Download - AP Calculus AB Semester I 2009
-
8/14/2019 AP Calculus AB Semester I 2009
1/137
-
8/14/2019 AP Calculus AB Semester I 2009
2/137
469 / 470 BC - 399 BC
Socratic IgnoranceSocratic IgnoranceSocratic IgnoranceSocratic Ignorance"I know that I know nothing"
-
8/14/2019 AP Calculus AB Semester I 2009
3/137
PITHAGORAS (600 BC) ZENO (500 BC) EUDOXUS (400 BC) EUCLID (300 BC) ARCHIMIDES (200 BC)*V
NEWTON (1700 AC)*V LIEBNIZ (1700 AC) BERNOULLIS (1700 AC) EULER (1700 AC) LAGRANGE (1800 AC)
GALILEO (1500 AC) FERMAT (1600 AC) CAVALIERI (1600 AC)
DESCARTES (1600 AC) ISAAC BARROW (1600
AC)
FOURIER (1830 AC) CAUCHY (1857 AC) RIEMANN (1866 AC)
WEIERSTRAUSS (1897 AC)
-
8/14/2019 AP Calculus AB Semester I 2009
4/137
Was born and worked in Syracuse (Greek city inSicily) 287 BCE and died in 212 BCE
Friend of King Hieron II
Eureka! (discovery of hydrostatic law)
Invented many mechanisms, some of which were used for the defence
of Syracuse Other achievements in mechanics usually attributed to Archimedes
(the law of the lever, center of mass, equilibrium, hydrostatic pressure)
Used the method of exhaustions to show that the volume of sphere
is 2/3 that of the enveloping cylinder
According to a legend, his last words were Stay away from my
diagram!, address to a soldier who was about to kill him
-
8/14/2019 AP Calculus AB Semester I 2009
5/137
was designed to find areas and volumes of
complicated objects (circles, pyramids,spheres) using
approximationsby simple objects
rec ang es, r ang es, pr smshaving known areas (or volumes)
-
8/14/2019 AP Calculus AB Semester I 2009
6/137
Approximating the circle Approximating the pyramid
-
8/14/2019 AP Calculus AB Semester I 2009
7/137
1) Inner polygons P1 < P2 < P3 Q2 > Q3 >
3) Qi Pi can be made arbitrary small
4) Hence Pi approximate C(R) arbitrarily closely
5 Elementar eometr shows that P isP1
P2
Let C(R) denote area of the circle of radius R
We show that C(R) is proportional to R2
proportional to R2 . Therefore, for two circleswith radii R and R' we get:Pi(R) : Ri (R) = R2:R2
6) Suppose that C(R):C(R) < R2:R2
7) Then (since Pi approximates C(R)) we can find isuch that Pi (R) : Pi (R) < R2:R2 whichcontradicts 5)
Q1
Q2
Thus Pi(R) : Ri (R) = R2:R2
-
8/14/2019 AP Calculus AB Semester I 2009
8/137
Triangles
1 , 2 , 3 , 4, Note that
+ = 1 4 1
Y S Z
Similarly4 + 5 + 6 + 7= 1/16
1and so on
3
4 7
6
2
5
O
Q
XP
Thus A = 1 (1+1/4 + (1/4)2
+) = 4/3 1
-
8/14/2019 AP Calculus AB Semester I 2009
9/137
Calculus appeared in 17th
century as a system ofshortcuts to results obtained by the method ofexhaustion
Calculus derives rules for calculations
Problems, solved by calculus include finding areas,volumes (integral calculus), tangents, normals andcurvatures (differential calculus) and summing of
This makes calculus applicable in a wide variety ofareas inside and outside mathematics
In traditional approach (method of exhaustions) areas
and volumes were computed using subtle geometricarguments
In calculus this was replaced by the set of rules forcalculations
-
8/14/2019 AP Calculus AB Semester I 2009
10/137
Cylon, a Crotoniate and leading citizen by birth,fame and riches, but otherwise a difficult, violent,
disturbing and tyrannically disposed man,eagerly desired to participate in the Pythagoreanway of life. He approached Pythagoras, then anold man, but was rejected because of the
.happened Cylon and his friends vowed to make astrong attack on Pythagoras and his followers.Thus a powerfully aggressive zeal activated Cylonand his followers to persecute the Pythagoreans
to the very last man. Because of this Pythagorasleft for Metapontium and there is said to haveended his days.
-
8/14/2019 AP Calculus AB Semester I 2009
11/137
Los cylonsson
una civilizacinciberntica queest en guerra con
de la humanidaden la pelcula yseries de
BattlestarGalactica ............
-
8/14/2019 AP Calculus AB Semester I 2009
12/137
"Imagination is more important than knowledge."
-
8/14/2019 AP Calculus AB Semester I 2009
13/137
ALBERT EINSTEIN WROTE THISRIDDLE EARLY DURING THE 19thCENTURY. HE SAID THAT 98% OFTHE WORLD POPULATION WOULD
NOT BE ABLE TO SOLVE IT.
ARE YOU IN THE TOP 2% OFINTELLIGENT PEOPLE IN THEWORLD?SOLVE THE RIDDLE AND FINDOUT.
-
8/14/2019 AP Calculus AB Semester I 2009
14/137
There are no tricks, just pure logic,
so good luck and don't give up.
1. In a street there are five houses,
painted five different colours.2. In each house lives a person ofdifferent nationality.
drink a different kind of beverage,smoke different brand of cigar andkeep a different pet.
THE QUESTION:WHO OWNS THE FISH?
