Download Math 0120 c

FinalExaminationSample
Math 0120 Final Examination
Sample
Name (Print)
ID #
Signature
.
.
Instructor (circle one):
Lecture time (circle one):
Instructions:
1. Show your University of Pittsburgh ID if requested.
2. Clearly print your name and PeopleSoft number and sign your name in the space above.
Circle the name of your lecturer and the time of your lecture.
3. Work each problem in the space provided. Extra space is available on the back of each
exam sheet. Clearly identify the problem for which the space is required when using the
backs of sheets.
4. Show all calculations and display answers clearly. Unjustified answers will receive no
credit.
5. Write neatly and legibly. Cross out any work that you do not wish to be considered for
grading.
6. No tables, books, notes, earphones, calculators, or computers may be used. All
derivatives and integrals are to be found by methods of calculus learned in this course.
DO NOT WRITE BELOW THIS LINE
_____________________________________________________________________
Problem Points
1
10
2
15
3
30
4
25
5
20
Score
Problem
6
7
8
9
10
Total
Points Score
12
25
30
21
12
200
1
FinalExaminationSample
1. (a) (3 pts.) Find an equation of the line with slope m = 2 that passes through the point (1, 3).
(b) (3 pts.) Find the vertex of the parabola y = 4x - x2.
(c) (4 pts.) Sketch the line and the parabola on the same set of axes.
.
2. (a) (5 pts.) Find lim x  1
x3  1
.
x 1
(b) (10 pts.) f ( x)  limh0
f ( x  h)  f ( x)
. Use this definition to find the derivative of f(x) = x2.
h
2
FinalExaminationSample
3. (30 pts.) Find the derivatives. (You need not simplify):
(a) g(x) =
x2  1
x3  2
.
(b) f ( x )  ( x 3  x )(2x 2  1) 3 . .
(c) x 2 y  xy 2  0. Find
dy
..
dx
3
FinalExaminationSample
4. (a) (13 pts.) A moving company wishes to design an open-top box with a base in which the length
is twice the width. The volume is to be exactly 36 cubic feet. Find the dimensions of the
box that will minimize the surface area. (If the width of the box is x, the length is 2x and the
height is y, the surface area is 2x2 + 6xy and the volume is 2x2y).
(b) (12 pts.) Country Motorbikes Incorporated finds that it costs $200 to produce each motorbike and that
fixed costs are $1500 per day. The price function is p(x) = 600 – 5x, where p is the price (in dollars) at
which exactly x motorbikes will be sold. Find the quantity Country Motorbikes should produce and the
price it should charge to maximize profit.
4
FinalExaminationSample
5. (20 pts.) f(x) = x5 + 5x4 = x4(x +5). f ( x ) = 5x4 + 20x3 = 5x3(x + 4), and f ( x ) = 20x3 + 60x2 = 20x2(x + 3).
Give a specific answer to each part:
(a) Construct sign charts for the first and second derivatives. (b) Find the critical numbers and the inflection
points of f. (c) Find all open intervals of increase and decrease and open intervals on which the graph is
concave up and concave down. (d) Classify each critical point as a relative maximum, relative minimum or
neither. (e) Sketch the graph of y = f(x) by hand, labeling only the relative extreme points, inflection points
and intercepts. Note that 34 = 81, 44 = 256 and use the factored form to evaluate the functional values.
5
FinalExaminationSample
6 (a) (6 pts.) f(x) = ln(2 – x) . Find an equation of the tangent line at x = 1.
(b) (6 pts.) A demand function is D(p) = 100 – p2 . Find an expression for E(p), the elasticity of demand.
Determine whether the demand is elastic, inelastic or unitary at p = 5.
7. (a) (10 pts.) A company’s marginal cost function is MC(x) =
1
2x  25
and its fixed costs are 100. Find the
cost function.
(b) (15 pts.) Find the area between the curves f(x) = x2 and g(x) = 2x on [-1,3]
.
6
FinalExaminationSample
8. (30 pts.) Find the following integrals:
(a)  (e  0.2x  x 3 
5
3
 1) dx
x
(b)  x 3 ln x dx
2
(c) .  xe x dx
7
FinalExaminationSample
9. (a) (6 pts.) Find the first-order partial derivatives for f(x, y) = ylnx + xey.
(b) (15 pts.) Find all critical points of f(x, y) = x2 + y3 - 6x - 12y and classify each as a relative maximum,
relative minimum, or saddle point.
8
FinalExaminationSample
10. (12 pts.) Use the method of Lagrange multipliers to minimize f(x, y) = x2 + y2 subject to the constraint
x + 2y = 30. (The minimum value exists).
9