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 r1 = 2 + 2sinθ and r2 = 1 for 0 ≤ θ ≤ 2π are shown below:
a) Write an integral expression for the area inside r2 and outside r1.
b) Write expressions for dx/dθ and dy/dθ in terms of θ for the r1 curve.
c) Write an equation in terms of x and y for the line tangent to the graph of the polar curve r1 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) = 23x. 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