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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
3. Consider the the function f (x) = ln(1 + x)
(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.
2
4. Complete the following
(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
3
6. Complete the following.
(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
4
7. (10 pts) Find the area of the region inside lemniscate r2 = 2 sin 2θ and outside the circle r = 1 .
Graph the curves in one coordinate plane and find all the intersection points first.
5
8. Complete the following. Sketch the graph for each problems.
(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.
