Download Math 2263 Midterm 1 Solutions - Fall 2008 - Prof. Robert D. Gulliver Ii and more Exams Calculus in PDF only on Docsity!
Math 2263 Name (Print):
Fall 2008 Student ID:
Midterm 1 Section Number:
October 2, 2008 Teaching Assistant:
Time Limit: 50 minutes Signature:
This exam contains 8 pages (including this cover page) and 7 problems. Check to see if any pages
are missing. Enter all requested information on the top of this page, and put your initials on the
top of every page, in case the pages become separated. Calculators may be used. Please turn off
cell phones.
Do not give numerical approximations to quantities such as sin 5, ฯ, or
- However, you should
simplify cos
ฯ
2
= 0, e
0 = 1, and so on.
The following rules apply:
- Show your work, in a reasonably neat and coherent way, in the space provided. All
answers must be justified by valid mathematical reasoning. To receive full credit on
a problem, you must show enough work so that your solution can be followed by someone
without a calculator.
- Mysterious or unsupported answers will not receive full credit. Your work should
be mathematically correct and carefully and legibly written.
- A correct answer, unsupported by calculations, explanation, or algebraic work
will receive no credit; an incorrect answer supported by substantially correct calculations
and explanations might still receive partial credit.
1 15 pts
2 15 pts
3 15 pts
4 15 pts
5 10 pts
6 15 pts
7 15 pts
TOTAL 100 pts
Math 2263 Fall 2008 Midterm 1- Page 2 of 8 October 2, 2008
- (15 points) The lines given parametrically by
ใx, y, zใ = ใ 5 โ t, 3 + 2t, 1 + 2tใ, โโ < t < โ
and
ใx, y, zใ = ใ5 + 2s, 3 + 2s, 1 โ sใ, โโ < s < โ
intersect at the point ใx, y, zใ = ใ 5 , 3 , 1 ใ. Find an equation for the plane which contains both
lines.
Math 2263 Fall 2008 Midterm 1- Page 4 of 8 October 2, 2008
- (15 points) Evaluate the limit
lim
(x,y)โ(0,0)
x
2 โ xy + y
2
x
2
2
or state that it does not exist, giving reasons.
Math 2263 Fall 2008 Midterm 1- Page 5 of 8 October 2, 2008
- (15 points) For the function
f (x, y) = e
3 y
cos 2x,
find the second partial derivatives
f xx
2 f
โx
2
, f xy
2 f
โyโx
and f yy
2 f
โy
2
Math 2263 Fall 2008 Midterm 1- Page 7 of 8 October 2, 2008
- (15 points) The point ใx, y, zใ = ใโ 2 , 1 , 0 ใ lies on the surface S:
x
2 โ y
2
2 = 1.
Find the equation of the tangent plane to the surface S at ใโ 2 , 1 , 0 ใ, in the form
ax + by + cz = d.
Math 2263 Fall 2008 Midterm 1- Page 8 of 8 October 2, 2008
- (15 points) (a)Find the gradient of the function f (x, y, z) = (x + z
2 ) sin(xy) at the point
ใx, y, zใ = ใ 1 ,
ฯ
2
(15 points) (b) Find the directional derivative of f at the point ใ 1 ,
ฯ
2
, 2 ใ in the direction
of the unit vector
~u =
i โ
j โ 2
k
Math 2263 Fall 2008 Midterm 1, WITH SOLUTIONS- Page 2 of 3 October 2, 2008
- (15 points) For the function
f (x, y) = e
3 y
cos 2x,
find the second partial derivatives
f xx
2 f
โx
2
, f xy
2 f
โyโx
and f yy
2 f
โy
2
SOLUTION: Compute f x
โf
โx
= โ 2 e
3 y sin 2x and f y
โf
โy
= 3e
3 y cos 2x. Then
f xx
= โ 4 e
3 y cos 2x = โ 4 f, f xy
= โ 6 e
3 y sin 2x and f yy
= 9e
3 y cos 2x = 9f.
- (10 points) Suppose z = f (x, y) is a function with partial derivatives fx(3, 4) = 3 and
f y
(3, 4) = โ2. If x and y are both functions of t: x = 4 โ t
2 and y = 3t + t
2 , find
dz
dt
d
dt
f
x(t), y(t)
at t = 1.
SOLUTION: The chain rule says that
dz
dt
= fx
dx
dt
dy
dt
Compute x(1) = 4 โ 1 = 3, y(1) = 3 + 1 = 4 so f x
and f y
are to be evaluated at x = 3, y = 4.
Compute
dx
dt
= โ 2 t so
dx
dt
(1) = โ2 and
dy
dt
= 3 + 2t so
dy
dt
(1) = 5. Finally,
dz
dt
- (15 points) The point ใx, y, zใ = ใโ 2 , 1 , 0 ใ lies on the surface S:
x
2
โ y
2
2
= 1.
Find the equation of the tangent plane to the surface S at ใโ 2 , 1 , 0 ใ, in the form
ax + by + cz = d.
SOLUTION: Compute the gradient of g(x, y, z) = x
2 โ y
2
2 :
โg = (2x + z)
i +
(โ 2 y + x)
j + (x โ 8 z)
k. Then
โg(โ 2 , 1 , 0) = โ 4
i โ 4
j โ 2
k is a normal vector to the surface S
given by g(x, y, z) = 1 at ใโ 2 , 1 , 0 ใ. The equation of the tangent plane to S at ใโ 2 , 1 , 0 ใ is
โ4(x + 2) โ 4(y โ 1) โ 2(z โ 0) = 0, or 2 x + 2y + z = โ 2.
Math 2263 Fall 2008 Midterm 1, WITH SOLUTIONS- Page 3 of 3 October 2, 2008
- (15 points) (a)Find the gradient of the function f (x, y, z) = (x + z
2 ) sin(xy) at the point
ใx, y, zใ = ใ 1 ,
ฯ
2
(15 points) SOLUTION: f x
= sin(xy) + (x + z
2 )y cos(xy); f y
= (x + z
2 )x cos(xy); and f z
2 z sin(xy). So the partial derivatives of f at ใx, y, zใ = ใ 1 ,
ฯ
2
, 2 ใ are fx = 1 +
5 ฯ
2
f y
= (โ2 + 1)(โ2)(0) = 0 and f z
= 4(1) = 4. Together, the gradient
โf (1,
ฯ
i + 4
k.
(b) Find the directional derivative of f at the point ใ 1 ,
ฯ
2
, 2 ใ in the direction of the unit
vector
~u =
i โ
j โ 2
k
SOLUTION: We know that D ~u
f (1,
ฯ
2
, 2) = ~u ยท
โf (1,
ฯ
2
1
3
i โ
j โ 2
k
i + 4
k
2 โ 8
3
