. Critical points are x = 2, -3, -1. Are they min/max?
9. dy/dx= xy find d^2y/dx^2
10. 3in (xy) = x^2y find dy/dx
11. s(t) = 3t^3+6t^2+12t+4. when is acceleration = 0.
12. f(x) = x (square root of) x^2+1. f ' (x) = ?

Answer
1. f'(x)<0 -> 2x^2-12x<0 -> x^2-6x<0 : 0<x<6
2. h'(3)=[f'(3)g(3)-f(3)g'(3)]/f(3)^2 = 2
3. (y-2)/(x-3)=1/2
4. (y-1)/(x-1)=-3/5   (" y'=-[2x+y]/[x+4y] ")  
5. a(t)=s''(t)=[(1-t)/(1+t)]''=1/(1+t)=0.25
6. r'(t)=2 , v=(4/3)πr^3 -> v'=4πr^2*r'-> v|r=4|=128π
7. ---------------------
8. 2:min  , -3:saddle  , -1:max
9. y(1+x^2)
10. -----------
11. s''(t)=0 -> t=-2/3
12. [x√(x^2+1)]'=(2x^+1)/√(x^2+1)


Alon.