Departmental final Exam Review Part 3
f/13
More Applications of the Derivative
1. Locate the absolute extrema of the function on the closed interval. (Section 4.1)
2
a) y  3x 3  2 x
[-1, 2]
b) f ( x)  x ln( x  3) [ 0, 3 ]
c) y  e x sin x
0, 
2. Determine whether Rolle’s Theorem can be applied to f on the closed interval [a,b]. if Rolle’s
Theorem can be applied, find all values of c in the open interval (a,b) such that f ' (c)  0 . (Section
4.6)
x2 1
a. f ( x) 
[-1, 1]
b) f ( x)  x  2 ln x [1,3]
x
3.
Determine whether the Mean Value Theorem can be applied to f on the closed
interval [a,b]. If the mean Value Theorem can be applied, find all values that
satisfy the conclusion of the Mean Value Theorem. (Section 4.6)
a) f ( x)  sin x on [0,  ].
b) f ( x)  e 2 x
[0,1]
Section 4.2 and 4.3—A Summary of Curve Sketching
For each of the following find
a) first derivative, intervals where the graph is increasing or decreasing
b) relative extrema
c) points of inflection
d) intervals where the graph is concave up or down
e) graph by hand.
x2  1
x2  9
4.
f ( x) 
5.
f ( x)  x 16  x 2
6.
f ( x)  x 5  5 x
7.
f ( x )  e3 x ( 2  x )
8.
9.
f ( x) 
x3
x2  1
Use the derivative 𝑓′ to determine the local minima and maxima of f and the intervals of increase and
decrease.Then find the x-values for the point(s) of inflection and intervals of concave up and concave
down. Sketch a possible graph of f (f is not unique) [Section 4.3]
a. 𝑓 ′ (𝑥) = (𝑥 − 1)(𝑥 + 2)(𝑥 + 4)
−4
b. 𝑓 ′ (𝑥) = 𝑥 ⁄5 (𝑥 + 1)
c. 𝑓 ′ (𝑥) = 5𝑠𝑖𝑛𝑥 𝑜𝑛 [0, 2𝜋]
Section 4.4—Optimization Problems
10. You must make a small rectangular box with a volume of 400 in 3 . Its bottom is a rectangle
whose length is twice its width. The bottom costs 7 cents per square inch; the top and four sides of
the box cost 5 cents pr square inch. What dimensions would minimize the cost of the box?
11. Each page of a book will contain 30 in 2 of print, and each page must have 2-in. margins at top
and bottom and 1-in margins at each side. What is the minimum possible area of such a page?
12. A farmer plans to fence a rectangular pasture adjacent to a river. The pasture must contain
180,000 square meters in order to provide enough grass for the herd. What dimensions would
require the least amount of fencing if no fencing is needed along the river?
13. Find two positive numbers such that the sum of the first and twice the second is 100 and the
product is a maximum.
14. A man is in a boat 2 miles from the nearest point on the coast. He is to go to a point Q, located
3 miles down the coast and 1 mile inland. He can row at 2 miles per hour and walk at 4 miles per
hour. Toward what point on the coast should he row in order to reach point q in the least time?
15. Captain Nemo is relaxing on the deck of his submarine, the Nautilus, when he notices a warship
6 miles due east of his position. He immediately orders a torpedo to be fired toward the warship.
Unfortunately, the good captain had been engaged ina thorough inspection of the submarine’s wine
cellar just a little while earlier and so failed to take into account that the warship was moving. If the
torpedo is traveling due east at a speed of 50 mph and the warship is traveling due south at a rate of
30 mph, what will be the closest the torpedo ever gets to the warship?
16. Evauate the limits analytically: (L’Hopitl’s Rules)
2x2 
a. lim 2
b. lim 2  5e x
x  x  4
x 

ln(𝑥)
𝑑. lim+
1
𝑥→0
√𝑥
𝑔. lim+ 𝑥
𝑥→1
1⁄
1−𝑥

𝑥2
𝑒. lim
𝑥→0 ln(𝑠𝑒𝑐𝑥)
ℎ. lim (𝑙𝑛𝑥)
c. lim
x  
𝑓. lim+ (
𝑥→1
x
x2  1
1
1
−
)
𝑥 − 1 𝑙𝑛𝑥
1⁄
𝑥
𝑥→∞
17. Sketch the graph of a continuous function that satisfies the following conditions.
a. 𝑓(2) = 𝑓(4) = 0
b. 𝑓 ′ (𝑥) < 0 𝑓𝑜𝑟 𝑥 < 2
𝑓(3)𝑖𝑠 𝑑𝑒𝑓𝑖𝑛𝑒𝑑
𝑓 ′ (𝑥) > 0 𝑓𝑜𝑟 𝑥 > 2
′ (𝑥)
𝑓
< 0 𝑖𝑓 𝑥 < 3
lim 𝑓(𝑥) = lim 𝑓(𝑥) = 3
𝑓 ′ (3)𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑒𝑥𝑖𝑠𝑡
𝑓 ′ (𝑥) > 0 𝑖𝑓 𝑥 > 3
𝑓 ′′ (𝑥) < 0 𝑤ℎ𝑒𝑛 𝑥 ≠ 3
𝑥→−∞
𝑥→∞