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Math 21C Practice Midterm II
1. (12 pts: Power Series) Determine the x values for which the following power series converges: X ∞ n=1 (−1)n 2 n n (x − 1)n .
2. (11 pts for each part: Taylor Polynomials) Find the first three nonzero terms for the following Taylor series associated to
(a) f(x) = √ x about x = 4
(b) f(x) = cos(√ x) sin(2x) about x = 0. Hint: For this one you can use shortcuts and not compute any derivatives.
3. (11 pts: Taylor Remainder)
Estimate the error if the Maclauren polynomial P1(x) = x associated to f(x) = R x t=0 e − sin(t)dt is used to make the estimate of 1 3 for the integral R 1 3 t=0 e − sin(t)dt.
4. (11 pts: Vectors)
Let u = h1, −2i and v = h3, 4i.
Find projvu.
5. (11 pts: Forces) Consider a 100N weight suspended by two wires with slopes −1 and 2. Find the magnitudes of the force vectors on the two wires.
6. (11 pts: Lines) Consider the following two intersecting lines: L1 is given by x = 1+t, y = 2t and z = −1+3t. L2 is given by x = 3+2s, y = 1+s and z = −2 − s.
(a) Find their point (x, y, z) of intersection.
(b) Find the angle between the lines.
7. (11 pts: Planes) Compute the distence from the origin (0, 0, 0) to the plane 2x + y − 2z = 6.
8. (11 pts: Functions) Consider the function f(x, y) = 4 − p y − x 2 .
(a) Determine and sketch the domain of f in the plane.
(b) Determine the range of f.
9. (10 pts: Extra Credit... you may skip this problem)