-
8/14/2019 AP Calculus AB Semester I 2009
15/137
1. The Brit lives in a red house.2. The Swede keeps dogs as pets.3. The Dane drinks tea.4. The Green house is next to, and on the left of
the White house.5. The owner of the Green house drinks coffee.6. The person who smokes Pall Mall rears birds.7. The owner of the Yellow house smokes Dunhill.8. The man living in the centre house drinks milk.9. The Norwegian lives in the first house.
.
keepscats.
11. The man who keeps horses lives next to the man whosmokes
Dunhill.
12. The man who smokes Blue Master drinks beer.13. The German smokes Prince.14. The Norwegian lives next to the blue house.15. The man who smokes Blends has a neighbour who drinks
water.
-
8/14/2019 AP Calculus AB Semester I 2009
16/137
Einstein's RiddleEinstein's RiddleEinstein's RiddleEinstein's Riddle ---- ANSWERANSWERANSWERANSWER
The German owns the fish.
-
8/14/2019 AP Calculus AB Semester I 2009
17/137
"Do not worry about your difficulties inMathematics.
I can assure you mine are still greater."
-
8/14/2019 AP Calculus AB Semester I 2009
18/137
CALCULUS: CALCULAE: STONES
TWO FUNDAMENTAL IDEAS OF CALCULUSDERIVATIVE-INTEGRAL
CALCULUS APPLICATIONS BOOK RESOURCES
TI 84 PLUS
-
8/14/2019 AP Calculus AB Semester I 2009
19/137
Calculus is deeply integrated in every branch of thephysical sciences, such as physics and biology. It is
found in computer science, statistics, andengineering; in economics, business, and medicine.Modern developments such as architecture, aviation,and other technologies all make use of what
.
Finding the Slope of a Curve Calculating the Area of Any Shape
Visualizing Graphs Finding the Average of a Function Calculating Optimal Values
-
8/14/2019 AP Calculus AB Semester I 2009
20/137
HOW TO FIND:
INSTANTANEOUS RATE OF CHANGE
AREA UNDER A CURVE
T
=
T
A B
-
8/14/2019 AP Calculus AB Semester I 2009
21/137
R= D / T
RATE = CHANGE IN DISTANCE/ CHANGE INTIMEDISTANCE THE AVERAGE RATE OF CHANGE
BETWEEN TWO POINTS =
TIME
CONNECTING THE TWO POINTS
THE INSTANTANEOUS RATE OFCHANGE =THE SLOPE OF THE TANGENT LINE
R = CHANGE IN D / CHANGE IN TR = O / O = UNDEFINED
BIG PROBLEM
-
8/14/2019 AP Calculus AB Semester I 2009
22/137
BLACKBOARD EXAMPLE:
From home to school.
SKETCHPAD
Rate o c ange
-
8/14/2019 AP Calculus AB Semester I 2009
23/137
DISTANCE
THE INSTANTANEOUS RATE OF CHANGE
THE DEFINITION OF THEDERIVATIVE
f(x) TD
R
=
TIMET
0
)()(lim'
+=
x
xxfxxff
x
THE DERIVATIVE OF f(x) AT x REPRESENTSTHE SLOPE OF THE TANGENT LINE AT A POINT x
-
8/14/2019 AP Calculus AB Semester I 2009
24/137
0
)()(lim'
+=
x
xxfxxff
THE DERIVATIVE OF f(x) AT x REPRESENTSTHE SLOPE OF THE TANGENT LINE AT A POINT x
THE INSTANTANEOUS RATE OF CHANGE
T
DR
=
-
8/14/2019 AP Calculus AB Semester I 2009
25/137
Given the graph of below, evaluate the following limits.
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
1
lim ( )x
f x
1lim ( )x
f x
4lim ( )
xf x
3lim ( )x
f x
4lim ( )
xf x
+
2
lim ( )x
f x+
lim ( )x
f x
4lim ( )
xf x
2lim ( )x f x
-
8/14/2019 AP Calculus AB Semester I 2009
26/137
1st Direct Substitution If it fails
2nd Factoring
3rd The Conjugate Method
-
8/14/2019 AP Calculus AB Semester I 2009
27/137
Algebraic Limits:Algebraic Limits:Algebraic Limits:Algebraic Limits:
(a) (b) (c)22
4lim
2x
x
x x
+ +
2
23
9lim
6x
x
x x
23
lim2x
x
x
+
(d) (e) (f)
( )
22
3lim
2x
x
x
+
3 5lim
2 4x
x
x
+2
4 7lim
3 5x
x
x x
+
+
-
8/14/2019 AP Calculus AB Semester I 2009
28/137
Workout the MAGIC (Algebra)
Review:
ECUATIONS, RELATIONS, AND FUNCTIONS TRIGONOMETRY
-
8/14/2019 AP Calculus AB Semester I 2009
29/137
-
8/14/2019 AP Calculus AB Semester I 2009
30/137
-
8/14/2019 AP Calculus AB Semester I 2009
31/137
-
8/14/2019 AP Calculus AB Semester I 2009
32/137
-
8/14/2019 AP Calculus AB Semester I 2009
33/137
-
8/14/2019 AP Calculus AB Semester I 2009
34/137
-
8/14/2019 AP Calculus AB Semester I 2009
35/137
-
8/14/2019 AP Calculus AB Semester I 2009
36/137
Let f be a function with f(1) = 4 such that for allpoints (x, y) on the graph of f. The slope is givenby 23 1
2
x +
where x= 1.
(b)Write an equation for the line tangent to the
graph off at x= 1 and use it to approximatef(1.2)
-
8/14/2019 AP Calculus AB Semester I 2009
37/137
The graph of the velocity v(t), in ft/sec, of a car traveling on astraight road, for is shown above.
