Ap Calculus Ab Semester I Ago-dic 2009 Ii

AP CALCULUS AB SEMESTER I AGO-DIC 2009

SOCRATES 469 / 470 BC - 399 BC

Socratic Ignorance "I know that I know nothing"

CALCULUS HISTORY             

PITHAGORAS (600 BC) ZENO (500 BC) EUDOXUS (400 BC) EUCLID (300 BC) ARCHIMIDES (200 BC)*V KEPLER (1500 AC) GALILEO (1500 AC) FERMAT (1600 AC) CAVALIERI (1600 AC) DESCARTES (1600 AC) ISAAC BARROW (1600 AC) NEWTON (1700 AC)*V LIEBNIZ (1700 AC)

Archimedes 

Was born and worked in Syracuse (Greek city in Sicily) 287 BCE and died in 212 BCE

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Friend of King Hieron II

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“Eureka!” (discovery of hydrostatic law)

n Invented many mechanisms, some of which were used for the defence of Syracuse n Other achievements in mechanics usually attributed to Archimedes (the law of the lever, center of mass, equilibrium, hydrostatic pressure) n Used the method of exhaustions to show that the volume of sphere is 2/3 that of the enveloping cylinder n According to a legend, his last words were “Stay away from my diagram!”, address to a soldier who was about to kill him

The Method of Exhaustion 

was designed to find areas and volumes of complicated objects (circles, pyramids, spheres) using

◦ approximations by simple objects (rectangles, triangles, prisms) having known areas (or volumes)

Examples Approximating the circle

Approximating the pyramid

Example: Area enclosed by a Circle

n Let C(R) denote area of the circle of radius R n We show that C(R) is proportional to R2 1)

Inner polygons P1 < P2 < P3 <…

2)

Outer polygons Q1 > Q2 > Q3 >…

P

3)

Qi – Pi can be made arbitrary small

2

4)

Hence Pi approximate C(R) arbitrarily closely

5)

Elementary geometry shows that Pi is proportional to R2 . Therefore, for two circles with radii R and R' we get: Pi(R) : Ri (R’) = R2:R’2

6)

Suppose that C(R):C(R’) < R2:R’2

7)

Then (since Pi approximates C(R)) we can find i such that Pi (R) : Pi (R’) < R2:R’2 which contradicts 5)

P1

Q1

Q2

Thus Pi(R) : Ri (R’) = R2:R’2

4.4 The area of a Parabolic Segment [Archimedes (287 – 212 BCE)] Y

S

Z

1 R

4

Q

6

5 O

Thus A = Δ1 (1+1/4 + (

Triangles Δ1 , Δ2 , Δ3 , Δ4,…

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Note that Δ2 + Δ3 = 1/4 Δ1

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Similarly Δ4 + Δ 5 + Δ6 + Δ7 = 1/16 Δ1 and so on

7

3

2

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P

X

1/4)2+…) = 4/3 Δ1

What is Calculus? 

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Calculus appeared in 17th century as a system of shortcuts to results obtained by the method of exhaustion Calculus derives rules for calculations Problems, solved by calculus include finding areas, volumes (integral calculus), tangents, normals and curvatures (differential calculus) and summing of infinite series This makes calculus applicable in a wide variety of areas inside and outside mathematics In traditional approach (method of exhaustions) areas and volumes were computed using subtle geometric arguments In calculus this was replaced by the set of rules for calculations

Pythagoras death 

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 Pythagorean way of life. He approached Pythagoras, then an old man, but was rejected because of the character defects just described. When this happened Cylon and his friends vowed to make a strong attack on Pythagoras and his followers. Thus a powerfully aggressive zeal activated Cylon and his followers to persecute the Pythagoreans to the very last man. Because of this Pythagoras left for Metapontium and there is said to have ended his days.

Cylon 

Los cylons son una civilización cibernética que está en guerra con las Doce Colonias de la humanidad en la película y series de Battlestar Galactica ...

