BC 1- 2
Spring 12 Semester Review
Name:
This is a beginning. It is not intended as a complete review. It is a reminder of the many types of
problems we have done this semester and it is a guide for active study. Remember to review old tests
and quizzes, problems from the book, and handouts. Pace yourself while studying during the coming
week or so. Remember topics that were clear in August still deserve some time now. Also, please spend
some time thinking about the conceptual meaning behind the problems on the following pages and not
just the mechanics of getting through each problem.
Final Exam: Tuesday, May 29, 2012, 8:00 am, in A133
Topics (NOT all-inclusive!):
Problems from the book:
Functions
 Exponential, log, trig, polynomial,
rational, inverse (including trig)
 Continuous
 IVT
 Approximating using Euler's Method
ch 1
P. 49–50: 12, 15, 16, 22, 23, 26, 27, 28
P. 58–9, 2, 3, 13, 14, 29, 33, some from 40–53
Limits






One-sided
Requiring algebraic manipulations
Involving infinity
Used to define continuity
Special trigonometric
Formal definition
Derivatives
 Limit definition
 Interpretations
 Rules for calculating
o Powers, exponential, trig, inverse
trig, log, and inverse functions
o Sums and differences
o Product and quotient rules
o CHAIN RULE!!!!!!
 Implicit
Using derivatives
 Properties of graphs
 Relationship between graphs of f and f ′.
 Linear approximations
 MVT, Race Track Principle
 Related rates
 Optimization and EVT
ch 2
P. 75–6: 3, 7, 10, 14, 15, 17–21, 24–28
P. 82–5: 10, 13, 17, 21, 22, 28, some from 32–47
P. 90–2: some from 3–16, 23, 25, some from 27–41,
43, 45–47
P. 95–7: 2, 6, 13, 15, 18, 27, 28
P. 102–4: 4–6, 15–21, some from 22–31
P. 107-108: 8, some from 10–16
ch 3
P. 121-123: some from 6–47, 55, 56, 58, 59, 62, 65,
66, 67, 71
P. 126–127: some from 1–26, 40, 42–45, 47,
P. 130–132: some from 3–30, 31, 32, 43, 46, 48, 54
P. 137–139: some from 1–50, 51, 54, 59, 61, 65, 67,
68, 71-76, 79, 81, 83
P. 143–144: some from 2–39, 40, 44, 47–50
P. 149–151: some from 1–33, 38, 45, 51–53, 56–58,
62, 63
P. 153–154: some from 1–27, 28–30, 33, 34, 36
P. 162–163: 11, 14–16, 19, 21, 22, 23, 25–27, 34, 35
P. 167–168: 1–5, 10, 14, 15, 18–22
P. 168–169: some from 1–79
ch 4:
P. 182–185: 1–6, 13, 17, 22–25, 28–33, 35–37, 41,
43, 47, 49, 50, 52
P. 190–193: some from 5–21, some from 23–31, 33,
34, 38, 41
BC 1-2 Semester Review p.1
SP 12
More goodies to do – to complement the problems in the book, not replace them:
(1)
(2)
Find each of the following limits.
3 x3  6 x
(a)
(b)
lim
x 1  2 x3
4 x3  5 x
(d)
(e)
lim
x 2 x 4  1
x4
lim
(g)
(h)

x2 2  x
1  cos 3 x
x 0
x
lim
(c)
lim
| x 5|
5 x
(f)
lim
x3
2
x  2x
(i)
x5
x2
Determine whether or not  is continuous. Justify
your answer using the definition of continuity.
sin 3 x
tan 2 x
lim
x 0
lim
x3
lim
x 2  x  12
2x  6
sin 3 4
 0
3
2 x  4

f ( x)   2
 2
x  3
x 1
x 1
x 1
(3)
Use the IVT to explain why f  x   x 4  6 x  2 has a root on the interval [1, 2].
(4)
Given lim x3  3  5 , find an interval  x1, x2  such that if x   x1, x2  , then f  x   L  0.1 .
x 2