A table of values for v(t), at 5 second intervals of time t, is shown0 50,t
to the right of the graph.
(a)During what intervals of time is the acceleration of the carpositive? Give a reason for your answer
(b)Find the average acceleration of the car, in ft/sec2
, over theinterval
(c)Find one approximation for the acceleration of the car, in ft/sec2,at t= 40. Show the computations you used to arrive
at your answer.
0 50,t
-
8/14/2019 AP Calculus AB Semester I 2009
38/137
An equation for the line tangent to the graphof at is:
(a) (b)
cos(2 )x= 4x =
14
y x
=
1 24
y x
=
(c) (d)
(e)
24
y x =
4y x
=
24
y x
=
-
8/14/2019 AP Calculus AB Semester I 2009
39/137
At what point on the graph ofis the tangent line parallel to the line ?
a (0.5, -0.5) b (0.5, 0.125) c (1, -0.25)d (1, 0.5) e (2, 2)
21
2y x=
2 4 3x y =
The following table gives US populations at time t:Estimate and interpret P(1996).
t P (t)
1992 255,002,000
1994 260,292,000
1996 265,253,000
1998 270,002,000
2000 274,634,000
-
8/14/2019 AP Calculus AB Semester I 2009
40/137
-
8/14/2019 AP Calculus AB Semester I 2009
41/137
ken s. . . It also helps you to practice and develop yourlogic/reasoning skills. Calculus throws you challengingproblems your way which make you think.Life after school and college will likewise undoubtedly throw
.
Although you may never use calculus ever again in yourlifetime or career, you will definitely hold on to the lessonsthat calculus taught you.Things like time management, how to be organized andneat, how to hand in things on time, how to perform under
pressure when tested, how to be responsible for your futureboss, how to be amongst people in your class (who areanalogous to your future clients and co-workers).Calculus on face-value may not seem important to you andmay seem useless, but the lessons and skills you arelearning will be with you your whole lifetime.
-
8/14/2019 AP Calculus AB Semester I 2009
42/137
Olivia J: learning advanced math helps youstrengthen your mind overall. Think of your
mind as a muscle. When you lift heavythings for a while, the lighter things seem
.
whats my name again: If you want to be amath teacher you can use it to torture a
whole other generation of kids.
-
8/14/2019 AP Calculus AB Semester I 2009
43/137
KillerLi...You may not use Calculus, but much ofour society relies on it.
The financial operation of our economy relies onforecastings and predictions that only Calculuscan provide. Electrical Engineers use Calculus tooptimize the processing power of the CPU thatruns your computer. ty p anners an surveyorsuse Calculus to find the exact areas of irregularregions of land. So calculus is very important inlife.
As for the meaning of life, Calculus gives noanswers, as it is strictly analytical, and notinterpretational.
-
8/14/2019 AP Calculus AB Semester I 2009
44/137
If you multiply two terms with the same base(here its x), add the powers and keep the base.
ILS AP CALCULUS AB
you v e wo erms w e same ase,
subtract the powers and keep the base.
A negative exponent indicates that a variable is inthe wrong spot, and belongs in the opposite part ofthe fraction, but it only affects the variable itstouching. Note that the exponent becomes positivewhen it moves to the right place.
-
8/14/2019 AP Calculus AB Semester I 2009
45/137
If an exponential expression is raised to a
power, you should multiply the exponents andkeep the base.
ILS AP CALCULUS AB
The numerator of the fractional power remains
the exponent. The denominator of the powertells you what sort of radical (square root, cuberoot, etc.).
-
8/14/2019 AP Calculus AB Semester I 2009
46/137
Example 4:Simplify
Solution:
ILS AP CALCULUS AB
First raise to the third power.
Then
Multiply the xs and ys together
-
8/14/2019 AP Calculus AB Semester I 2009
47/137
Problem 4:
Simplify the expressionusing exponential rules.
ILS AP CALCULUS AB
-
8/14/2019 AP Calculus AB Semester I 2009
48/137
-
8/14/2019 AP Calculus AB Semester I 2009
49/137
-
8/14/2019 AP Calculus AB Semester I 2009
50/137
-
8/14/2019 AP Calculus AB Semester I 2009
51/137
-
8/14/2019 AP Calculus AB Semester I 2009
52/137
-
8/14/2019 AP Calculus AB Semester I 2009
53/137
-
8/14/2019 AP Calculus AB Semester I 2009
54/137
-
8/14/2019 AP Calculus AB Semester I 2009
55/137
-
8/14/2019 AP Calculus AB Semester I 2009
56/137
-
8/14/2019 AP Calculus AB Semester I 2009
57/137
-
8/14/2019 AP Calculus AB Semester I 2009
58/137
-
8/14/2019 AP Calculus AB Semester I 2009
59/137
-
8/14/2019 AP Calculus AB Semester I 2009
60/137
-
8/14/2019 AP Calculus AB Semester I 2009
61/137
-
8/14/2019 AP Calculus AB Semester I 2009
62/137
-
8/14/2019 AP Calculus AB Semester I 2009
63/137
-
8/14/2019 AP Calculus AB Semester I 2009
64/137
-
8/14/2019 AP Calculus AB Semester I 2009
65/137
-