"Imagination is more important than knowledge."

Einstein’s Riddle ALBERT EINSTEIN WROTE THIS RIDDLE EARLY DURING THE 19th CENTURY. HE SAID THAT 98% OF THE WORLD POPULATION WOULD NOT BE ABLE TO SOLVE IT. ARE YOU IN THE TOP 2% OF INTELLIGENT PEOPLE IN THE WORLD? SOLVE THE RIDDLE AND FIND OUT.

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 of different nationality 3. These five homeowners each drink a different kind of beverage, smoke different brand of cigar and keep a different pet. THE QUESTION: WHO OWNS THE FISH?

HINTS

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. 10. The man who smokes Blends lives next to the one who keeps cats. 11. The man who keeps horses lives next to the man who smokes 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 neighbourwho drinks water.

Einstein's Riddle - ANSWER

The German owns the fish.

"Do not worry about your difficulties in Mathematics. I can assure you mine are still greater."

CALCULUS INTRO CALCULUS: CALCULAE: STONES TWO FUNDAMENTAL IDEAS OF CALCULUS DERIVATIVE-INTEGRAL  CALCULUS APPLICATIONS  BOOK  RESOURCES  TI 84 PLUS 

CALCULUS APLICATIONS 

Calculus is deeply integrated in every branch of the physical sciences, such as physics and biology. It is found in computer science, statistics, and engineering; in economics, business, and medicine. Modern developments such as architecture, aviation, and other technologies all make use of what calculus can offer.

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Finding the Slope of a Curve Calculating the Area of Any Shape Visualizing Graphs Finding the Average of a Function Calculating Optimal Values

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THE TWO BIG QUESTIONS OF CALCULUS HOW TO FIND: 

INSTANTANEOUS RATE OF CHANGE

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AREA UNDER A CURVE

A

B

INSTANTANEOUS RATE OF CHANGE 

R= D / T RATE = CHANGE IN DISTANCE/ CHANGE IN TIME THE AVERAGE RATE OF CHANGE BETWEEN TWO POINTS = THE SLOPE OF THE SECANT LINE CONNECTING THE TWO POINTS

DISTANC E

THE INSTANTANEOUSRATE OF CHANGE = THE SLOPE OF THE TANGENT LINE TIME R = CHANGE IN D / CHANGE IN T R = O / O = UNDEFINED “BIG PROBLEM”

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BLACKBOARD EXAMPLE: From home to school.

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SKETCHPAD Rate of change

LIMITS THE INSTANTANEOUS RATE OF CHANGE

f (x)DISTANC E

THE DEFINITION OF THEDERIVATIVE

x TIME THE DERIVATIVE OF f(x) AT x REPRESENTS THE SLOPE OF THE TANGENT LINE AT A POINT x

The Derivative

THE DERIVATIVE OF f(x) AT x REPRESENTS THE SLOPE OF THE TANGENT LINE AT A POINT x

THE INSTANTANEOUS RATE OF CHANGE

UNDERSTANDING LIMITS

Given the graph of below, evaluate the following limits.

   

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Evaluating Limits 

1st Direct Substitution ◦ If it fails…

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2nd Factoring ◦ If it fails…

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3rd The Conjugate Method

Algebraic Limits: (a)

(b)

(c)

(d)

(e)

(f)

Limits 

Workout the MAGIC (Algebra)

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Review: ◦ ALGEBRA ◦ ECUATIONS, RELATIONS, AND FUNCTIONS ◦ TRIGONOMETRY

From 1998 AB4 Let f be a function with f(1) = 4 such that for all points (x, y) on the graph of f . The slope is given by

(a)Find the slope of the graph of f at the point where x = 1. (b)Write an equation for the line tangent to the graph of f at x= 1 and use it to approximate f(1.2)

From 1998 AB3

The graph of the velocity v(t), in ft/sec, of a car traveling on a straight road, for is shown above. A table of values for v(t), at 5 second intervals of time t, is shown to the right of the graph.   (a)During what intervals of time is the acceleration of the car positive? Give a reason for your answer (b)Find the average acceleration of the car, in ft/sec2, over the interval (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.

