Math 1071 Final Review Sheet

Math 1071 Final Review Sheet
The following are some more review questions to help you study. Some may be repeats from earlier review
sheets. You can use this review sheet plus the review sheets from the first two midterms to help you study
for the final exam. The problems below are a very minimal set of problems. You should know the general
topics, concepts and problems discussed in class, the textbook or assigned as homework. As well as be able
to do problems related to those found in the two midterm review sheets, homework problems and quizzes.
Problems on the exam may be on topics not covered in this review sheet and topics here may not appear on
the exam.
(1) Find the derivative of the following functions.
(a) h(x) =
14
x10
− 12x7 +
√
x
(b) r(x) = x ln x
(c) f (x) = (x2 + 3x − 2)100 (2x + 5)8
(d) f (x) = ln(x3 + 2x − 1)
(e) g(x) = 3(x
4
−7x)
+
√
x2 − 1
2
(f) k(x) =
(g) q(x) =
(h) j(x) =
e3x +5
x+5
√
13x2 − 5x + 8
8x−x6
x3
− 54
(i) m(x) = 3 log5 (17 − x) + 9
(j) n(x) = 5ex
2
−7x+12
(k) b(x) = (ln(8x + 1))4
(l) g(x) = e−9x
5
+12x2 +7x+6
(2) Find the equation of the tangent line to f (x) =
1
x2 +1
at x = 1.
(3) Write 2 log x + log y − log z as a single logarithm.
(4) Write the following in terms of log x, log y, and log z:
√
xy
log 2 .
z
(5) Solve the following equation for x:
2 log3 (x + 7) + 5 = log4 16
(6) Solve the following equation for x:
3 · 52−x = 4
(7) Solve for x: 53x = 1254x−4
1
(8) Solve for x: ln(3 − 4x) − ln(1 − x) = ln 3
(9) A demand function for an item is given by p = −.45x + 270. Find the revenue function for this item,
and determine how many items must be sold to maximize revenue.
(10) Find the following limits:
x12 − 10x
x→−∞ 3 − 4x7
6x9 − 3x4 + 2x
(b) lim
x→∞ 10 − 5x5 − 4x9
x6 − 2x + 1
(c) lim
x→−∞ 3x + 7x10
x2 − 3x + 2
(d) lim
x→2
x−2
1
(e) lim
x→∞ 5 − 3e−x
1
(f) lim
x→−∞ 1 + ex
(
x if x > 0
. Find limx→0− f (x), limx→0+ f (x), and limx→0 f (x).
(11) Let f (x) = 1
if x ≤ 0
x
( 2
x −9
if x < 3
(12) Find the following limits if f (x) = 3x−9
x − 1 if x ≥ 3
(a) lim+ f (x)
(a)
lim
x→0
(b) lim− f (x)
x→0
(c) lim f (x)
x→0
(d) lim f (x)
x→3−
(e) lim f (x)
x→3
(13) If f (x) is given by the picture below, find the following limits:
(a) lim f (x)
x→0+
(b) lim− f (x)
x→0
(c) lim f (x)
x→0
(d)
(e)
lim f (x)
x→−2+
lim f (x)
x→−2−
(f) lim f (x)
x→−2
(g) lim+ f (x)
x→3
(h) lim f (x)
x→3−
(i) lim f (x)
x→3
(j) lim+ f (x)
x→4
(k) lim f (x)
x→4−
(l) lim f (x)
x→4
(m) lim f (x)
x→5
(n) lim f (x)
x→∞
(o) Also, find all values a where lim does not exist.
x→a
(p) Find all values a where f 0 (a) does not exist.
(14) For each of the following functions, find critical values of f , find the intervals where f is increasing
and decreasing, and any relative extrema of f . Find all inflection values, and intervals where f is
concave up and concave down.
(a) Let f (x) = 2x3 + 3x2 − 12x + 6.
(b) f (x) =
x+1
.
x+2
(15) Find the absolute maximum and minimum value of h(x) = x3 − 6x2 on the following intervals:
(a)
(b)
(c)
(d)
[−1, 2]
[2, 3]
[−1, 5]
(−∞, 10]
(16) Calculate
the following antiderivatives:
Z
5
3
(a)
2x − √ + 10x − e dx
x
Z
1
(b) (3x2 − √ + 10) dx
x
Z √
√
3
(c)
( t − t5 ) dt
Z
1
(d)
(5x + + 2ex ) dx
x
Z
√
(e)
x(x + 5) dx
Z
(f) (2x3 − x)(x4 − x2 + 6)5 dx
Z
6x2 − 2
(g)
dx
x3 − x
Z
2
(h)
xex dx
Z
2
(i) (x − 1)ex −2x dx
Z
√
(j)
x x − 3 dx
Z
√
x − 3 dx
(k)
Z
x2
√
(l)
dx
x3 + 2
Z
(m)
(2 − 3x)5 dx
√
e x
√ dx
(n)
x
Z
x+1
(o)
dx
x2 + 2x
Z
(17) Calculate the following definite integrals:
Z 1
(a)
2x + 1 dx
0
Z
4
x2 −
(b)
√
x + 1 dx
1
Z
2
(2x − 1)(x2 − x)5 dx
(c)
1
Z
(d)
0
1
2
(x − 1)ex
−2x
dx
Z
1
(e)
√
1 − x dx
0
Z
1
4x(x2 + 1)9 dx
(f)
0
Z
2
(g)
1
4x
dx
+1
x2
(18) Let’s say you’re given the following graph of f (x). What is
R4
0
f (x)dx? What about
R4
2
f (x)dx?
(19) Find each of the following definite integrals by finding the area of an appropriate geometric figure:
Z 6
(a)
(4 − x)dx
Z−1
2 p
(b)
4 − x2 dx
−2
(20) Find the distance traveled by an object with velocity v = f (t) = |2 − t| on the time interval [1, 4] by
finding the area of the appropriate geometric region.
(21) Find the average value of the function f (x) =
dy
(22) In each of the following exercises, find
:
dx
√
2
2
(a) x y − y + x = −13
(b) x2 y 2 + xy 4 = 2x − 5
1
2x+1
on the interval [1, 4].
(23) You want to paint a picture. You have enough wood to make a frame that is 400 cm around. What’s
the largest picture you can make?
(24) A fence is to be built around a 300 square foot rectangular field. One side costs twice as much per
unit length as the other three. Find the dimensions of the enclosure that minimizes total cost.
(25) The manager of a large apartment complex knows from experience that 110 units will be occupied
if the rent is 296 dollars per month. A market survey suggests that, on the average, one additional
unit will remain vacant for each 4 dollar increase in rent. Similarly, one additional unit will be occupied for each 4 dollar decrease in rent. What rent should the manager charge to maximize revenue?
(26) A closed rectangular box of volume 324 cubic inches is to be made with a square base. If the material
for the bottom costs twice per square inch as much as the material for the sides and top, find the
dimensions of the box that minimize the cost of the materials.
(27) A deli sells 320 sandwiches per day at a price of 4 each. A market survey shows that for every 0.10
reduction in the price, 20 more sandwiches will be sold. How much should the deli charge in order
to maximize the revenue?
(28) Suppose that a farmer has 1000 yards of fencing. The farmer’s land has a river running through it,
and he wishes to fence in a pasture using a rectangular shape where one side is formed by the river
(assuming the river is straight). What are the dimensions of the largest pasture that the farmer can
fence?