Math 1100-6
October 23, 2008
Page 1
Math 1100 Test #1
Name:SOLUTIONS
UID:
Instructions:
1. Do not look at this test until the strange guy with the red hair says it’s okay
to do so.
2. No calculators, books, or other aids are permitted for this test.
3. You will have until 10:30pm to finish the test. I will remind you when you have 20 min,
5 min, and 2 min remaining.
4. Please remain in your seat if you finish the test in the last 5 minutes so that
your peers are not distracted in the final minutes of the test.
5. Please raise your hand if you have any questions and/or would like some more paper.
6. Please show all your work for full credit.
7. Solutions to this test will be posted on-line following class and the test will be returned
next class.
8. You are strongly encouraged to celebrate after finishing the test.
Question
1a,b
2
3
4
5a,b,c,d,e
6
7
Silly Bonus
Presentation
Max Marks
3,4
4
5 each (10 total)
7 each (14 total)
3,4,2,3,3
17
+5
+1
3
Earned Marks
Bolded numbers are bonus marks.
Total =
70
Math 1100-6
October 23, 2008
Page 2
1. Evaluate the following limits:
(a)
x2 + 5x − 14
x→2 x − 3x + 2
lim
=
lim
x→2
x+7
=9
x−1
(b)
lim f (x)
x→−3
where f (x) =
(
x2 +
x3 −4
4
x
x
µ
¶
4
2
lim f (x) =
lim
x +
=
x
x→−3−
x→−3−
µ 3
¶
x −4
=
lim f (x) =
lim
x
x→−3+
x→−3+
: for x ≤ −3
: for x > −3
23
3
31
3
Because the left and right hand limits are not equal...
lim f (x) = DN E
x→−3
2. Using the figure below, put a square around the point(s) where the function is NOT
continuous and put a triangle around the point(s) where the function is NOT differentiable.
1.6
1.4
1.2
f(x)
1
0.8
0.6
0.4
0.2
0
−4
−3
−2
−1
0
1
2
3
4
5
x
3. Differentiate the following functions. Please simplify your answer fully.
(a)
f (x) = (x2 + 4x + 7)2
f 0 (x) = 2(x2 + 4x + 7)(2x + 4)
(b)
1
1
+ 2x 5
3
x
2
3
0
3
f (x) = 4x − 4 +
4
x
5x 5
f (x) = x4 +
Math 1100-6
October 23, 2008
Page 3
4. Differentiate two of the following three functions. Please simplify your answer fully. If you
attempted all of them, please circle the questions you want to be graded otherwise the first
two will be graded.
(a)
f (x) =
f 0 (x) =
(x2 − 1)3
2x + 1
2(x2 − 1)2 (5x2 + 3x + 1)
(2x + 1)2
(b)
√
f (x) = 2x2 4 3x − 1
x(27x − 8)
f 0 (x) =
3
2(3x − 1) 4
(c)
√
f (x) = (x2 − 2)2 (3 x + 2)
3
f 0 (x) =
(x2 − 2)(27x2 + 16x 2 − 6)
√
2 x
5. Answer the following questions using the total revenue and cost functions given below:
R(x) = x3 + 40x
C(x) = x3 + x2 − 5x + 200
(a) How many sales are needed to break even?
When R = C
x3 + 40x = x3 + x2 − 5x + 200
→
0 = x2 − 45x + 200 = (x − 40)(x − 5)
Either x = 5 or x = 40 are required to break even.
(b) Compute the minimum cost?
Set C 0 = 0 = 3x2 + 2x − 5
√
−2 + 4 − 4 · 3 · 5
→
x =
2·3
−2 + 8
=
=1
6
∴ C(1) = 1 + 1 − 5 + 200 = 197
(c) What is the profit function?
P (x) = R(x) − C(x) = −x2 + 45x − 200
Math 1100-6
October 23, 2008
Page 4
(d) Determine an equation for the marginal profit. What does this function tell you about
the profit?
P 0 (x) = −2x + 45
This tells us that the profit increases until x = 22.5 units are sold, then the profit
decreases. The marginal profit is how much the profit per unit changes as the
number of units sold increases.
(e) Find the point of diminishing returns. What does this tell you about the profit?
P 00 (x) = −2
This shows us that there is no point of diminishing returns and that the marginal
profit is a decreasing function of x.
6. Provide a complete sketch of the following function using the steps discussed in class. Note
that the first and second derivatives of the function are given.
f (x) =
3x3
x3 − 8
f 0 (x) = −
72x2
(x3 − 8)2
f 00 (x) =
288x(x3 + 4)
(x3 − 8)3
Step 1: General Observations
Note that f (0) = 0 and the domain is all x 6= 2.
Step 2: HA & VA
lim f (x) =
x→±∞
3x3
=3
x→±∞ x3
lim
and note that there is a VA at x = 2, so note that
lim f (x) = −∞
x→2−
lim f (x) = ∞
x→2+
(Numerator> 0 and denominator< 0)
(Numerator and denominator> 0)
Step 3: CP & Max/Min
f 0 (x) = 0 when x = 0, and f 0 (x) is undefined at x = 2. Also note that f 0 (x) < 0 for all x
for which it is defined, therefore, f (x) is always decreasing and x = 0 is a horizontal POI.
Step 4: POI & Curvature
√
f 00 (x) = 0 when x = 0, − 3 4 ≈ −1.59, and f 00 (x) is undefined at x = 2. Therefore, we
note the following:
x
x < −1.59 x = −1.59 −1.59 < x < 0 x = 0 0 < x < 2 x = 2 2 < x
sign of f 00 (x)
−
0
+
0
−
DN E
+
y
CD
CU
0
CD
DN E CU
Math 1100-6
October 23, 2008
Page 5
Step 5: Sketch
7. (BONUS) A developer plans to fence a rectangular region and then divide it into two
identical rectangular lots by putting a fence down the middle. Suppose that the fence for the
outside boundary costs $20/ft and the fence for the middle costs $4/ft. If the field must
contain 22,000 square ft, what are the dimensions of the region that will minimize the costs?
22000
x
= $20 · 2x + $20 · 4y + $4 · x
A = 22000 = xy
Cost
y=
→
C = 44x + 80y
80 · 22000
= 44x +
x
80 · 22000
0
Set C = 0 = 44 −
x2
r
80 · 22000 √
→
x =
= 20 · 2000 = 200
44
22000
→
y =
= 110
200
Therefore, the rectangular regions need to be 200 ft × 110 ft each.
Silly Bonus (+1): In what country can you find the Royal Canadian Mounted Police (RCMP)?
.