1. The velocity vector of a particle moving in the (x,y) plane has components given by:

a) For 0 < t < 2, find all values of *t* at which the line tangent to the path of the particle is horizontal.

b) Write an equation for the line tangent to the path of the particle at t = 1, if the position of the particle at t = 0 is (1,0).

c) Find the speed of the particle at t = 1.

d) For 0 < t < 1, find all values of *t* for which the speed of the particle is equal to 0.

2. The graphs of the polar curves r_{1} = 2 + 2sinθ and r_{2} = 1 for 0 ≤ θ ≤ 2π are shown below:

a) Write an integral expression for the area inside *r _{2}* and outside

*r*.

_{1}b) Write expressions for

*dx/dθ*and

*dy/dθ*in terms of

*θ*for the

*r*curve.

_{1}c) Write an equation in terms of

*x*and

*y*for the line tangent to the graph of the polar curve r

_{1}at the point where

*θ = 0*.

3. Let f be a function derivabile on the (0, 0.6) interval and f '(x) = 3 + ln(x + 1).

a) If f(.1) = 2, what is f(.6)?

b) Let R be the region in the first quadrant bounded by the y axis and the graphs of f'(x) and g(x) = 2^{3x}. Region R is the base of a solid. For this solid, each cross-section perpendicular to the x-axis is a square. Find the volume of this solid.

c) Find the area of R.

Verify your solutions