BC 1,2
Spring 13 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 28, 2013, 8:00-9:30 am, in the Ac. Pit
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 Fast Semester Review p.1
SP 13
BC 1,2
Spring 13 Semester Review
Name:
Derivative applications
 Critical points, local/global maxs/mins
 Optimization
 Related rates
 L'Hôpital's Rule
 Parametric equations – graphs, derivatives,
tangents, motion, speed
Integrals
 Definition – limit of a sum, signed area
 Fundamental Theorem of Calculus (both
versions)
 Properties
 Riemann sums (left, right, midpoint),
trapezoidal approx., calculator approx.
 Antiderivatives (constructed graphically,
numerically, analytically)
 Differential equations (basic concept)
Integral Applications
 Distance traveled, average value
 Signed area, area between curves
ch 4
P. 207–9: 4, 5, 11, 22, 28, 30, 32, 33, 40, 41
P. 224-7: 6, 11, 20, 27, 28, 33, 36, 37, 38, 42, 44
P. 234: 18–21, some from 22–35, 39, 43-47, 53
P. 242-5: 8, 13, 14, 19, 21, 23, 26, 37, 45, 52
ch 5
P. 262–4: 3, 5, 9, 14, 16, 18, 23, 26, 27
P. 269–71: 4, 7, 17, 21, 24, 32, 36
P. 278–281: 7, 16, 21ab, 27, 31, 33, 36, 37, 39
P. 288–9: 2, 8, 21–26, 34, 37, 39, 42, 43
ch 6
P. 303–5: 6, 11, 15, 17, 22, 24
P. 310–1: some from 1–63, 67, 71, 74, 77, 81
P. 315–7: 5, 9, 14, 15, 17, 21, 25
P. 320–1: 3, 11, 22, 27, some from 29–36, 38
P. 325-27: 46,52,55,56,59-68,70,71,74
ch 7: assorted integrals (lots) to practice techniques
P. 338–40: some from 3–40, some from 47–62,
some from 69–76, 89, 94, 99, 109
P. 346–7: some from 2–28, 36, 37, 39, 46, 56, 58
P. 359–60: some from 8–22, some from 27–58
Integration (techniques)
 Powers, polynomials, exponential, trig,
inverse trig, log
 Substitution (including changing limits!)
 Integration by parts
 Partial fractions
BC Fast Semester Review p.2
SP 13
More goodies to do – to complement the problems in the book, not replace them:
(1)
(2)
Find each of the following limits.
3x3  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  cos3 x
x 0
x
(c)
lim
lim
x 5
| x 5|
5 x
(f)
x3
lim 2

x 2 x  2 x
(i)
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]. 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   x2  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 Fast Semester Review p.3
SP 13
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 Fast Semester Review p.4
SP 13
(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 Fast Semester Review p.5
SP 13
(34)
5
3
Consider the function f  x   x  5ax  b , where a and b are constants.
(a)
Find all critical points of f.
(b)
For what values of a and b does f have exactly one critical point? What are the
coordinates of this one critical point, and is it a local maximum, local minimum, or neither?
(c)
For what values of a and b does f have exactly three critical points? What are the
coordinates of these critical points? Which are local maxima, which are local minima, and which
are neither?
(35)
Draw the graph of a continuous function that has three
different types of critical points on the interval [–5, 5].
(36)
Six squares are cut out of a piece of cardboard 24 cm by
16 cm. The sides are folded up and the top is folded
around to form a closed box with double walls on 3 sides.
Find the value of x to maximize the volume of the box.
x
16
24
bottom
top
(37)
A rectangular plot of land containing 216 square meters of area is to be enclosed by a fence and
divided into three equal parts by two other fences that will both run parallel to the same side of
the rectangular plot. What dimensions for the entire rectangular plot require the smallest total
amount of fencing? How much total fencing will be needed?
(38)
A commuter train carries 600 passengers each day from a suburb to a city. The cost to ride the
train is $4.50 per person. Market research reveals that 40 fewer people would ride the train for
each 25 cent increase in fare and 40 more for each 25 cent decrease. What fare should be
charged to generate the largest possible revenue?
(39)
Determine each of the following limits.
(a)
(40)
Find
(a)
e x
lim
x 
 1
ln  1  
x

1
lim x  tan  
x 
 x
(b)
dy
for each of the following:
dx
x  e , y  4e
t
2t
(b)
3

lim 1  
x  
x
1

1
lim  

x 0  x
ln  x  1 

2x
(c)
(d)
 
x  t sin  t  , y  cos t
BC Fast Semester Review p.6
2
SP 13
(41)
(42)
Consider the curve given by the parametric equations x  2sin   and y  cos  4  .

