Sample Examination 1 Multivariable Calculus
1.
Let f Hx, yL = y2 + x sin Ix2 yM.
a.
Find the gradient of f Hx, yL.
SOLUTION. We have that
f
f Hx, yL = J x ,
f
y
N = Isin Ix2 yM + 2 x2 y cosIx2 yM, 2 y + x3 cos Ix2 yMM.
Find the directional derivative o
Review Sheet 3 Multivariable Calculus
Final Examination: Monday of Exam Week at 7:309:30 AM in class room
.I. Triple Integral
A. Continuous functions on rectangles
1.
Riemann sums
B. Continuous functions on closed and bounded sets
C. Iteration by cross s
Sample Examination 2 Multivariable Calculus
Find all critical points of f Hx, yL = 3  x2 + 2 x  y2  4 y.
1.
SOLUTION. We set
f Hx, yL = H2 x + 2,  2 y  4L = 0
and get a single critical point Hx, yL = H1, 2L.
1
The critical points of f Hx, yL = x2
Solution Examination 2 Multivariable Calculus
1. Find the double integral of f Hx, yL = x3 y over the region between the curves y = x2 and y = x H1  xL.
1
2
y = x2

1
2
1

1
2
y = x H1  xL
1
2
SOLUTION. The intersection of the curves is given by x2 =
Solutions Final Examination Calculus 4
Evaluate the line integral C y2 x3 x + 2 x y z3 y + 3 x y2 z2 z where C is the curve C : H2 cos t, 3, 2 sin tL
for 0 t 2 p. HINT: Stokes' Theorem.
1.
z
x
y
SOLUTION. The curve is the boundary of the region with param
l Review I Multivariable Calculus
I.
II.
Vector valued functions of a real variable
A. Components
B. Vector operations
1. Addition, Scalar Multiplication
2. Dot Products
3. Cross Products
C. Calculus Operations
1. Limits
2. Differentiation
a.
Velocity v =
l Review I Multivariable Calculus
I.
II.
Vector valued functions of a real variable
A. Components
B. Vector operations
1. Addition, Scalar Multiplication
2. Dot Products
3. Cross Products
C. Calculus Operations
1. Limits
2. Differentiation
a.
Velocity v =
Solutions Examination 1 Multivariable Calculus
Let w = x2 + y2 + z2 , x = s t, y = s cos t, z = s sin t. Use the Chain Rule to find
1.
w
t
when s = 1 and t = p 2.
a. Find the gradient of w.
SOLUTION. We have
w = Iwx , wy , wz M = 2 Hx, y, zL.
b. USE THE
Solutions Extended Quiz 1 Multivariable Calculus
1.
Draw the graph of rHtL = Hcos t, sin t, t2 L for 0 t 2 p on the axes below.
z
y
x
FOR PROBLEMS 2, 3, 4, USE rHtL = Ht2 , 2 t, ln tL at t = 1.
2.
Find the velocity and acceleration vectors of the graph at
Extended Quiz 2 Solutions Multivariable Calculus
1. Let f Hx, yL = Hx  yL2 + Hx + 2 y + 1L2  8 x y.
a. Find the critical points. DO NOT TEST THE CRITICAL POINTS.
SOLUTION. The critical points Hx, yL are those points with f Hx, yL = 0. We have
f Hx, yL
Solutions Final Examination Multivariable Calculus
1
Find
S
1 + x2 + y2 S
where
S
is
the
surface
with
parametric
equations
rHu, vL = u cos v + u sin v + v for 0 u 1, 0 v p.
z
x
y
SOLUTION. We have that
ru = Hcos v, sin v, 0L and rv = H u sin v, u cos v,
Sample Extended Quiz 3 Multivariable Calculus
1.
Find the surface area of the part of the sphere x2 + y2 + z2 = 2 that lies inside the paraboloid z = x2 + y2 .
SOLUTION. The intersection of the two surfaces is given by
x2 + y2 + z2 = x2 + y2 + Hx2 + y2 L2