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qoqs10
math_practice.pdf
Name: Practice Test 3 6/15
1a) Convert the following Cartesian coordinates to polar coordinates:
( 1
4 ,
√ 3
4 ), (0, 8), (−
√ 2, −
√ 2).
1b) Convert the following polar coordinates to Cartesian coordinates:
( 3;
π
2
) ,
( 1;
4π
3
) ,
( 1
2 ; π
6
) .
1c) Convert the polar equation r = cot2 θ csc θ to a Cartesian equation.
2a) Find the fourth degree Taylor polynomial for f(x) = e−x.
2b) Find a formula for the remainder of this polynomial.
2c) On the interval [0, 1], how accurate would this approximation be?
3a) Find the second derivative of
∞∑ k=0
xk.
3b) Using this, find the power series for x4
(1 − x)3 .
3c) Evaluate ∞∑ k=0
k(k − 1) ( 1
2
)k+3 .
4) Find the area shared by r = 1 and r = 1 + sin θ.
5) Find lim x→0
3 tan−1 −3x + x3
x5 .
6a) Graph the parametric equation x(t) = 8 sin t, y(t) = cos t, 0 ≤ t ≤ π/2. Label start and end points with t, x, and y values and positive orientation.
6b) Find the line tangent to this curve at t = π/4.
7) Find the angle at which horizontal and vertical tangent lines occur for the curve r = 6 √ 3 + 6 sin θ.