8/14/2019 AP Calculus AB Semester I 2009
66/137
ssssssss worksheetworksheetworksheetworksheet 3333
-
8/14/2019 AP Calculus AB Semester I 2009
67/137
-
8/14/2019 AP Calculus AB Semester I 2009
68/137
-
8/14/2019 AP Calculus AB Semester I 2009
69/137
-
8/14/2019 AP Calculus AB Semester I 2009
70/137
-
8/14/2019 AP Calculus AB Semester I 2009
71/137
-
8/14/2019 AP Calculus AB Semester I 2009
72/137
-
8/14/2019 AP Calculus AB Semester I 2009
73/137
-
8/14/2019 AP Calculus AB Semester I 2009
74/137
ExponentialExponentialExponentialExponential FormFormFormForm 101010103333 = 1,000= 1,000= 1,000= 1,000 33333333 = 27= 27= 27= 27 77770000 = 1= 1= 1= 1
LogarithmicLogarithmicLogarithmicLogarithmic FormFormFormForm log101,000 = 3 log416 = 2 log55 = 1
ExponentialExponentialExponentialExponential FormFormFormForm 4444----2222 = 1/16= 1/16= 1/16= 1/16 101010102222 = 100= 100= 100= 100 3333----2222 = 1/9= 1/9= 1/9= 1/9
LogarithmicLogarithmicLogarithmicLogarithmic FormFormFormForm log255 = 1/2 log31 =0
ExponentialExponentialExponentialExponential FormFormFormForm
LogarithmicLogarithmicLogarithmicLogarithmic FormFormFormForm log28+= Log41= log1010,000= log101/100= Log327=
-
8/14/2019 AP Calculus AB Semester I 2009
75/137
ExponentialExponentialExponentialExponential FormFormFormForm 101010103333 = 1,000= 1,000= 1,000= 1,000 44442222 = 16= 16= 16= 16 33333333 = 27= 27= 27= 27 55551111 = 5= 5= 5= 5 77770000 = 1= 1= 1= 1
LogarithmicLogarithmicLogarithmicLogarithmic FormFormFormForm log101,000 = 3 log416 = 2 log327 = 3 log55 = 1 log71 = 0
ExponentialExponentialExponentialExponential FormFormFormForm 252525251/21/21/21/2 = 5= 5= 5= 5 4444----2222 = 1/16= 1/16= 1/16= 1/16 101010102222 = 100= 100= 100= 100 3333----2222 = 1/9= 1/9= 1/9= 1/9 33330 =0 =0 =0 = 1111
LogarithmicLogarithmicLogarithmicLogarithmic FormFormFormForm log255 = 1/2 log4(1/16) =-2
log10100 = 2 Log3 1/9 = -2
log31 =0
ExponentialExponentialExponentialExponential FormFormFormForm22223333 = 8= 8= 8= 8 44440000 = 1= 1= 1= 1 101010104444 = 10000= 10000= 10000= 10000 10101010----2222 = 1/100= 1/100= 1/100= 1/100 33333333 = 27= 27= 27= 27
LogarithmicLogarithmicLogarithmicLogarithmic FormFormFormForm log28+= 3 Log41= 0 log1010,000 = 4 log101/100= -2 Log327= 3
-
8/14/2019 AP Calculus AB Semester I 2009
76/137
Greatest Common FactorsGreatest Common FactorsGreatest Common FactorsGreatest Common FactorsFactoring using the greatest common factor is the easiestmethod of factoring and is used whenever you see terms thathave pieces in common.Take, for example, the expression 4x + 8. Notice that bothterms can be divided by 4, making 4 a common factor.Therefore, you can write the expression in the factored form of4(x + 2).In effect, I have pulled out the common factor of 4, and whats
left behind are the terms once 4 has been divided out of each.In these type of problems, you should ask yourself, What doeach of the terms have in common? and then pull that greatestcommon factor out of each to write your answer in factoredform.
Problem 5: Factor the expression
-
8/14/2019 AP Calculus AB Semester I 2009
77/137
You should feel comfortable factoring
trinomials such as x + 5x + 4 = 0using whatever method suits you.
os peop e p ay w nom a pa rs nom a pa rs nom a pa rs nom a pa rs un
they stumble across some-thing that works,in this case
(x + 4)(x + 1)
-
8/14/2019 AP Calculus AB Semester I 2009
78/137
There are some patterns that you should have memorized: Difference of perfect squares: a b = (a + b)(a b)
Explanation: A perfect square is a number like 16, whichcan be created by multiplying something times itself. Inthe case of 16, that something is 4, since 4 times itself is16. If ou see one erfect s uare bein subtracted fromanother, you can automatically factor it using the pattern
above. For example, x 25 is a difference of x and 25, and both
are perfect squares. Thus, it can be factored as(x + 5)(x 5).
You cannot factor the sum of perfect squaresso whereas x 4 is factorable, x + 4 is not!
-
8/14/2019 AP Calculus AB Semester I 2009
79/137
Sum of perfect cubes:a + b = (a + b)(a ab + b)
Explanation: Perfect cubes are similar toerfect s uares. The number 125 is a erfect
cube because5 5 5 = 125. This formula can be altered
just slightly to factor the difference of perfectcubes, as illustrated in the next bullet. Other
than a couple of sign changes, the process isthe same.
-
8/14/2019 AP Calculus AB Semester I 2009
80/137
Difference of perfect cubes:a b = (a b)(a2 + ab + b2)
Example 5: Factor x 27 using the difference of
perfect cubes factoring pattern. Solution: Note that xis a perfect cube since x x x = x, and 27 is also,since 3 3 3 = 27. Therefore, x 27 corresponds
, .Now, all thats left to do is plug a and b into theformula:
You cannot factor (x + 3x + 9) any further,so you are finished.
