16. Consider the perimetrically defined curve r(t) = t - sin 2t, y = 4 - 3cost, 0 0. Thus the conic consists of all those points P = (r, y) such that [PF] - e(PL|. Find the polar equation of the conic section. (With the origin as the pole, and the positive r-axis as the polar axis.) 6. (a) Identify and sketch the graph of the conic section defined by 2 - 4 cos 0 in the r, y, plane without converting it to cartesian coordinates. Clearly label all foci and vertices. (Hint: You may use the equation r = ep 1 - e cose without deriving it.) (b) Convert the polar coordinates equation given above to a cartesian coor- dinates equation of the form Ar' + Br + Cy' + Dy = E. 7. (a) Find an equation of the set of all points (r, y) that satisfy this condition: The distance from (r, y) to (5,0) is exactly half the distance from (z, y) to the line x = -5. (b) Simplify your answer from part (a) enough to be able to tell what type of conic section it is. 8. Derive the equation of the set of all points P(z, y) that are equidistant from the point A(1, 0) and the line r = -5. Provide a diagram with your work. Simplify the equation.1. For each of the following ten statements answer TRUE or FALSE as appro- priate: (a) If f is differentiable on [-1, 1] then f is continuous at a = 0. (b) If f'(x) 0 for all r then f is concave down. (c) The general antiderivative of f(x) = 3x3 is F(x) = r'. (d) In z exists for any z > 1. (e) Inr = # has a unique solution. (f) e" is negative for some values of r. (g) Iner = r for all z. (h) f(x) = [x| is differentiable for all r. (i) tan r is defined for all r. (j) All critical points of f(x) satisfy f'(z) = 0. 2. Answer each of the following either TRUE or FALSE. (a) The function /(x) = 3+ 1-7 In-2 if x * 2 is continuous at all real if x =2 numbers r. (b) If f'(x) = g(x) for 0 0 on /, then g(?) = 7(x) - is decreasing on I. 81 82 CHAPTER 5. TRUE OR FALSE AND MULTIPLE CHOICE PROBLEMS (d) There exists a function f such that f(1) = -2, f(3) = 0, and f'(x) > 1 for all r. (e) If f is differentiable, then - /(V) = f(2) 2VI 10' = 2107-1 (f) de (g) Let e = exp(1) as usual. If y = e then 1/ = 2e. (h) If f(x) and g(r) are differentiable for all r, then -f(g(x)) = f(g(x));(r). (i) If g(x) = 2', then lim 9(#) - 9(2) 1-2 = 80. (j) An equation of the tangent line to the parabola y = a' at (-2,4) is y - 4 = 2r(x + 2). ) - tan'n sec' r (1) For all real values of a we have that - (x* + x| = 12x + 1/. (m) If f is one-to-one then f (x) = 7(x) (n) If r > 0, then (Inx) =6inz. (o) If lim f(x) = 0 and ling(x) = 0, then lim / (2) does not exist. i g(1) (p) If the line r = 1 is a vertical asymptote of y = f(x), then f is not defined at 1. (q) If f'(c) does not exist and f'(r) changes from positive to negative as r increases through c, then /(r) has a local minimum at r = c. (r) va- = a for all a > 0. (s) If f(c) exists but f'(c) does not exist, then I = c is a critical point of f(x). (t) If f" (c) exists and f"(c) > 0, then /(r) has a local minimum at r = c. 3. Are the following statements TRUE or FALSE. (a) lim vr - 3=, lim (z - 3). ( In 2VI (b) de =0 (c) If f(x) = (1+x)(1+ x?) (1+ 23)(1 + x ), then f'(0) = 1.Exercise 2 - The paradox of saving Consider the following model: Y = C+G+/ C = co+ gild T = 50, G =150. / = 100 with co=10 and c1=0.5. a) Compute equilibrium income, consumption and savings. b) Suppose the government asks consumers to save more. The consumers abide and reduce co by 5. What happens to equilibrium income, consumption and saving? Explain the intuition