Docsity
Docsity

Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity


Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan


Guidelines and tips
Guidelines and tips

16 Questions for Old Exam 2 - Calculus and Analytical Geometry III | MTH 231, Exams of Analytical Geometry and Calculus

Material Type: Exam; Class: Calculus/Analytic Geom III; Subject: Mathematics; University: Marshall ; Term: Spring 2006;

Typology: Exams

Pre 2010

Uploaded on 07/30/2009

koofers-user-4ke
koofers-user-4ke 🇺🇸

10 documents

1 / 2

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
Exam 2 MTH 231 Summer 2006 Total Pts:100 8/3/2006
Name: Total Received:
Show all work for full credit. Do not write anything on question papers except your name.
1. Find and sketch the domain of the function f(x, y) = x+y+1
x+1 . Find f(2,3). (6 Pts)
2. Sketch the level curves of the function f(x, y) = xy for k= 1,2. (5 Pts)
3. For the limit lim(x,y)(0,0) x2
x2+y2, compute the following and determine whether the
above limit exists.
(a) lim(x,y)(0,0) x2
x2+y2along the line y= 0.
(b) lim(x,y)(0,0) x2
x2+y2along the line y=x.
(c) lim(x,y)(0,0) x2
x2+y2along the curve x=y2. (7 Pts)
4. Determine the set of points at which the function f(x, y) = ln(x2+y21) is
continuous and sketch. (5 Pts)
5. For the function f(x, y) = x2+2xy+y3x2y3, find fx(x, y ), fy(x, y), fxx(0,1), fyy (1,0),
and fxy(0,0). (7 Pts)
6. If f(x, y) = xex2y, find f
∂x and ∂f
∂y . (7 Pts)
7. Find an equation of the tangent plane to the surface z= 4x2y2+ 2yat (1,-1,1).
(7 Pts)
8. Find ∂z
∂x and ∂z
∂y if x3+y3+z3+ 6xyz = 1 using the formula ∂z
∂x =Fx
Fzand ∂z
∂y =Fy
Fz.
(6 Pts)
9. Use the chain rule to find ∂z
∂s and ∂z
∂t where z=x2+xy +y2,x=s+t, y =st.
(7 Pts)
10. Find the directional derivative of the function f(x, y, z) = y
x+zat (4,1,1) in the
direction of the vector
v=<1,2,2>. What is the maximum value of this
derivative at the point (4,1,1)? (8 Pts)
11. Find the equation of the tangent plane to the surface x2+ 2y2+ 3z2= 21 at the
point (4,-1,1). (6 Pts)
(Hint: Fx(x0, y0, z0)(xx0) + Fy(x0, y0, z0)(yy0) + Fz(x0, y0, z0)(zz0) = 0)
12. Find the local maximum and minimum values and saddle points of the function
f(x, y) = 9 2x+ 4yx22y2using the Second Derivative Test. (7 Pts)
13. Calculate the iterated integral R2
0Rπ
2
0xsin y dy dx. (6 Pts)
14. Find the volume of the solid that lies under the plane 3x+ 2y+z= 12 and above
the rectangle R={(x, y)|0x1,2y3}. (7 Pts)
pf2

Partial preview of the text

Download 16 Questions for Old Exam 2 - Calculus and Analytical Geometry III | MTH 231 and more Exams Analytical Geometry and Calculus in PDF only on Docsity!

Exam 2 MTH 231 Summer 2006 Total Pts:100 8/3/

Name: Total Received:

Show all work for full credit. Do not write anything on question papers except your name.

  1. Find and sketch the domain of the function f (x, y) =

√x+y+ x+1. Find^ f^ (2,^ 3).^ (6 Pts)

  1. Sketch the level curves of the function f (x, y) = xy for k = 1, 2. (5 Pts)
  2. For the limit lim(x,y)→(0,0) (^) x 2 x+^2 y 2 , compute the following and determine whether the above limit exists. (a) lim(x,y)→(0,0) (^) x 2 x+^2 y 2 along the line y = 0. (b) lim(x,y)→(0,0) (^) x 2 x+^2 y 2 along the line y = x. (c) lim(x,y)→(0,0) (^) x 2 x+^2 y 2 along the curve x = y^2. (7 Pts)
  3. Determine the set of points at which the function f (x, y) = ln(x^2 + y^2 − 1) is continuous and sketch. (5 Pts)
  4. For the function f (x, y) = x^2 +2xy+y^3 −x^2 y^3 , find fx(x, y), fy(x, y), fxx(0, 1), fyy(1, 0), and fxy(0, 0). (7 Pts)
  5. If f (x, y) = xex^2 y, find ∂f∂x and ∂f∂y. (7 Pts)
  6. Find an equation of the tangent plane to the surface z = 4x^2 − y^2 + 2y at (1,-1,1). (7 Pts)
  7. Find ∂z∂x and ∂z∂y if x^3 +y^3 +z^3 +6xyz = 1 using the formula ∂z∂x = −F Fxz and (^) ∂y∂z = −F Fyz. (6 Pts)
  8. Use the chain rule to find ∂z∂s and ∂z∂t where z = x^2 + xy + y^2 , x = s + t, y = st. (7 Pts)
  9. Find the directional derivative of the function f (x, y, z) = (^) x+yz at (4,1,1) in the direction of the vector −→v =< 1 , 2 , − 2 >. What is the maximum value of this derivative at the point (4,1,1)? (8 Pts)
  10. Find the equation of the tangent plane to the surface x^2 + 2y^2 + 3z^2 = 21 at the point (4,-1,1). (6 Pts) (Hint: Fx(x 0 , y 0 , z 0 )(x − x 0 ) + Fy(x 0 , y 0 , z 0 )(y − y 0 ) + Fz (x 0 , y 0 , z 0 )(z − z 0 ) = 0)
  11. Find the local maximum and minimum values and saddle points of the function f (x, y) = 9 − 2 x + 4y − x^2 − 2 y^2 using the Second Derivative Test. (7 Pts)
  12. Calculate the iterated integral

0

∫ π 2 0 x^ sin^ y dy dx.^ (6 Pts)

  1. Find the volume of the solid that lies under the plane 3x + 2y + z = 12 and above the rectangle R = {(x, y) | 0 ≤ x ≤ 1 , − 2 ≤ y ≤ 3 }. (7 Pts)
  1. Find the volume of the solid that lies under the paraboloid z = x^2 + y^2 and above the region D in the xy-plane bounded by the line y = 2x and the parabola y = x^2. (7 Pts)
  2. Evaluate the integral

0

3 y e

x^2 dx dy by reversing the order of the integration. Sketch the region. (7 Pts)

Note: Extra 5 points. If you miss a problem of up to 5 points, you can still get total score of 100 points.

Good Luck!