-
8/14/2019 AP Calculus AB Semester I 2009
81/137
Problem 6:Factor the expression 8x + 343
-
8/14/2019 AP Calculus AB Semester I 2009
82/137
MethodMethodMethodMethod OneOneOneOne:::: FactoringFactoringFactoringFactoring
Met o Two: Comp eting t e SquareMet o Two: Comp eting t e SquareMet o Two: Comp eting t e SquareMet o Two: Comp eting t e Square
Method Three: The Quadratic FormulaMethod Three: The Quadratic FormulaMethod Three: The Quadratic FormulaMethod Three: The Quadratic Formula
-
8/14/2019 AP Calculus AB Semester I 2009
83/137
To begin, set your quadratic equation equal to 0; If the resulting equation is factorable, factor it and set each
individual term equal to 0. These equations will give you thesolutions to the equation. Thats all there is to it.
Example 6: Solve the equation 3x + 4x = 1 by factoring
Solution: Always start the factoring method by setting theequation equal to 0.
3x + 4x + 1 = 0.
Now, factor the equation and set each factor equal to 0.(3x + 1)(x + 1) = 0
3x + 1 = 0 x + 1 = 0x = - 1/3 x = - 1
This equation has two solutions: x = -1/3 or x = 1You can check them by plugging each separately into theoriginal equation, and youll find that the result is true.
-
8/14/2019 AP Calculus AB Semester I 2009
84/137
Factoring: 3x + 4x +1
Solution 3x 1 = x (3x + 1)
x 1 = 3x (x + 1)
4x
(3x + 1)( x + 1) = 0
(3x + 1) = 0 , (x + 1) =0
x = - 1/3 , x = -1
-
8/14/2019 AP Calculus AB Semester I 2009
85/137
Example 7: Solve the equation 2x + 12x 18 = 0by completing the square.
Solution:Move the constant to the right side of the equation:2x + 12x = 18
This is im ortant: For com letin the s uare to work thecoefficient of x2 must be 1. Divide every term in the equation by
2: x + 6x = 9Heres the key to completing the square: Take half of thecoefficient of the x term, square it, and add it to both sides.In this problem, the x coefficient is 6, so take half of it (3) andsquare that (3 = 9). Add the result (9) to both sides of theequation:
x + 6x + 9 = 9 + 9x + 6x + 9 = 18
-
8/14/2019 AP Calculus AB Semester I 2009
86/137
At this point, if youve done everything correctly, theleft side of the equation will be factorable. In fact, itwill be a perfect square!
(x + 3)(x + 3) = 18(x + 3) = 18
To solve the equation, take the square root of boths es. a w cance ou e exponen . enever
you do this, you have to add a sign in front of theright side of the equation. This is always done whensquare rooting both sides of any equation:
(x + 3) = 18x + 3 = 18
Solve for x, and thats it. It would also be good formto simplify into : x= -3 18
x = -3 3 2x = - 3 + 3 2 x = - 3 - 3 2
-
8/14/2019 AP Calculus AB Semester I 2009
87/137
The quadratic formulaSet the equation equal to 0, and youre halfwaythere. Your equation will then look like this:
ax + bx + c = 0
where a b and c are the coefficients asindicated.
Take those numbers and plug them straight intothis formula :
Youll get the same answer you would achieve bycompleting the square.
a
acbbx
2
42 =
Solve the equation 2x + 12x 18 = 0
i h d i f l
-
8/14/2019 AP Calculus AB Semester I 2009
88/137
using the quadratic formula.Solution: The equation is already set equal to 0,
in form
ax + bx + c = 0, and a = 2, b = 12, and c = 18
the quadratic formulaand simplify:
-
8/14/2019 AP Calculus AB Semester I 2009
89/137
Problem 7: Solve the equation3x + 12x = 0
three times, using all the methods you havelearned for solving quadratic equations.
-
8/14/2019 AP Calculus AB Semester I 2009
90/137
Basic equation solving is an important skill incalculus.
Reviewing the five exponential rules willprevent arithmetic mistakes in the long run.
You can create the equation of a line with justa little information using point-slope form.
There are three major ways to solve quadraticequations, each important for different reasons.
-
8/14/2019 AP Calculus AB Semester I 2009
91/137
WHEN IS AN ECUATION A FUNCTION? IMPORTANT FUNCTION PROPERTIES FUNCTION SKILLS THE BASIC PARAMETRIC ECUATIONS
GO TO THE TEXTBOOKGO TO THE TEXTBOOKGO TO THE TEXTBOOKGO TO THE TEXTBOOK
-
8/14/2019 AP Calculus AB Semester I 2009
92/137
-
8/14/2019 AP Calculus AB Semester I 2009
93/137
The Rule of four:Tables, Graphs, Formulas, and Words.
CricketCricketCricketCricket ChirpChirpChirpChirp RateRateRateRate versusversusversusversus TemperatureTemperatureTemperatureTemperature
Temperature F 40 75 100 136
Chirps per minute 0 140 240 384
C=4T - 160
The Chirp Rate is a Functionof Temperature C(T)=4T-160
0
100
200300
400
500
40 75 100 136
C (Chirps per minute)
T (F)
300
400
500
C (Chirps per minute)
CricketCricketCricketCricket ChirpChirpChirpChirp RateRateRateRate versusversusversusversus TemperatureTemperatureTemperatureTemperature
C=4T - 160
-
8/14/2019 AP Calculus AB Semester I 2009
94/137
Domain (inputs) =All T values between 40F and 136F
0
100
200
300
40 75 100 136
T (F)
CricketCricketCricketCricket ChirpChirpChirpChirp RateRateRateRate versusversusversusversus TemperatureTemperatureTemperatureTemperatureTemperature F 40 75 100 136
Chirps perminute
0 140 240 384
=All T values with 40x136 =[40,136]
Range (outputs) =All C values from 0 to 384 =All C value with 0C384 =[0,384]
-
8/14/2019 AP Calculus AB Semester I 2009
95/137
This function, called g, accepts any realnumber input. To find out the output g gives,
you plug the input into the x slot.
