to approximate tan−1 1/5, and the polynomial of degree 1 to approximate tan−1 1/239.
(c) Use part (b) to evaluate the expression in (a).
(d) Explain how the approximation for π/4 given here compares with that obtained using π/4 =
tan−1 1.
BC ONLY
- (a) Show that the series converges.
(b) How many terms of the series are needed to get a partial sum within 0.1 of the sum of the
whole series?
(c) Tell whether the series is absolutely convergent, conditionally convergent, or
divergent. Justify your answer.
- Given = ky(10 − y) with y = 2 at t = 0 and y = 5 at t = 2:
(a) Find k.
(b) Express y as a function of t.
(c) For what value of t will y = 8?
(d) Describe the long-range behavior of y.
- An object P is in motion in the first quadrant along the parabola y = 18 − 2x^2 in such a way that
at t sec the x-value of its position is
(a) Where is P when t = 4?
(b) What is the vertical component of its velocity there?
(c) At what rate is its distance from the origin changing then?
(d) When does it hit the x-axis?
(e) How far did it travel altogether?
- A particle moves in the xy-plane in such a way that at any time t ≥ 0 its position is given by x(t)
= 4 arctan t,
(a) Sketch the path of the particle, indicating the direction of motion.
(b) At what time t does the particle reach its highest point? Justify.
(c) Find the coordinates of that highest point, and sketch the velocity vector there.
(d) Describe the long-term behavior of the particle.
- Let R be the region bounded by the curve r = 2 + cos 2θ, as shown.
(a) Find the dimensions of the smallest rectangle that contains R and has sides parallel to the x-
