Problem 1: Find dy/dx if (Do NOT simplify your answer.).
(a) y=sin(x)-ln(2): dy/dx=cos(x)+0=cos(x)
(b) y=x*cos(ln(3x)): dy/dx=cos(ln(3x))+x*(-sin(ln(3x))*1/(3x)*3
(c) y=\tan^{-1}(e-2x)=arctan(e-2x): dy/dx=1/(1+(e-2x)2)*e-2x*(-2)
(d) \cos(xy)+y4+x5=11: Take d/dx, -sin(x*y)*(y+x*y')+4y3*y'+5x4=0, so
y'= (-5x4+sin(x*y)*y)/(4y3-x*sin(x*y))
Problem 2: Find the absolute maximum and absolute minimum of f(x)= 3x4+8x3-18x2+3 on the interval [-1,1].
f '(x)= 12x3+24x2-36x=12x*(x2+2x-3)=12x*(x+3)*(x-1), so the critical values are x=0, x=1, x=-3. Only x=0 and x=-1 lie in [-1,1], so we need only check the endpoints -1,1, and 0: f(-1)=3-8-18+3=-20, f(0)=3, f(1)=3+8-18+3=-4, so the abs max is 3 and the absolute min is -20.
Problem 3: A balloon of the shape of a right circular cylinder of radius r and height h has volume V=\pi r2 h. How fast is the height increasing if the volume increases at a rate of 200 (ft)^3/min, the radius increases at a rate of 3 ft/min, r=2 ft and h=4 ft?
dV/dt= \pi*2r dr/dt*h+\pi*r2*dh/dt, so
200=\pi*2*2*3*4+\pi*22*dh/dt, so
dh/dt= (200-48*\pi)/(4*\pi).
Problem 4: True or False? (Please give your reason in complete sentences. An answer alone is worth zero.) If f(x) is differentiable at x=1, and f '(1)=0, then f(x) must have either a local maximum or a local minimum at x=1.
FALSE. Let f(x)=(x-1)3. Then f '(1) =0, but f does not have a local max or min at x=1. (Note: f(x)=constant does have both a local max and a local min at x=1, so this is not a counterexample!)
Problem 5: Evaluate the following limit, and explain your reasoning. (Here are possible hints: use the Mean Value Theorem, or try a linear approximation of the numerator for small h.)
\lim_{h\to 0^+} (sin(4+2h)-sin(4-h))/h.
From the MVT, sin(4+2h)-sin(4-h)=cos(c_h)*3h, where c_h is between 4-h and 4+2h. As h approaches 0, c_h is squeezed, and must approach 4. So the limit is 3*cos(4).
or if you use linear approximations, for small h,
sin(4+2h) is close to sin(4)+2*cos(4)*h,
sin(4-h) is close to sin(4)-cos(4)*h
so the numerator is close to
3*cos(4)*h, so the limit is 3*cos(4).