Real life examples
A persons height is a function of time Other examples (by ss)
-
8/14/2019 AP Calculus AB Semester I 2009
96/137
Sometimes youll plug more than a number into afunctionyou can also plug a function intoanother function. This is called composition of
functions.
Example 1: Iff(x) = and g(x) = x + 6,evaluate g( f (25)).
Solution: In this case, 25 is plugged into f, andthat output is in turn plugged into g.Evaluate f(25).
Now, plug this result into g: g(5) = 5 + 6 = 11Therefore, g(f (25)) = 11.
-
8/14/2019 AP Calculus AB Semester I 2009
97/137
Pi i d fi d f ti
-
8/14/2019 AP Calculus AB Semester I 2009
98/137
Piecewise-defined function
f(1)= f(2)= f(3)= f(10)=
f(0)=
-
8/14/2019 AP Calculus AB Semester I 2009
99/137
The last important thing you should knowabout functions is the vertical line testvertical line testvertical line testvertical line test.
This test is a way to tell whether or not agiven graph is the graph of a function or not.
-
8/14/2019 AP Calculus AB Semester I 2009
100/137
-
8/14/2019 AP Calculus AB Semester I 2009
101/137
Linear functionsy=f(x)=b +mx
y-y=m(x-x)
Nmero de habitantes
-
8/14/2019 AP Calculus AB Semester I 2009
102/137
Nmero de habitantes En el II Conteo de Poblacin y Vivienda 2005,
realizado por el INEGI, se contaron 103 263 388103 263 388103 263 388103 263 388habitantes en Mxico.
Por ello, Mxico est entre los once pases mspoblados del mundo, despus de:China, India, Estados Unidos de Amrica, Indonesia,Brasil, Pakistn, Rusia, Bangladesh, Nigeria y Japn.
-
8/14/2019 AP Calculus AB Semester I 2009
103/137
THE GENERAL EXPONENTIAL FUNCTION PPPP is an exponential funtion oftttt with base aaaa if
Where Pis the initial quantity (when t=0) and a is
taPP 0=
e ac or y w c c anges w en ncreases y .
If a>1, we have exponential growth If 0
-
8/14/2019 AP Calculus AB Semester I 2009
104/137
1981 69.13 1.75
1982 70.93 1.80
1983 72.77 1.841984 74.66 1.89
1985 76.60 1.940
200
400
600
1 3 5
1986 78.59 1.99 1 510
Calculate the Exponential Function: taPP 0=
t (years since 1980)
What is the initial quantity?
What is the Growth Rate?
Evaluate and Interpret P(2005):P(2009):
For what year was the Populationestimated in 100 million people?
EliminationEliminationEliminationElimination of aof aof aof a DrugDrugDrugDrug fromfromfromfrom thethethetheBodyBodyBodyBody
t (hours) Q (mg)
0 250200250
300
Q (mg)
ExponentialExponentialExponentialExponential DecayDecayDecayDecay
-
8/14/2019 AP Calculus AB Semester I 2009
105/137
1 150
2 90
3 54
4 32.4
5 19.40
50
100
150
200
0 1 2 3 4 5t (hours)
ExponentialExponentialExponentialExponential DecayDecayDecayDecay
Calculate the Exponential Function:t
aQQ 0=What is the initial quantity?
What is the Growth Rate?
Evaluate and Interpret Q(10):
How many hours does it takefor the drug to decrease to 0.001mg?
Example 1
Suppose that Q=f(t) is an exponential function of t.
-
8/14/2019 AP Calculus AB Semester I 2009
106/137
pp pIf f(20)=88.2 and f(23)=91.4
a. Find the base b. Find the growth rate c. Evaluate f(25)
Any exponential Growth function can be written, for somea>1 and k>0 in the form
-
8/14/2019 AP Calculus AB Semester I 2009
107/137
a>1 and k>0, in the form
or
And any exponential Decay function can be written, forsome 0
-
8/14/2019 AP Calculus AB Semester I 2009
108/137
The graph of a function is concaveconcaveconcaveconcave upupupup if it bendsupward as we move from left to right;
It is concaveconcaveconcaveconcave downdowndowndown if it bends downward.
Exercises pg. 14: 1,2,3,4,5,6,7,8,9,10,1112,17,23,24,25,26,27,37,39
Shifts and StretchesM lti l i f ti b t t t t h th h ti ll
-
8/14/2019 AP Calculus AB Semester I 2009
109/137
Multiplying a function by a constant, c, stretches the graph vertically(if c>1). Or shrinks the graph vertically (if 0
-
8/14/2019 AP Calculus AB Semester I 2009
110/137
a. f(g(2))
b. g(f(2))
c. f(g(x))
d. g(f(x))
Exmp 2. Express the following function as a composition.h(t)=(1+t)
Odd and Even Functions: Symmetry
The graph of any polynomial involving only even powersof x has symmetry about the x-axis.(E f i E f( ) )
-
8/14/2019 AP Calculus AB Semester I 2009
111/137
(Even functions. E.g. f(x)=x)
Polynomials with only odd powers of x are symmetricabout the origin.(Odd functions. E.g. g(x)=x)
f(x)=x g(x)=x
For any function f,f is an Even function if f(-x)=f(x) for all x.f is an Odd function if f(-x)=-f(x) for all x.
