Practice Problems: Midterm Exam
1. Evaluate the following limits. You must justify your answers to receive full credit.
√
2n
n n + (n + 1)3
(a) lim 3
(b) lim
n→∞ n + 1
n→∞ 3n ln n + en
cos(x3 ) − 1
(d) lim
x→0
x6
n + 7n
(c) lim n
n→∞ 3 + n2
2. Evaluate the following integral:
Z ∞
1
2
3
4
+ 3 + 4 + 5 + · · · dx.
x2
x
x
x
3
3. Determine whether each of the following series converges or diverges. You must justify your
answer to receive full credit.
∞
1
(a) ∑ 2
n=1 n + 3
∞
1
(b) ∑
n=1 2n + 1
(−1)n
∑
n=1 n + 3
(e)
n2
√
∑ 4
n=1 n + 1
(h)
n3
∑ 2n
n=1
∞
1
∑ n(ln n)3
n=2
(f)
(−1)n
∑
−n
n=1 1 + e
(i)
n!
∑ 2n
n=1
∞
∞
(g)
(c)
∞
∞
(d)
∞
∞
1
∑ n ln(2n)
n=1
4. In the following picture, the large square has a side length of 1, each of the three squares
attached to it has a side length of 1/3, and so forth.
Find the total area of all of the squares.
∞
5.
(a) Find the sum of the series
∑ x3+4n, assuming −1 < x < 1.
n=0
3n n
∑ 2x .
n=1 n
∞
(b) Find the radius of convergence of the power series
(c) Find the Taylor series for the function f (x) = x3 sin(x2 ). Express your answer in summation notation.
(d) Find the first three terms of the Taylor series for the function f (x) =
6.
p
x + 25.
(a) Find the distance between the planes x + 8y + 4z = 12 and x + 8y + 4z = 30.
(b) Find the equation of the plane containing the lines
r(t) = (2, 1 + t, 3 + 2t)
r(t) = (2 − t, 1 + 4t, 3 + t).
and
(c) Find the point on the plane 2x + 3y + 6z = 6 that is closest to the point (2, 1, 1).
7. Let L be the line r(t) = (2 + 3t, 3 − 2t, 1 + t), and let P be the plane x + 2y + z = 4. Find the
equation of the plane perpendicular to P that contains the line L.
8. In the following picture, a rectangle is tangent to the ellipse x2 + 4y2 = 13 at the point (3, 1).
Find the coordinates of the point P.
4
2
H3, 1L
9.
P
2
2
(a) Find the equation of the tangent plane to the surface z = x2 + y2 at the point (1, 1, 2).
(b) A unit sphere is tangent to the outside of the surface z = x2 + y2 at the point (1, 1, 2). Find
the coordinates of the center of the sphere.
10.
∂f
if f (x, y) = x sin(xy).
∂x
(b) Find ∇f (3, 5) if f (x, y) = x 2 y.
(a) Find
(c) Find a function f (x, y) so that
∂ 3f
= x 2.
∂x ∂y 2
(d) Let f : R2 → R3 be the function f(x, y) = y3 , x2 , xy . Compute Df(x, y).
(e) Find the directional derivative of the function f (x, y) = 5x2 sin(2y) in the direction of the
unit vector (3/5, 4/5).
(x + y)2
(x,y)→(0,0) x2 + y2
(f) Use polar coordinates to evaluate
lim
11. The following figures show the graphs of f (x, y) as a function of x for several different values
of y.
y = 0.4
y = 0.5
y = 0.6
30
30
30
25
25
25
f Hx,yL 15
f Hx,yL 15
f Hx,yL 15
10
10
10
5
5
5
0
0
0
0
20
20
0.25
0.5
0.75
1
20
0.25
x
0.5
0.75
0
0
1
0.25
x
Use this information to estimate the values of
0.5
0.75
1
x
∂f
∂f
(0.5, 0.5) and
(0.5, 0.5).
∂x
∂y
12. Let f : R2 → R2 , and suppose that
f(3, 5) = (−1, 2)
and
Df(3, 5) =
"
#
1 0
5 2
.
(a) Use a linear approximation to estimate the value of f(3.01, 5.04).
(b) Estimate the values of x and y for which f(x, y) = (−1.02, 2.06).
"
#
1
2
(c) Let g : R2 → R2 be another function, and suppose that Dg(−1, 2) =
. Use the chain
3 4
rule to compute D(g ◦ f)(3, 5)
13.
(a) Find the tangent plane to the surface xz3 − yz = 4 at the point (1, 2, 2).
(b) Find parametric equations for the normal line to the surface at this point.