MA172/Worksheet 3
Name:
1. Given the parametric equation 𝑥 = 3 cos 𝑡 , 𝑦 = 3 sin 𝑡 0 ≤ 𝑡 ≤ 2𝜋.
a) Eliminate the parameter and find the Cartesian equation for it.
b) Sketch the curve and indicate the direction of motion.
c) Find the length of the curve.
2. Convert the equation to rectangular and simplify, and graph the equation.
a) 𝑟 = 2
b) 𝑟 = 1 + sin 𝜃
3. Find the length of the curve given by the polar coordinate equation.
𝑟 = 1 − cos 𝜃
4. Find an equation for a sphere whose center is 𝑃(4,3, −1) and radius 3.
5. Give a geometric description of the set of points in space whose coordinate satisfy the given pairs of
equations.
𝑥 2 + 𝑦 2 + (𝑧 + 3)2 = 25, 𝑧 = 0
6. Let 𝑢 be the vector with the initial point 𝑃(3, −1) and terminal point 𝑄(−3,9).
a) Graph 𝑢 in the coordinate plane.
b) Express 𝑢 in term of 𝑖 and 𝑗.
c) Find a unit vector that has the same direction as 𝑢.
7. Let 𝑢 = ⟨3,2⟩ and 𝑣 = ⟨5, −1⟩. Find the following
a) 𝑢 − 3𝑣.
b) |𝑢 + 𝑣|
c) |𝑢|
d) |−3𝑢|
8. Given 𝑢 and 𝑣. Copy the vectors 𝑢 and 𝑣 as needed to sketch the following vectors.
a) – 𝑢
b) 𝑢 + 𝑣
c) 𝑢 − 𝑣
𝑢
𝑣
9. Find the radius of convergence and the interval of convergence of the series

 (1)
n 1
n
( x  2) n
3
n
.
10. a) Find the power series representation for 𝑓(𝑥) =
1
1+𝑥
.
b) Use your answer from part a) to find power series representation for
𝑓(𝑥) =
1
16+𝑥 4
.
11. Find the first four nonzero terms of the Taylor series generated by the
function 𝑓(𝑥) = √3 + 𝑥 2 at 𝑥 = −1.
12. Find the Taylor series for the functions at 𝑥 = 0.
1
a) 𝑓(𝑥) =
2
3+𝑥
b) 𝑓(𝑥) = 𝑒 −𝑥
2