1997MC AB10 

An equation for the line tangent to the graph of at is:

  (a)

(b)

(c)

(d)

(e)

1997MC AB12  At what point on the graph of is the tangent line parallel to the line a (0.5, -0.5) d (1, 0.5)

b (0.5, 0.125) e (2, 2)

c (1, -0.25)

The following table gives US populations at time t: Estimate and interpret P’(1996).

?

ILS AP CALCULUS AB WORKSHEET 3

secretus...What

is the importance of calculus in life???

ken s. . . It also helps you to practice and develop your logic/reasoning skills. Calculus throws you challenging problems your way which make you think. Life after school and college will likewise undoubtedly throw you problems which you will have to learn to solve. Although you may never use calculus ever again in your lifetime or career, you will definitely hold on to the lessons that calculus taught you. Things like time management, how to be organized and neat, how to hand in things on time, how to perform under pressure when tested, how to be responsible for your future boss, how to be amongst people in your class (who are analogous to your future clients and co-workers). Calculus on face-value may not seem important to you and may seem useless, but the lessons and skills you are learning will be with you your whole lifetime.

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Olivia J: learning advanced math helps you strengthen your mind overall. Think of your mind as a muscle. When you lift heavy things for a while, the lighter things seem really easy.

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whats my name again: If you want to be a math teacher you can use it to torture a whole other generation of kids.

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KillerLi...You may not use Calculus, but much of our society relies on it. The financial operation of our economy relies on forecastings and predictions that only Calculus can provide. Electrical Engineers use Calculus to optimize the processing power of the CPU that runs your computer. City planners and surveyors use Calculus to find the exact areas of irregular regions of land. So calculus is very important in life. As for the meaning of life, Calculus gives no answers, as it is strictly analytical, and not interpretational.

EXPONENTIAL RULES AND PROPERTIES If you multiply two terms with the same base (here it’s x), add the powers and keep the base.

If you divide two terms with the same base, subtract the powers and keep the base.

A negative exponent indicates that a variable is in the wrong spot, and belongs in the opposite part of the fraction, but it only affects the variable it’s touching. Note that the exponent becomes positive when it moves to the right place. ILS AP CALCULUS AB

If an exponential expression is raised to a power, you should multiply the exponents and keep the base.

The numerator of the fractional power remains the exponent. The denominator of the power tells you what sort of radical (square root, cube root, etc.). ILS AP CALCULUS AB

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Example 4: Simplify Solution: First raise to the third power. Then Multiply the x’s and y’s together

ILS AP CALCULUS AB

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Problem 4: Simplify the expression using exponential rules.

ILS AP CALCULUS AB

LOGARITHMIC RULES AND PROPERTIES

ss worksheet 3

Complete the table

Answers

FACTORING POLYNOMIALS 

Greatest Common Factors Factoring using the greatest common factor is the easiest method of factoring and is used whenever you see terms that have pieces in common. Take, for example, the expression 4x + 8. Notice that both terms can be divided by 4, making 4 a common factor. Therefore, you can write the expression in the factored form of 4(x + 2). In effect, I have “pulled out” the common factor of 4, and what’s left behind are the terms once 4 has been divided out of each. In these type of problems, you should ask yourself, “What do each of the terms have in common?” and then pull that greatest common factor out of each to write your answer in factored form. Problem 5: Factor the expression

Special Factoring Patterns 

You should feel comfortable factoring trinomials such as x² + 5x + 4 = 0 using whatever method suits you. Most people play with binomial pairs until they stumble across some-thing that works, in this case (x + 4)(x + 1)

Special Factoring Patterns There are some patterns that you should have memorized:  ◆ Difference of perfect squares: a² – b² = (a + b)(a – b) 

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Explanation: A perfect square is a number like 16, which can be created by multiplying something times itself. In the case of 16, that something is 4, since 4 times itself is 16. If you see one perfect square being subtracted from another, you can automatically factor it using the pattern above. For example, x² – 25 is a difference of x² and 25, and both are perfect squares. Thus, it can be factored as (x + 5)(x – 5). You cannot factor the sum of perfect squares so whereas x² – 4 is factorable, x² + 4 is not!

