Final Exam Practice Math 232, Calculus III,
1. Find the tangent line to the curve r(t) = hsin t, cos t, ti at (0, 1, 0). Find tangent line
at t = ⇡/4. Find the length of the curve of r(t) over the interval 0 t ⇡/2.
2. Determine the value of the limit. If it exists, find the value. If it does not show why.
y x
(a) lim p
(x,y)!(0,0) x2 + y 2
2xy
(b) lim p
(x,y)!(0,0) x2 + y 2
@z
3. If z = ex tan y, where x = s2 + t2 and y = st, find when s = 1 and t = 0.
@x
4. Find an equation of the plane through the point (1, 5, 4) and perpendicular to the line
x = 1 + 7t, y = t, z = 23t.
5. Let P (1, 2, 3), Q(1, 1, 2), and R(0, 0, 0) be three points in R3 .
(a) Find an equation of the plane through P , Q, and R.
(b) Find the area of the triangle formed by P QR.
(c) Find the equation of the line though P that is perpendicular to the plane from
(a).
@z
6. Given xy + exyz z ey = 0, use implicit partial derivative to find at the point
@x
P (1, 1, 1).
7. Let f (x, y) = x2 5xy
(a) Find rf (x, y).
(b) Find the directional derivative at (2, 1) in the direction of ~v = i + 3j.
(c) Find the equation of the tangent line on f (x, y) at (2, 1).
(d) Find the linearization L(x, y) of f at (2, 1).
(e) Use the linearization to approximate f (1.9, 0.9).
1 3 1 3
8. Find the local max, min, and saddle points for f (x, y) = x + y xy + 4 (if any
3 3
exist).
9. Find the local max, min, and saddle points for f (x, y) = 2x3 + xy 2 + 5x2 + y 2 .
10. Use Lagrange Multipliers to find the maximum and minimum of f (x, y) = x2 y subject
to the constraint x2 + y 2 = 1
Z 2Z 1
2
11. Evaluate ex dx dy by changing the order of integration.
0 y/2
12. Setup the triple integral in the order of dz dx dy and again as dz dy dx to find the
volume of the solid tetrahedron which is bounded by 3x+y +z = 1 and the coordinate
planes (i.e., the first octant).
,13. Set up, do not evaluate the triple integral in rectangular, cylindrical, and spherical
coordinates to find the volume of the solid
p in the first octant bounded above by
x2 + y 2 + z 2 and bounded below by z = x2 + y 2 .
14. Find the Jacobian for the transformation x = u2 v + v 2 and y = uv 2 u2 .
Z Z
15. Evaluate (3x y)3/2 (x + y)5 dA where D is the region bounded by y = x,
D
y = x + 1, y = 3x, and y = 3x 1. Use the change of variables u = 3x y and
v = x + y.
Z
16. Evaluate the line integral (xz + 2y) dS, where C is the line segment from (0, 1, 0)
C
to (1, 0, 2).
17. Let F (x, y) = (xy 2 + 2y)i + (x2 y + 2x + 2)j be a vector field.
(a) Show F is conservative.
(b) Find f such that rf = F .
Z
(c) Evaluate F dr where C is defined by r(t) = het , 1 + ti, 0 t 1.
C
Z
(d) Evaluate F dr where C is a closed curve r(t) = h2 sin(t), cos(t)i, 0 t 2⇡.
C
Z
(e) Use Green’s Theorem to evaluate y 3 dx + x3 dy where C is a circle given
C
by r(t) = h2 cos t, 2 sin ti, 0 t 2⇡.
Z
18. Use Green’s Theorem to evaluate y 3 dx + x3 dy where C is a circle given by
C
r(t) = h2 cos t, 2 sin ti, 0 t 2⇡.
Z
19. Use Green’s Theorem to evaluate (ex +y 2 ) dx+(ey +x2 ) dy where C is the positively
C
oriented boundary of the region in the first quadrant bounded by y = x2 and y = 4.
