MATH224 – Calculus III Honors
Midterm Exam Review Problems (Fall Term, 2015)
1. Consider surface defined by the graph of the equation x 2 + 4y2 + z = 4 .
(a) Draw the trace of this surface in each of the coordinate planes:
(i) xy-plane:
(ii) yz-plane:
(iii) xz-plane:
(b) Identify and draw the graph of this surface in Â3 .
x 2 y2 z 2
2. Consider the surface defined by 2 + 2 - 2 =1, where a, b and c are all different. Which of
a b c
the following traces is not possible? Explain your reasoning.
(a)
(b)
(c)
(d)
3. Identify/name the following surfaces:
(a) x 2 + y2 = z 2
(b) x 2 + y 2 = z
(c) x 2 + y2 =1+ z 2
4. Find and sketch the domain of the function g(x, y) =
ln(x 2 + y 2 -1) - 9 - x 2 - y 2
.
x2 - 4
5. Find the domain and range of f (x, y) = ln(x 2 + y2 - 4) .
6. Find and describe (or sketch) the domain of f (x, y, z) = 4 - x 2 - y 2 - 4z 2
7. Provide an example of a function whose limit at the origin will produce different values along
the curves y = x 2 and y = x 3 .
e- x -y x 3 y 2
8. Determine the limit lim
or prove that it does not exist.
( x,y)®(0,0) x 5 + 2y 5
2
2
x+ z
is differentiable. [Note: For
ln(y - x 2 + z 2 )
functions of several variables, the partials must exist and be continuous for a function to be
differentiable.]
9. Find the region where the function f (x, y, z) =
10. Find
¶f
¶f
and
given
¶y
¶x
f (x, y) = cos(xy)× e xy
11. Given sin(2y) = xy - yx - x , find
2
3
2
2
.
dy
at the point (0,π) using the implicit differentiation
dx
result from the multivariate chain rule.
12. Find where the partials of f (x, y) =
x3
7
,
+ ye x + xy are equal to fx = -12
7e2
fy = 1-2e
e2
2
.
𝜕𝑧
𝜕𝑧
13. Use the chain rule to find 𝜕𝑠 and 𝜕𝑡 where 𝑧 = 4 cos(𝑥 2 𝑦) and 𝑥 = 𝑠 2 + 𝑡, 𝑦 = 4 − 𝑠𝑡.
𝜕𝑧
14. Find 𝜕𝑥 where 𝑧 is defined implicitly by the equation 𝑥 sin( 𝑦𝑧 2 ) = 𝑒 cos(𝑥𝑦) + 𝑥 2 𝑦 2 .
15. Given some arbitrary function 𝑓(𝑥, 𝑦) where 𝑥 is defined by an arbitrary function 𝑔(𝑢, 𝑣) and
𝑦 is defined by an arbitrary function ℎ(𝑢, 𝑣), find the general form of
𝜕2 𝑓
𝜕𝑢2
.
16. The Cobb-Douglass production function is given by P( L, K )  L K 1 where a and b are
positive constants and 0 < a <1. Find an expression for the total differential for P and the
equation of the tangent plane to the graph of P(L, K) at the point (sin a, cos b ) .
(
)
17. Find the tangent plane to the sphere x 2 + y2 + z 2 =16 at the point 2, 2, 8 .
18. Find the tangent plane and linearization for 5xyz - 2yz + 3x = 21 at P = (1, 2, 3) . (Hint: Using
implicit differentiation might work nicely.)
19. Find the directional derivative of the function 𝑓(𝑥, 𝑦) = 𝑥 𝑦 2 − 4𝑥 at the point (1,-1) in the
⃑ = 3 𝐢̂ + 4 𝒋̂.
direction of the vector 𝒗
20. Given 𝑓(𝑥, 𝑦) = 𝑥 4 − 𝑦 3 + 3𝑥 2 𝑦 − 2𝑦. Find the maximum rate of change of 𝑓 at the point
(3,2).
21. Suppose you are hiking a 14’er, the Gray’s and Torrey’s modeled by the graph of 𝐹(𝑥, 𝑦) =
3 − 3𝑥𝑦 −
𝑥4
4
−
𝑦4
4
. Gray’s Peak is to the left of Torrey’s peak as seen in the picture below.
a. Find the position of the two peaks.
b. You reach a landmark that says “Point (-1/2,1/2)!!” and wonder about the rate of ascent
to Gray’s peak. Find the rate of change in altitude at the point (-1/2,1/2) towards Gray’s
peak.
c. After summiting Gray’s peak, you hike towards Torrey’s, but realize that you forgot to
lock your car and decide to return to the parking lot. In which direction does the altitude
decrease the fastest from the point (1/4,-1/4)?
22. For the function f (x, y) = 2x 3 + xy2 + 5x 2 + y2 + 56 , find the critical points and classify them as
local maximum, local minimum, or saddle points.
23. For the function f (x, y, z) = (x -1)2 + (y - 2)2 + (z - 3)2 , find the critical points.
24. For the function f (x, y) = x 3 + y3 - 3xy , find the critical points and classify them as local
maximum, local minimum, or saddle points.
25. Use the method of Lagrange Multiplier to determine the point on the plane
x y z
+ + =1 that is
a b c
closest to the origin.
26. A fish tank, in the shape of a rectangular prism without a top, must have a volume of K ft3.
Use Lagrange multipliers to find the dimensions of this tank that would use the minimum amount
of glass (assuming the thickness of the glass is negligible in the design). Use this answer to
provide the dimensions of the optimal 10ft3, 25 ft3, and 30 ft3 tanks.
27. Suppose a company’s production is modeled by the Cobb-Douglass Function
f (x, y) =100x 4/5 y1/5 where x represents the units of labor and y represents the units of capital. If
each unit of labor costs $50 and each unit of capital costs $100 and the total expenses for labor and
capital cannot exceed $10,000 then the total production is constrained by the equation
50x +100y =10000 .
(a) Use the method of Lagrange multipliers to determine the maximum production for this
company.
(b) In this model, the Lagrange multiplier  represents the percent of each additional dollar
spent that will turn into production. Solve for l at the optimal point in (a) and
determine the additional production that would result from investing an additional
$5,000.