MATH 220 – Written Homework 4
Instructions: There are 8 problems. Try to show key steps. Correct solution without showing proper work may receive no credit.
1. (10 pts) Find the radius and the interval of convergence for following the power series. Show all needed justification.
(a)
∞
∑
k=0
k10(2x − 4)k
10k .
2. (10 pts) Compute the limit using Taylor series. lim
x→0
−x − ln(1 − x)
x2
1
(a) (8 pts) Find the Maclaurin series (Which is the Taylor series centered at 0). Show all details.
(b) (3 pts) Use the series for ln(1 + x) to find the Maclaurin series for the function ln(1 + x2)
(c) (9 pts) Estimate the value of the integral
∫ 0.4
0
ln(1 + x2) dx with error at most 10−4 using Taylor series.
(a) (8 pts) Find parametric equations for the line segment starting at P(−1, −3) and ending at (6, −16)
(b) (2 pts) Eliminate the parameter t of the parametric equation you found in (a) to obtain an equation in xy.
5. Consider the parametric equation x = sint, y = cost
(a) (3 pts)Find dy dx in terms of t
(b) (7 pts)Find the equation of tangent line at the point t = π/4
(a) (4 pts) Express the point with polar coordinates P(2, 7π 4 ) in Cartesian coordinates.
(b) (4 pts) Express the point with Cartesian coordinates P(1, √3) in polar coordinates in two different ways.
(c) (7 pts) Graph the polar equation r = f (θ ) = 4 + 4 cos θ
(a) (5 pts) Find the equation of the parabola with vertex (0, 0) symmetric about the x-axis and passes through the point (2, −3). Specify the location of the focus and the equation of directrix, and graph the parabola.
(b) (5 pts) Find the equation of the ellipse centered at the origin with its foci on the x-axis, a major axis of length 8, and a minor axis of length 6. Graph the ellipse, label the coordinates of vertices and the foci.
(c) (5 pts) Find the equation of the hyperbola centered at the origin with vertices V1 and V2 at (±5, 0) and foci F1 and F2 at (±7, 0). Find the equations of the asymptotes and graph the hyperbola.