,
,
1. Find the tangent line to the curve r(t) = hsin t, cos t, ti at (0, 1, 0). Find tangent line
at t = ⇡/4. Find the length of the curve of r(t) over the interval 0 t ⇡/2.
2. Determine the value of the limit. If it exists, find the value. If it does not show why.
y x
(a) lim p
(x,y)!(0,0) x2 + y 2
2xy
(b) lim p
(x,y)!(0,0) x2 + y 2
@z
3. If z = ex tan y, where x = s2 + t2 and y = st, find when s = 1 and t = 0.
@x
4. Find an equation of the plane through the point (1, 5, 4) and perpendicular to the line
x = 1 + 7t, y = t, z = 23t.
5. Let P (1, 2, 3), Q(1, 1, 2), and R(0, 0, 0) be three points in R3 .
(a) Find an equation of the plane through P , Q, and R.
(b) Find the area of the triangle formed by P QR.
(c) Find the equation of the line though P that is perpendicular to the plane from
(a).
@z
6. Given xy + exyz z ey = 0, use implicit partial derivative to find at the point
@x
P (1, 1, 1).
7. Let f (x, y) = x2 5xy
(a) Find rf (x, y).
(b) Find the directional derivative at (2, 1) in the direction of ~v = i + 3j.
(c) Find the equation of the tangent line on f (x, y) at (2, 1).
(d) Find the linearization L(x, y) of f at (2, 1).
(e) Use the linearization to approximate f (1.9, 0.9).
1 3 1 3
8. Find the local max, min, and saddle points for f (x, y) = x + y xy + 4 (if any
3 3
exist).
9. Find the local max, min, and saddle points for f (x, y) = 2x3 + xy 2 + 5x2 + y 2 .
10. Use Lagrange Multipliers to find the maximum and minimum of f (x, y) = x2 y subject
to the constraint x2 + y 2 = 1
Z 2Z 1
2
11. Evaluate ex dx dy by changing the order of integration.
0 y/2
12. Setup the triple integral in the order of dz dx dy and again as dz dy dx to find the
volume of the solid tetrahedron which is bounded by 3x+y +z = 1 and the coordinate
planes (i.e., the first octant).
,13. Set up, do not evaluate the triple integral in rectangular, cylindrical, and spherical
coordinates to find the volume of the solid
p in the first octant bounded above by
x2 + y 2 + z 2 and bounded below by z = x2 + y 2 .
14. Find the Jacobian for the transformation x = u2 v + v 2 and y = uv 2 u2 .
Z Z
15. Evaluate (3x y)3/2 (x + y)5 dA where D is the region bounded by y = x,
D
y = x + 1, y = 3x, and y = 3x 1. Use the change of variables u = 3x y and
v = x + y.
Z
16. Evaluate the line integral (xz + 2y) dS, where C is the line segment from (0, 1, 0)
C
to (1, 0, 2).
17. Let F (x, y) = (xy 2 + 2y)i + (x2 y + 2x + 2)j be a vector field.
(a) Show F is conservative.
(b) Find f such that rf = F .
Z
(c) Evaluate F dr where C is defined by r(t) = het , 1 + ti, 0 t 1.
C
Z
(d) Evaluate F dr where C is a closed curve r(t) = h2 sin(t), cos(t)i, 0 t 2⇡.
C
Z
(e) Use Green’s Theorem to evaluate y 3 dx + x3 dy where C is a circle given
C
by r(t) = h2 cos t, 2 sin ti, 0 t 2⇡.
Z
18. Use Green’s Theorem to evaluate y 3 dx + x3 dy where C is a circle given by
C
r(t) = h2 cos t, 2 sin ti, 0 t 2⇡.
Z
19. Use Green’s Theorem to evaluate (ex +y 2 ) dx+(ey +x2 ) dy where C is the positively
C
oriented boundary of the region in the first quadrant bounded by y = x2 and y = 4.
,
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