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◆ Sum of perfect cubes: a³ + b³ = (a + b)(a² – ab + b²) Explanation: Perfect cubes are similar to perfect squares. The number 125 is a perfect cube because 5 ⋅ 5 ⋅ 5 = 125. This formula can be altered just slightly to factor the difference of perfect cubes, as illustrated in the next bullet. Other than a couple of sign changes, the process is the same.

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◆ Difference of perfect cubes: a³ – b³ = (a – b)(a2 + ab + b2) Example 5: Factor x³ – 27 using the difference of perfect cubes factoring pattern. Solution: Note that x is a perfect cube since x ⋅ x ⋅ x = x³, and 27 is also, since 3 ⋅ 3 ⋅ 3 = 27. Therefore, x³ – 27 corresponds to a³ – b³ in the formula, making a = x and b = 3. Now, all that’s left to do is plug a and b into the formula:

You cannot factor (x² + 3x + 9) any further, so you are finished.

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Problem 6: Factor the expression

8x³ + 343

Solving Quadratic Equations 

Method One: Factoring

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Method Two: Completing the Square

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Method Three: The Quadratic Formula

Method One: Factoring  

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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 the solutions to the equation. That’s all there is to it. Example 6: Solve the equation

3x² + 4x = –1

by factoring

Solution: Always start the factoring method by setting the equation equal to 0. 3x² + 4x + 1 = 0. Now, factor the equation and set each factor equal to 0. (3x + 1)(x + 1) = 0 3x + 1 = 0 x+1=0 x = - 1/3 x=-1 This equation has two solutions: x = -1/3 or x = –1 You can check them by plugging each separately into the original equation, and you’ll find that the result is true.

Factoring: 3x² + 4x +1 Solution 3x x

1=x 1 = 3x 4x

(3x + 1) (x + 1)

(3x + 1)( x + 1) = 0 (3x + 1) = 0 , (x + 1) =0 x = - 1/3 , x = -1

Method Two: Completing the Square 

Example 7: Solve the equation 2x² + 12x – 18 = 0 by completing the square. Solution: Move the constant to the right side of the equation: 2x² + 12x = 18 This is important: For completing the square to work, the coefficient of x2 must be 1. Divide every term in the equation by 2: x² + 6x = 9 Here’s the key to completing the square: Take half of the coefficient 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) and square that (3² = 9). Add the result (9) to both sides of the equation: x² + 6x + 9 = 9 + 9 x² + 6x + 9 = 18

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At this point, if you’ve done everything correctly, the left side of the equation will be factorable. In fact, it will be a perfect square! (x + 3)(x + 3) = 18 (x + 3)² = 18 To solve the equation, take the square root of both sides. That will cancel out the exponent. Whenever you do this, you have to add a ± sign in front of the right side of the equation. This is always done when square rooting both sides of any equation: √(x + 3) ² = ± √18 x + 3 = ± √18 Solve for x, and that’s it. It would also be good form to simplify into : x = -3 ± √18 x = -3 ± 3 √2 x = - 3 + 3 √2 x = - 3 - 3 √2

Method Three: The Quadratic  The quadratic formula Formula Set the equation equal to 0, and you’re halfway there. Your equation will then look like this: ax² + bx + c = 0

where a, b, and c are the coefficients as indicated. Take those numbers and plug them straight into this formula : You’ll get the same answer you would achieve by completing the square.