Even function Odd function
Inverse Functionsf(y)=x means y=f(x)
A function has an inverse if (and only if)2
-
8/14/2019 AP Calculus AB Semester I 2009
112/137
A function has an inverse if (and only if)its graph intersects any horizontal line at most once.
In other words,
each y-value correspond to a unique x-value
= ==
],
14,
22,
25,
26,
5
Find the Inverse function. C=f(T)=4T-160 y=xf(C)
Exer
cises.
Pg
21.
[1,
8
The logarithmlogarithmlogarithmlogarithm to base 10 of x, written log x,
is the power of 10 we need to get x.
-
8/14/2019 AP Calculus AB Semester I 2009
113/137
log x = c means 10^c = x
The naturalnaturalnaturalnatural logarithmlogarithmlogarithmlogarithm of x, written ln x,
is the power of e needed to get x.
ln x = c means e^c = x
Properties of Logarithms
1. Log (AB)=log A + log B2. Log (A/B)=log A log B3. Log A^p= p log A4. Log 10^x= x
5. 10 log x= x
1. Ln (AB)=log A + log B2. Ln (A/B)=log A log B3. Ln A^p= p log A4. Ln e^x= x
5. e ln x= x
Log x and Ln x are not defined when x is negative or 0.Log 1=0 Ln 1=0
EX 1. Find t such that 72 =t
-
8/14/2019 AP Calculus AB Semester I 2009
114/137
EX 2. Find when the population of Mexico
reaches 200 million by solving tP )026.1(38.67=
EX 3. What is the half life of ozone?(Decaying exponentially at a continuous rate of 0.25% per year)
-
8/14/2019 AP Calculus AB Semester I 2009
115/137
kteQQ = 0
EX 4. The population of Kenya was 19.5 million in
1984, and 21.2 million in 1986. Assuming itincreases exponentially find a formula for the
-
8/14/2019 AP Calculus AB Semester I 2009
116/137
increases exponentially, find a formula for thepopulation of Kenya as a function of time.
ktePP 0=
Give a formula for the inverse of the following
function. (Solve for tin terms ofP)ttfP )0261(3867)(
-
8/14/2019 AP Calculus AB Semester I 2009
117/137
tfP )026.1(38.67)( ==
Exercises pg 27: 1,7,8,9,11,17,25,28,29,41
An angle of 1 radianradianradianradian is defined to be the angle
-
8/14/2019 AP Calculus AB Semester I 2009
118/137
= 1 Radian
at the center of a unit circle which cuts off anarc of length 1. (measured counterclockwise)
Arc length= 1 180 = radians 1 radian = 180 /
The Unit CircleThe Unit CircleThe Unit CircleThe Unit Circle
=Fundamental Identity: cos t + sin t = 1
Amplitude, Period, and Phase
For any Periodic function of time
AmplitudeAmplitudeAmplitudeAmplitude is half the distance between the maximum and the minimumvalues. (if it exists)P i dP i dP i dP i d i th ll t ti d d f th f ti t t l t
-
8/14/2019 AP Calculus AB Semester I 2009
119/137
PeriodPeriodPeriodPeriod is the smallest time needed for the function to execute one completecycle.
PhasePhasePhasePhase is the difference a periodic function is shifted with respect to other.
Amplitude =1
Period = 2
Sine and Cosine graphsare shifted horizontally /2
cos t = sin(t+ /2)
sin t = cos(t /2)
The phase difference orphase shift between
sin t and cos t is /2Phase = /2
To describe arbitrary amplitudes and periods of Sinusoidal functions:
f(t)=A sin( B t ) and g(t)=A cos( B t )
Where |A| is the amplitude and 2/|B| is the period
The graph of a sinusoidal function is shifted horizontally by a distance |h|
-
8/14/2019 AP Calculus AB Semester I 2009
120/137
The graph of a sinusoidal function is shifted horizontally by a distance |h|when t is replaced by t-h or t+h.
Functions of the form f(t)=A sin (Bt) + C and g(t)=A cos( Bt) + Chave graphs which are shifted vertically and oscillate about the value C.
a) y=5 sin(2t) b) y=-5 sin(t/2) c) y=1 + 2sin t
EX 2. Find possible formulas for the following sinusoidal functions
g(t) f(t) h(t)
-
8/14/2019 AP Calculus AB Semester I 2009
121/137
-6 6t
3
-1 3t
2
-5 7 t
3
-3 -2 -
-
8/14/2019 AP Calculus AB Semester I 2009
122/137
The tangent functionThe tangent functionThe tangent functionThe tangent function
-
8/14/2019 AP Calculus AB Semester I 2009
123/137
tan t=sin t / cos t
The inverse trigonometric functionsThe inverse trigonometric functionsThe inverse trigonometric functionsThe inverse trigonometric functions
arcsine y=x means sin x=y with -/2 x /2
(sin)
arctan y=x means tan x=y with -/2
-
8/14/2019 AP Calculus AB Semester I 2009
124/137
Where k and p are constant.
kxxf =)(
Ex: the volume, V, of a sphere of radius r is given by
V= g(r)=4/3 r
Ex2: Newtons Law of Gravitation
F=k/r or F=kr
PolynomialsPolynomialsPolynomialsPolynomialsare the sums of power functions with nonnegative integer exponents
n is a nonnegative integer called the degree of the polynomial
01
1
1 ...)( axaxaxaxpyn
n
n
n ++++==
-
8/14/2019 AP Calculus AB Semester I 2009
125/137
n is a nonnegative integer called the degree of the polynomial.
degree of the function=_____
752)( 23 == xxxxpy
A leading negative coefficient turns the graph upside down.The quadratic (n=2) turns around once.The cubic (n=3) turns around twice.The quartic (n=4) turns around three times.An degree polynomial turns around at most n-1 times.**There may be fewer turns**
thn
n=2 n=3n=4
n=5
f(x)4
g(x) h(x)
EX1: Find possible formulas for the polynomials.
