Assignment 7 (MATH 214 B1)
1. (a) Find parametric equations of the curve of intersection of the plane z = 1 and the
sphere x2 + y 2 + z 2 = 5.
Solution. The curve of intersection has parametric equations
x = 2 cos t,
y = 2 sin t,
0 t 2.
z = 1,
(b) At what
Assignment 4 (MATH 214 B1)
1. Let A = (1, 1, 3), B = (2, 3, 5) and C = (3, 0, 1).
(a) Find the distance between A and B.
Solution. We have
|AB| =
(2 1)2 + (3 (1)2 + (5 3)2 = 9 = 3.
(b) Find the vectors AB and AC.
Solution. We have
AB = (2 1) i + (3 (1) j
Assignment 5 (MATH 214 B1)
1. Let u = 2 i + 3 j 2 k, v = i k, and w = 2 i + j 3 k.
(a) Find u v.
Solution. We have
i j
uv = 2 3
1 0
k
2 = 3 i 3 k.
1
(b) Find v u.
Solution. We have
v u = 3 i + 3 k.
(c) Find the triple product (u v) v.
Solution. We have
(u
Assignment 6 (MATH 214 B1)
1. Let C be the circle given by the equation (x + 2)2 + (y 1)2 = 25.
(a) For each of the following points, determine whether it is inside the circle, on the
circle, or outside the circle:
(0, 0),
(1, 3),
(1, 4),
(2, 5),
(2, 6).
Assignment 9 (MATH 214 B1)
1. Find equations of the tangent plane and the normal line to the given surface at the
specied point.
(a) z = 4 x2 2y 2 , (1, 1, 1).
Solution. Let f (x, y) = 4 x2 2y 2 . We have
fx =
f
x
=
x
4 x2 2y 2
and fy =
f
2y
=
.
y
4 x2 2y
Exponential and Logarithmic Functions
An exponential function has the form f (x) = ax , where
a > 0. If x = n is a positive integer, then
an = a a a .
n
factors
If x = 0, then a0 = 1, and if x = n, where n is a positive
integer, then an = 1/an .
If x = r
Assignment 8 (MATH 214 B1)
1. Find the rst-order partial derivatives of the given function.
(a) f (x, y) =
9 x2 y 2 , x2 + y 2 < 9.
Solution. We have
f
x
=
x
9 x2 y 2
and
f
y
.
=
y
9 x2 y 2
x
(b) g(x, y) =
, (x, y) = (0, 0).
x2 + y 2
Solution. We have
(
Real Numbers
A real number has a representation of the form
k + 0.d1 d2 d3 ,
where k is an integer and each digit dj belongs to
cfw_0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
We use IR to denote the set of all real numbers.
Integers such as 3, 0, 5 are real numbers.
A
Assignment 1 (MATH 214 B1)
1. Determine whether the given sequence converges or not. If it converges, nd its limit.
n1
n2 + 1
3n2 5
(c) cn =
2 + 3n 4n2
n2 2
n+3
(1)n
(d) dn =
n
(a) an =
(b) bn =
Solution. (a) We have
lim an = lim
n
n
n(1 1/n)
1 (1 1/n)
n
Assignment 3 (MATH 214 B1)
1. (a) Find a power series representation for f (x) = 1/(1 x). What is the interval of
convergence?
(b) Use term-by-term dierentiation to nd a power series representation for the
function g(x) = 1/(1 x)2 . What is the interval o
Assignment 10 (MATH 214 B1)
1. Determine whether the origin (0, 0) is a local maximum, a local minimum, or a saddle
point of the given function.
(a) p(x, y) = x2 + xy + y 2
(b) q(x, y) = x2 3xy + y 2
(c) r(x, y) = 6x2 + xy 2y 2
(d) s(x, y) = 4x2 + 5xy 3y
Assignment 2 (MATH 214 B1)
1. Use the Comparison Test or Limit Comparison Test to determine whether the
given series converges or diverges.
n + n
n + n
(a)
(b)
n + n2
n + n3
n=1
n=1
1 + 3n
(c)
1 + 2n
n=1
Solution.
(a) Let an := (n +
n)/(n + n2 ) and bn