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TEST 2 Solutions
Math 105 3/16/12 Name: (^) ︸ ︷︷ ︸ by writing my name I swear this work is my own
Read all of the following information before starting the exam:
- Show all work, clearly and in order, if you want to get full credit. I reserve the right to take off points if I cannot see how you arrived at your answer (even if your final answer is correct).
- Circle or otherwise indicate your final answers.
- Please keep your written answers brief; be clear and to the point. I will take points off for rambling and for incorrect or irrelevant statements.
- This test has 6 problems and is worth 100 points, It is your responsibility to make sure that you have all of the pages!
- Good luck!
1. (14 points)
x f (x) g(x) j(x) f ′(x) g′(x) j′(x) -2 0 1 -1 3 2 1 -1 1 3 2 -1 3 - 0 2 1 1 2 -2 2 1 3 1 -1 0 3 1 2 -2 2 1 3 0 3 3 -1 1 -1 1 -2 0
a. (7 pts) H(x) = f (j(x)) + 2g(x). Find H′(2).
H′(x) = f ′(j(x)) · j′(x) + 2g′(x) H′(2) = f ′(j(2)) · j′(2) + 2g′(2) = 0
b. (7 pts) F (x) = xj(x) f (x)^2
. Find F ′(0).
F ′(x) = (xj′(x) + j(x))f (x)^2 − 2 f (x)f ′(x)xj(x) f (x)^4
F ′(0) =
(1)(2)^2 − 2(2)(2)(0)(1)
2. (21 points) Find y′^ in 3 of 4 of the following. If you do more than four, then clearly mark which
three you want graded. If you don’t, the worst three will be chosen for you.
- y =
sin^4 (x) tan^2 (x) (x^2 + 1)^2 using logarithmic differentiation
y′^ =
4 cos x sin x
2 sec^2 x tan x
4 x x^2 + 1
sin^4 x tan^2 x (x^2 + 1)^2
- y = log 6 (x^2 ex) y′^ = 2 xex^ + x^2 ex ln(6)x^2 ex
- y = arcsin(x^2 − 1) + 4π^2 +
x^2
y′^ =
2 x √ 1 − (x^2 − 1)^2
x
- y =
arctan(2x) x^2 + 1
y′^ =
1 + 4x^2
(x^2 + 1) − (2x)(arctan 2x)
(x^2 + 1)^2
5. (20 points) Evaluate the following limits. Only use L’Hˆopital’s rule when appropriate. Show your
work!!
a. (5 pts) lim x→∞
x
(ln x)^2
lim x→∞
x
(ln x)^2 =
(ln x)^2 x
(T Y P E
= lim x→∞
2 ln x x
(T Y P E
= lim x→∞
x
b. (5 pts) lim x→ 0
4 x^2 + 2x + 1 3 x^2 + 1 1
c. (5 pts) lim x→∞
4 x^2 + 2x + 1 3 x^2 + 1
lim x→∞
4 x^2 + 2x + 1 3 x^2 + 1
(T Y P E
= lim x→∞
8 x + 2 6 x
(T Y P E
= lim x→∞
d. (5 pts) lim x→∞ x ln(^1 x)
y = lim x→∞ x ln(^1 x)
ln y = lim x→∞
ln(x) ln(x)
y = e
6. (21 points) Speeders Beware!
It has been found that for every 5mph you travel over 55mph, you decrease your gas mileage by 7%. You are renting a car for a 400 mile trip. The car rental costs $15/hour. Gas is $4.25/gallon. When traveling up to 55mph the gas mileage is 27miles/gallon. After 55mph, the car’s gas mileage drops by 7% for each 5mph over 55mph. a. (2 pts) If you travel at a constant speed of 55mph, how many hours will you travel to complete the trip? 400 55
= 7. 27 hours
b. (3 pts) How much would the trip cost if you travelled 55mph the entire trip?
Cost = $15 ∗ (7.27) +
c. (7 pts) You would like to minimize the cost on a trip with constant speed. Write the function for cost. Let t be the number of hours and x be the speed.
Cost = 15t + 4. 25
x − 55 5
d. (3 pts) What is the constraint? 400 = xt e. (3 pts) Write the objective function in terms of a single variable and simplify as much as possible. Two options:
Cost = 15t + 4. 25
400 t −^55 5
Cost = 15
x
x − 55 5
f. (3 pts) Describe, DO NOT CALCULATE, how you would finish the problem. I would find the first derivative of the cost function and determine when it is 0. Then I would check these critical points using the second derivative to determine whether the function has a minimum or a maximum at those critical values. I want the second derivative to be positive which will imply a minimum. I would then determine the speed.
