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Math 241
Name:
Midterm Exam
1. [12 points (4 pts each)] Find the sum of each of the following series.
(a) 1 −
1
1
1
1
1
+ −
+
−
+ ···
3
9
27
81
243
∞
(b)
∑ x 2n+5
n=1
(c)
1
1
1
1
1
1
−
+
−
+
−
+ ···
0!
1!
2!
3!
4!
5!
2. [12 points] State whether each of the following series converges or diverges. You do not need to justify
your answers. (Note: You will not receive any credit for the first four correct answers. Each additional
correct answer is worth 3 points.)
en
∑
n=1 n!
∞
∞
(a)
(b)
n=1
∞
(c)
1
∑ n√n + 1
n=1
(e)
∞
(d)
ln n
n=2 n
∑ n2 − 4
(f)
(−1)n
n=2 ln n
3n
∑ n 5
n=1 2 · n
(h)
∞
n
∑
∞
n=3
∞
(g)
1
∑ 1 + sin2 n
∑
∞
1
∑ ln n
n=2
3. [10 points] Find the Taylor series for the function f (x) =
mation notation.
1
. Express your answer in sum(1 + x)2
4. [8 points (4 pts each)]
(a) Find
∂f
if f (x, y) = x ln x 2 + y 2 .
∂x
(b) Find ∇f (3, 0, 1) if f (x, y, z) = xz 3 e 2y .
"
#
2
2xy
x
5. [6 points] Find a function f : R2 → R2 such that Df(x, y) =
.
2x e y
6. [5 points] Let f : R2 → R2 be a differentiable function. Given that
f(3, 4) = (2, 1),
∂ f1
(3, 4) = 3,
∂x
∂ f2
(3, 4) = 5,
∂x
∂ f1
(3, 4) = 4
∂y
and
∂ f2
(3, 4) = 2,
∂y
estimate the value of f(3.02, 3.97).
7. [5 points] Let f : R2 → R2 and g : R2 → R be differentiable functions. Given that
"
#
2 1
f(1, 1) = (1, 1),
g(1, 1) = 4,
Df(1, 1) =
and
∇g(1, 1) = (−2, 3).
5 3
use the chain rule to compute ∇(g ◦ f)(1, 1).
8. [12 points (6 pts each)] The following figure shows a contour plot for a function f : R2 → R.
5
0
1
2
3
4
6
7
5
8
20
4
5
4
10
18
16
3
12
14
14
3
16
2
2
6
4
2
12 14
8 10
18
1
0
1
0
1
2
3
4
5
(a) At which of the points (2, 1), (4, 1), and (5, 3) is the value of
(b) Estimate the value of
credit.
6
7
0
∂f
the greatest? Explain.
∂x
∂f
(5, 4). Your answer must be correct to within 15% to receive full
∂y
9. [12 points] Find the angle between the surfaces z = x 2 + y and xz2 + yz = 6 at the point (1, 1, 2).
10. [8 points] Let P be the point (6, −2, 15), and let Q be the point obtained by reflecting P across the
plane 2x − 3y + 6z = 10. Find the coordinates of Q.
11. [10 points] Let P be the plane 2x + 2y + z = 5, let L be the line on P that goes through the
points (1, 0, 3) and (0, 2, 1), and let M be the line on P perpendicular to L that goes through the
point (1, 0, 3). Find parametric equations for M.