ap calculus ab semester i 2009

8/14/2019 AP Calculus AB Semester I 2009

1/137


2/137

469 / 470 BC - 399 BC

Socratic IgnoranceSocratic IgnoranceSocratic IgnoranceSocratic Ignorance"I know that I know nothing"


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)


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


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)


6/137

Approximating the circle Approximating the pyramid


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


8/137

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


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


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.


11/137

Los cylonsson

una civilizacinciberntica queest en guerra con

de la humanidaden la pelcula yseries de

BattlestarGalactica ............


12/137

"Imagination is more important than knowledge."


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.


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?


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.


16/137

Einstein's RiddleEinstein's RiddleEinstein's RiddleEinstein's Riddle ---- ANSWERANSWERANSWERANSWER

The German owns the fish.


17/137

"Do not worry about your difficulties inMathematics.

I can assure you mine are still greater."


18/137

CALCULUS: CALCULAE: STONES

TWO FUNDAMENTAL IDEAS OF CALCULUSDERIVATIVE-INTEGRAL

CALCULUS APPLICATIONS BOOK RESOURCES

TI 84 PLUS


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


20/137

HOW TO FIND:

INSTANTANEOUS RATE OF CHANGE

AREA UNDER A CURVE

T

=

T

A B


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


22/137

BLACKBOARD EXAMPLE:

From home to school.

SKETCHPAD

Rate o c ange


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


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

=


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


26/137

1st Direct Substitution If it fails

2nd Factoring

3rd The Conjugate Method


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

+

+


28/137

Workout the MAGIC (Algebra)

Review:

ECUATIONS, RELATIONS, AND FUNCTIONS TRIGONOMETRY


29/137


30/137


31/137


32/137


33/137


34/137


35/137


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)


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


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

=


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


40/137


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.


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.


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.


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.


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.).


46/137

Example 4:Simplify

Solution:

ILS AP CALCULUS AB

First raise to the third power.

Then

Multiply the xs and ys together


47/137

Problem 4:

Simplify the expressionusing exponential rules.

ILS AP CALCULUS AB


48/137


49/137


50/137


51/137


52/137


53/137


54/137


55/137


56/137


57/137


58/137


59/137


60/137


61/137


62/137


63/137


64/137


65/137


66/137

ssssssss worksheetworksheetworksheetworksheet 3333


67/137


68/137


69/137


70/137


71/137


72/137


73/137


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=


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


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


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)


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!


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.


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.


81/137

Problem 6:Factor the expression 8x + 343


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


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.


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


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


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


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


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:


89/137

Problem 7: Solve the equation3x + 12x = 0

three times, using all the methods you havelearned for solving quadratic equations.


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.


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


92/137


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


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]


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)


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.


97/137

Pi i d fi d f ti


98/137

Piecewise-defined function

f(1)= f(2)= f(3)= f(10)=

f(0)=


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.


100/137


101/137

Linear functionsy=f(x)=b +mx

y-y=m(x-x)

Nmero de habitantes


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.


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


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


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.


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


107/137

a>1 and k>0, in the form

or

And any exponential Decay function can be written, forsome 0


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


109/137

Multiplying a function by a constant, c, stretches the graph vertically(if c>1). Or shrinks the graph vertically (if 0


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( ) )


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


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.


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


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)


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


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)(


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


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


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|


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)


121/137

-6 6t

3

-1 3t

2

-5 7 t

3

-3 -2 -


122/137

The tangent functionThe tangent functionThe tangent functionThe tangent function


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


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 ++++==


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.


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


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.


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


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


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)


131/137

t

f(t)

17. Find k so that the following function is continuous on any interval:

xx

xkxxf


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.


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)


134/137

If it fails (0/0 Indeterminate form)

2nd Factoring

3rd The Conjugate Method

Algebraic Limits:Algebraic Limits:Algebraic Limits:Algebraic Limits:


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


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)


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

ap calculus ab semester i 2009

Documents