Download Practice Questions for Midterm Exam 1 - Business and Economics Calculus | MATH 112 and more Exams Mathematics in PDF only on Docsity!
NAME: ________________________________ Student ID #: _________________
QUIZ SECTION:______
Math 112 B
Midterm I
April 25, 2006
Problem 1 15
Problem 2 15
Problem 3 20
Total: 50
- You are allowed to use a calculator, a ruler, and one sheet of notes.
- Your exam should contain 4 pages in total and 3 problems.
Make sure you have a complete test.
- Unless otherwise noted, you must show how you get your answers.
Correct (or incorrect) answers with no supporting work may result in little or no credit.
- If an algebraic method is available, answers obtained by guessing, approximating, or plug-and-
check will get little or no credit.
- Write your final answer in the indicated spaces. Unless otherwise noted, round your answer
to two decimal digits.
- If you need more room, use the backs of pages and indicate to the reader that you have done so.
- Raise your hand if you have a question.
GOOD LUCK!
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- (15 points)
a) (6 points) Compute the derivative f’(x) of the function (^)
2
x
f x x x
Rewrite the function in terms of powers of x:
3 1 2
2 2
− = +
= + x x x
f x x x
Then use the differentiation rules: 2
x
f x = x − x = x −
−
Answer: 2
x
f x = x − x = x −
−
b) (5 points) Compute the derivative dt
dy of the function 0. 452
= − t +
t
y
Rewrite the function in terms of powers of t: 2 0. 452
3
1 2
1 = − +
− y t t
Then use the differentiation rules: 0 3
3
2 2
3 = − − +
− − t t dt
dy
Answer: 3 3 2
3
2 2
3
t t
t t dt
dy = − − =− −
− −
c) (4 points) To the right you are given the derived graph g’(x) of a
function g(x). Which of the graphs labeled A through D below is the
graph of the original function g(x)?
(No need to justify your answer)
Answer: The graph labeled g’(x) above
is the derived graph of graph___B______.
Reason: The given derived graph should come from a function with a constant positive slope to the left of the y-axis (that is,
an line which goes up), and with a graph that is first increasing, then decreasing on the right (since the derived graph is
first positive, then negative).
A
g’(x)
B
C D
- (20 Points)
To the right are the Total Revenue (TR) and Total Cost (TC)
graphs for manufacturing and selling trinkets.
The corresponding formulas are:
TR ( q ) q 5. 75 q 9. 5 q
3 2 = − +
2 TCq = q +
where the quantity q is given in hundreds of trinkets, and both TR and TC are measured in hundreds of dollars.
a) Find formulas in terms of q for the Marginal Revenue and the Marginal Cost.
Use the differentiation laws to compute these.
ANSWER: MR(q)= 3 11. 5 9. 5
2 q − q +
MC(q)= q
MR and MC are measured in (circle one) : dollars OR hundreds of dollars.
b) What quantity q between 0 to 4 hundred trinkets will result in the largest profit?
Set MR=MC and solve the resulting quadratic equation via the quadratic formula.
3 q − 11. 5 q + 9. 5 = q
2
2 q − q + =
q=1 or q 3.
Which one gives the max profit?! It must be a transition point from MR>MC to MR<MC, or, even easier, we
can see from the TR/TC graphs above which root q has TR(q)>TC(q): the first (smaller) one. The other root
corresponds to the minimum profit (max loss).
ANSWER: Profit is maximal for q = 1 hundred trinkets.
c) Find the longest interval over which the Total Revenue is increasing but the Marginal Revenue is
decreasing.
Total Revenue increasing corresponds to MR>0. So we’re looking for the longest interval over which MR is
positive and decreasing. Since MR is a quadratic whose graph is a parabola that opens upward, it is positive
(above the x-axis) outside its roots, and it decreases up to its vertex. Also, we’re only interested in positive q
(since it’s a quantity!)
Roots: Quadratic formula on 3 11. 5 9. 5 0
2 q − q + = gives approximatively q=1. 2 and q=2.63.
The vertex is at q=1.9.
ANSWER: From q= 0 to q= 1. 2 hundred trinkets.
d) For what quantity q larger than 1 hundred trinkets is the Total Revenue minimal?
The max/min of TR are among the roots of its derivative. Set TR’=0 We already computed these roots in part c).
Looking at the provided graph, we see that the larger root (q=2.63) corresponds to the min of TR for q>1.
ANSWER: q=2.63 hundred trinkets.
TR
TC
