It's calculus2

1. Find the general solution of the differential equation. Write your solution explicitly. y" = (j:2 + 3,2 cos x)" 2. Consider the initial value problem of df=y2-5y+6.y(0)=1 where y represents the population of a species (in thousands), and it represents time (in years). a) Solve this initial value problem to nd the population as a function of time. Write your solution explicitly. b) What number does the population approach in the long term? (i.e., nd an equilibrium solution by computing Ilim y(t)) c) What number would the population approach if one could, theoretically, go back in time? (i.e., nd an equilibrium solution by computing tlirn y(t)). 3. Consider the parametric equations x = t3 121\" + 1,}! = 2t - t2. a) Find the slope of the tangent line at I = -l. b) Find the points where the tangent is vertical or horizontal. 4. Consider the parametric equations as = 2:?- + a]? = 8x5, 1 S t S 3. a) Find the exact length of the curve. b) Find the area of the surface obtained by rotating the curve about the x-axis. 5. Consider the polar curve 1' = GCOSH - 4sin9,{} S 9 S It. a) Find the cartesian equation for the curve. b) Sketch the curve. c) Find the slope of the tangent line to the curve at 6' = E. 6. Find the area inside the larger loop and outside the smaller loop of the curve 7' = 1 - sin 9