-
8/14/2019 AP Calculus AB Semester I 2009
126/137
-2 2x
-3 2x
1 -3 2x
-12
Rational functionsRational functionsRational functionsRational functions
are ratios of polynomials, p and q: )()(
)( xq
xp
xf =y y=0 is a Horizontal AsymptoteHorizontal AsymptoteHorizontal AsymptoteHorizontal Asymptote1=y1
3
-
8/14/2019 AP Calculus AB Semester I 2009
127/137
x
ory0 as x and y0 as x-
42 +xy
x=K is a Vertical AsymptoteVertical AsymptoteVertical AsymptoteVertical Asymptoteif
,7,
8,
9,
10,
12,
y or y- as x K
x
y
K
The graphs in Rational functionsmay have vertical asymptoteswhere the denominator is zero.
Rational functionshave horizontal asymptotesif f(x) approaches a finite number
as x or x-.
E
xercises
pg
42:5
A function is said to be continuouscontinuouscontinuouscontinuouson an intervalif its graph has no breaks, jumps or holes in that interval
A continuouscontinuouscontinuouscontinuousfunction has a graph which can be drawnwithout lifting the pencil from the paper.
-
8/14/2019 AP Calculus AB Semester I 2009
128/137
To be certain that a function has a zero in an interval
on which it changes sign, we need to know that the functionis defined and continuouscontinuouscontinuouscontinuousin that interval.
-1 1x
f(x)=3x-x+2x-15
-5
-1
1x
f(x)=1/x
No zero for -1x1although f(-1) and f(1)have opposite signs
Zero for 0x1F(0)=-1 and f(1) =3have opposite signs
A continuous function cannot skip values
-
8/14/2019 AP Calculus AB Semester I 2009
129/137
The function f(x)=cos x -2x must have a zerobecause its graph cannot skip over the x-axis.
f(x) has at least one zero in the interval 0.6x0.8since f(x) changes from positive to negative on thatx
f(x)=cos x -2x1
interva .
-1
0.4 0.6 0.8 1
The Intermediate Value TheoremThe Intermediate Value TheoremThe Intermediate Value TheoremThe Intermediate Value Theorem
Suppose f is a continuous function on a closed interval [a, b].If k is any number between f(a) and f(b), then there is at leastOne number c in [a, b] such that f(c)=k.
x 1.9 1.99 1.999 2.001 2.01 2.1
x 3 61 3 96 3 996 4 004 4 04 4 41
EX: Investigate the continuity of f(x)=x at x=2
-
8/14/2019 AP Calculus AB Semester I 2009
130/137
x 3.61 3.96 3.996 4.004 4.04 4.41
The values of f(x)=x approach f(2)=4 as x approaches 2.Thus f appears to be continuous at x=2
ContinuityContinuityContinuityContinuityThe functionfis continuous at x=c iffis defined at x=cand if
)()(lim
cf
cx
xf=
Exercises pg 47: 15, 17,15. An electrical circuit switches instantaneously from a 6 volt batteryto a 12 volt battery 7 seconds after being turned on. Graph the battery
voltage against time. Give formulas for the function represented byyour graph. What can you say about the continuity of this function?
f(t)
-
8/14/2019 AP Calculus AB Semester I 2009
131/137
t
f(t)
17. Find k so that the following function is continuous on any interval:
xx
xkxxf
-
8/14/2019 AP Calculus AB Semester I 2009
132/137
if the values off(x) approach Las xapproaches c.
)(lim xf )(lim xf)(lim xf
general limit right-hand limit left-hand limit
+ 2x 2x2x
When Limits Do Not ExistWhenever there is no number Lsuch that
cx
Lxf
=)(lim
+ 2
)(lim
x
xf 2
)(lim
x
xfExercises
pg
55:1,
2
-
0
0
Given the graph of below, evaluate the following limits.
-
8/14/2019 AP Calculus AB Semester I 2009
133/137
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)1
lim ( )x
f x
1lim ( )x
f x
4lim ( )
xf x
3lim ( )x
f x
4lim ( )x f x+
2lim ( )
xf x
+
lim ( )x
f x
4lim ( )x f x
2lim ( )
xf x
1st Direct Substitution If it fails (0/0 Indeterminate form)
-
8/14/2019 AP Calculus AB Semester I 2009
134/137
If it fails (0/0 Indeterminate form)
2nd Factoring
3rd The Conjugate Method
Algebraic Limits:Algebraic Limits:Algebraic Limits:Algebraic Limits:
-
8/14/2019 AP Calculus AB Semester I 2009
135/137
(a) (b) (c)22 4lim 2xx
x x+ +
2
239lim
6xx
x x
23lim2x
xx +
(d) (e) (f)
(g)
( )22
3lim
2x
x
x
+
3 5lim
2 4x
x
x
+2
4 7lim
3 5x
x
x x
+
+
4
2lim
4x
x
x
-
8/14/2019 AP Calculus AB Semester I 2009
136/137
Concepts are key to AP Examsslope of the tangent line
limit of slopes of secant lines
i f h
A derivative is
Continuity lim f(x)=f(x)
-
8/14/2019 AP Calculus AB Semester I 2009
137/137
Functions
instantaneous rate of change
limit of average rates of change
Exponential functions
New from old functions Logarithmic functions
Trigonometric functions
Powers, Polynomials, and Rational functions
Continuity
Limits