(5)
Suppose y  x2  4 x  3 on the interval [–1, 2]. Using Euler’s Method, graph an approximation
to y' and corresponding approximation for y using step size 1 and y(–1) = –5 to determine an
approximation for y(2).
(6)
State the limit definition of the derivative of the function f at the point x = a.
(7)
Use the limit definition of a derivative to find derivatives for each of the following functions.
1
f  x   x 2  3x  1
k  x 
g  x  x  2
(a)
(b)
(c)
x2
(8)
At what value(s) of x are each of the following functions NOT differentiable?
(a)
(9)
(10)
f  x   x 2  3x  4
3
(b)
g  x   x 2  3x  10
Draw the graphs of 3 different types of continuous functions that are not differentiable at x = 2.
Note: This means one and only one of your functions can have a corner. (Hint: other than a
corner, how else can the graph of a function not have a derivative at a point?)
(a)
t
 3 at time t  0.
t 1
Find the average velocity during the first 2 seconds.
(b)
Find the instantaneous velocity at t = 2 sec.
(c)
When, if ever, is the particle at rest?
A particle travels with position x  t  
BC 1-2 Semester Review p.2
SP 12
x 1
(11)
Use your calculator to approximate (1) correct to 3 decimal places if f  x    sin  x  
(12)
Find
(13)
A normal line is the line perpendicular to the tangent line to a curve at a point. Write the
.
 
dy
if y  sec x  sin 2 e x .
dx


equation of the normal line to the graph of f  x   ln x2  3x at x = 2.
(14)
Given the graph of a function , sketch the graph of .
(a)
(15)
(b)
The graph of a function ƒ' is shown on the open interval (–0.5, 2.5). Use the given x-values to
answer the questions below.
(a)
Where does ƒ have a local maximum?
Why?

0.3
0.15
0.5
0.84
1.4
(b)
Where is ƒ decreasing? Why?
(c)
Where is ƒ concave down? Why?
(d)
Where does ƒ have points of inflection?
Why?
(e)
Sketch a possible graph of ƒ.
2
(16)
What are critical points?
(17)
Without graphing, find the global (absolute) minimum value and the global (absolute) maximum


value of g  x   ln x2  3x  4 on the interval [0, 3].
(18)
Let f  x   3sin  x   2sin 3  x  . Find all global and local maxima and minima of f on the
interval 0, 2  . Use calculus to justify your answers.
BC 1-2 Semester Review p.3
SP 12
(19)
Use calculus to find the maximum value of f  x   x 1  x .
(20)
Find all values of x (approximate) such that the tangent line to y  x3  sin  x  is horizontal.
(21)
Find the local maximum and minimum values of the derivative of f  x   x 4  6 x3  7 x  3 .
(22)
If f  3  6, f   3  2, f  5  9, f   5  3 , and g is the inverse of f, find g   9  (approximate).
(23)
If f  x   2 x5  3x  1 , and g is the inverse of f, find g   9  .
(24)
Given the curve defined by the equation 9 x 2  6 y  x 2 y  17 , find the following:
(25)
(a)
The equations of all the tangent lines to the curve at x = 1.
(b)
Any point(s) on the curve where the tangent line is horizontal.
Suppose f  x   x  2  x  1 .
(a)
(b)
(26)
What does the MVT tell you about f on the interval [–3, 5]?
What does the EVT tell you about f on interval [–3, 5]? on the interval (–3, 5)?
Find all values of c which satisfy the MVT for h  x   x3  6 x  2 on the interval [–1, 3].
(27)
Draw the graph of a function that satisfies the conclusions of each of the following theorems but
NOT the hypotheses.
(a)
IVT
(b)
MVT
(c)
EVT
(28)
Sand is being emptied from a truck at the rate of 12 ft3/s and forms a conical pile whose height is
always three times the radius. At what rate is the radius increasing when the height is 4 ft?
(29)
A kite is 300 ft in the air and is being blown horizontally at the speed of 8 ft/s away from the
person holding the kite. How fast is the string being let out at the instant when 500 ft of string is
already out? From ground level, how fast is the angle of elevation to the kite changing? (Not
nice numbers, here.)
(30)
A cylinder has a fixed volume of 50 ft3. At the moment when the height is 2 ft and the radius is
5 ft, the height is decreasing by 0.25 ft/s. How fast is the radius changing?
(31)
If g(4) = 6 and g(4) = –3, estimate the value of g(4.2). If g(x) > 5 for 2  x  6 , will your
estimate for g(4.2) be greater than or less than the actual value of g(4.2)? Why?
(32)
If (2) = 5 and 1  (x)  4, find the possible values for (7). Justify carefully.
(33)
A poster is to contain 50 square inches of printed matter with 4–inch margins at top and bottom
and 2–inch margins on each side. What dimensions for the poster would use the least paper?
BC 1-2 Semester Review p.4
SP 12