Solution: 2738. Comparison tests Use the Comparison Test or Limit Comparison Test to

Calculus BC Final Prep Study Guide 1. Given the parametrically defined function that represents the movement of a particle x 4sint , y 5cost where 0 ≤ t≤ 2π : Find the Cartesian equation Graph the function on the given interval indicating on the graph the direction of movement. Find the velocity vector d) Find the speed of the particle at t= π 4 e) Find the length of the curve from t= π to t= π 6 4 2 d y Find dx2 for the function defined parametrically as x=t-sint, y=1-cost Find the value of each limit without using L’Hopital’s Rule: a) lim x _8 4 x­>2 x _16 b) lim _2x _ 4 3 2 x­>­ x _ 2x c) lim tan 3x 4x x­>0 tan d) lim sin(1 _ cos(t) ) t ­>0 1 - cos t 4. Answer each question using f(x) on the interval from 0 d x d 5 where 2 f(x)= x for 0 ≤ x ≤1, 3 for x= 1, 2 for 1 < x < 4, x-2 for4 < x ≤ 5 a) lim f (x) _ x­>4 b) lim f (x) x­>4 c) lim f (x) x­>4 f(4) What value should be assigned to f(x) to make it continuous at x=4 Give a value for δ < 0 _εx satisfying 0 _| x _ x |_ G the inequality |f(x)-L|< H holds. 0 f(x)=2x-2 L=-6 x0 =-2 H =0.02 6. Without a calculator, show that the equation x 3 _15 x _1 0 has 3 solutions in the interval [-4,4]. What theorem guarantees the existence of these three solutions 3 _ 1 7. Find the value of the limit (now you can use L’Hopital’s rule): lim x x xo2 _1 1 8. Find the value of the limit: lim(ln x) xof 1 x 9. Find the value of the limit: lim(1 _ ) xof x 3 2 2 ³ x sin ( x _1)dx tan x sec xdx ³ Answers: x 2 y 1a) _ 1 b) graph of an ellipse c) <4cost, -5sint> d) 41 e) 1.151 2 16 25 _1 3 _1 3 2) 3a) b) c) d) 1 2 (1 _ cos t) 8 2 4 4a) 2 b) 2 c) 2 d) dne e) 2 5) .01 6) x(-4)=-3, x(0)=1, x(1)=-13, x(4)=5 therefore by the Intermediate Value Theorem there must be one solution in each of the following intervals: (-4,0), (0,1), (1,4). 3 3 7) ln 3 8) 1 9) e 10) 1 ( x _1) _ 1 sin 2( _1) _ c 11) 1 3 tan x _ c 2 2 ln 2 3 6 3 2 12a) _ x _ 1 x _ b) y= e x _ ln x_1 S 15) 1 ln |1 _ 4ln x | _c 2 13) 17 14) 18 2 8