(a)
Write the equation of the tangent line to this curve when t 
(b)
If P is a point moving along this parametric curve, find the speed of P when t 
(c)
Eliminate the parameter t and write the Cartesian (x-y) equation of the curve.
6
.

6
.
t

Consider the cycloid given by the parametric equations x  a   sin  t   and
2

y  a 1  cos  t   . Determine all values of x (exact) for which the tangent to this curve is:.
(a)
horizontal
(b)
vertical
(43)
A rocket, rising vertically, is tracked by a radar station that is on the ground 5 miles from the
launch pad. When the rocket is 4 miles high, its distance from the radar station is increasing at a
rate of 2000 mph.
(a)
How fast is the rocket rising at this time?
(b)
How fast must the radar angle of elevation change at this time to keep the radar aimed at
the rocket?
(44)
A spherical ball 8 inches in diameter is coated with a layer of ice of uniform thickness. When
this layer of ice is 2 inches thick, it is melting at a rate of 10 cubic inches per minute.
(a)
How fast is the thickness of the ice decreasing at this time?
(b)
How fast is the outer surface area of the ice decreasing at this time?
(45)
Frodo is racing from Sauron’s tower in Mordor at a rate of 5 meters per second. He is running in
a straight line away from a light that is 12 meters directly above the base of the door through
which Frodo escaped. If Frodo is 1.25 meters tall, at what rate is the length of his shadow
changing when he is 16 meters away from this door?
(46)
Given the definite integral
 e  1 dx , complete the following.
2
x
0
(a)
Using n = 4, approximate the value of this integral with a left–hand Riemann sum, a
midpoint Riemann sum, and a trapezoidal approximation. Write out these sums clearly.
(b)
Which of the approximations used in part (a) overestimates the value of the integral?
Which underestimates? Clearly explain your reasoning.
(c)
Use your calculator to approximate the integral more accurately to 3 decimal places.
(d)
Use sigma notation to express the right-hand Riemann sum for this integral for n = 40.
BC Fast Semester Review p.7
SP 13
(47)
You and your bff are driving along a twisty dirt road in your old "beater" car. The speedometer
on this car works but not the odometer. To determine how long this particular road is, you record
the car's speed, s, in feet per second (being conscientious mathematics students, you converted
from miles per hour as you recorded the data) at different times, t, in seconds during the twominute time interval you were on the road. Those results are shown in the table below.
t
0
10
20
30
40
50
60
70
80
90
100 110 120
s
0
44
35
15
30
44
35
15
22
35
44
30
35
Using a Riemann sum with 6 equal subintervals, provide the largest possible upper estimate for
the length of this road. Explain why this is the largest upper estimate for all Riemann sums with
6 equal subintervals.
(48)
Express

4
x 2  e x dx as a limit.
1
n
(49)
(50)
The limit lim
n 
Let F  x  


i 1
x
2
2
 
 4
6i 
7  3    5  can represent an integral. What integral?
n
 
 n
f  t  dt where f is the function
y
graphed at the right. Note: The graph of f
consists of 4 line segments and a quarter-circle,
and the domain of f is [–6, 5].
(a)
Determine F(–5) and F(5).
(b)
Determine F   2 and F   2 .
(c)
How many zeros does F have on [–6, 5]?
(d)
On what intervals of x is F increasing?
(e)
At what values of x does F have a local maximum?
(f)
At what values of x does F have inflection points?
(g)
On what intervals of x is F concave down?
(51)
Find each of the following:
x

d 
10
(a)

sin t dt  .

dx  


(52)
(b)
d 

dx 


x

4
t  1 dt 

ln  x 

3 x 1
Evaluate each of the following integral WITHOUT using the Fundamental Theorem of Calculus!
2
5
(a)
5  2t  sin t  dt
(b)
5  2  x  dx (c)
BC Fast Semester Review p.8

0
4
16  y 2 dt
SP 13
(53)
2
The velocity of a particle is given by v  t   t  3t . Use integrals to find each of the following.
(a)
(a)
(54)
(55)
the displacement of the particle for 0  t  4.
the total distance traveled by the particle for 0  t  4.
Find the average value of f  x   x sin  x  on 0,2  .
Find the (positive) area enclosed by the graphs of:
(a)
y  x  x  2 x  5 and the x–axis
(b)
y  1  x and y  x
2
BC Fast Semester Review p.9
SP 13