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Solve the equation 2x² + 12x – 18 = 0 using the quadratic formula. Solution: The equation is already set equal to 0, in form ax² + bx + c = 0, and a = 2, b = 12, and c = –18 Plug these values into the quadratic formula and simplify:

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Problem 7: Solve the equation 3x² + 12x = 0 three times, using all the methods you have learned for solving quadratic equations.

The Least You Need to Know 

◆ Basic equation solving is an important skill in calculus.

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◆ Reviewing the five exponential rules will prevent arithmetic mistakes in the long run.

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◆ You can create the equation of a line with just a little information using point-slope form.

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◆ There are three major ways to solve quadratic equations, each important for different reasons.

ECUATIONS, RELATIONS, AND FUNCTIONS WHEN IS AN ECUATION A FUNCTION?  IMPORTANT FUNCTION PROPERTIES  FUNCTION SKILLS  THE BASIC PARAMETRIC ECUATIONS 

GO TO THE TEXTBOOK

BOOK: 1 A LIBRARY OF FUNCTIONS

FUNCTIONS 

The Rule of four: Tables, Graphs, Formulas, and Words.

C=4T - 160

The Chirp Rate is a Function of Temperature C(T)=4T-160

T (°F)

FUNCTIONS

C=4T - 160

T (°F)

◦ Domain (inputs) ◦ =All T values between 40°F and 136°F ◦ =All T values with 40≤x≥136 ◦ =[40,136] ◦ Range (outputs) ◦ =All C values from 0 to 384 ◦ =All C value with 0≤C≥384 ◦ =[0,384]

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This function, called g, accepts any real number input. To find out the output g gives, you plug the input into the x slot.

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Real life examples… ◦ A person’s height is a function of time ◦ Other examples (by ss)…

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Sometimes you’ll plug more than a number into a function—you can also plug a function into another function. This is called composition of functions.

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Example 1: If f(x) = and g(x) = x + 6, evaluate g( f (25)). Solution: In this case, 25 is plugged into f, and that output is in turn plugged into g. Evaluate f(25). Now, plug this result into g: g(5) = 5 + 6 = 11 Therefore, g(f (25)) = 11.

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Piecewise-defined function

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Evaluate ◦ ◦ ◦ ◦ ◦

f(1)= f(2)= f(3)= f(10)= f(0)=

Vertical line test 

The last important thing you should know about functions is the vertical line test. This test is a way to tell whether or not a given graph is the graph of a function or not.

1.1 FUNCTIONS AND CHANGE 

Linear functions y=f(x)=b +mx y-y₁=m(x-x₁)

1.2 EXPONENTIAL FUNCTIONS  Número de habitantes

◦ En el II Conteo de Población y Vivienda 2005, realizado por el INEGI, se contaron 103 263 388 habitantes en México.

◦ Por ello, México está entre los once países más poblados del mundo, después de: China, India, Estados Unidos de América, Indonesia, Brasil, Pakistán, Rusia, Bangladesh, Nigeria y Japón.

1.2 EXPONENTIAL FUNCTIONS

THE GENERAL EXPONENTIAL FUNCTION  P is an exponential funtion of t with base a if 

◦ Where P₀ is the initial quantity (when t=0) and a is the factor by which P changes when t increases by 1. ◦ If a>1, we have exponential growth ◦ If 0
Examples

◦ Population in Mexico ◦ Elimination of a drug from the body

P (Population in millions)

Exponential Growth

t (years since 1980)

Calculate the Exponential Function: What is the initial quantity? What is the Growth Rate? Evaluate and Interpret P(2005): P(2009): For what year was the Population estimated in 100 million people?

Exponential Decay

t (hours)

Calculate the Exponential Function: What is the initial quantity? What is the Growth Rate? Evaluate and Interpret Q(10): How many hours does it take for the drug to decrease to 0.001mg?

Example 1 

Suppose that Q=f(t) is an exponential function of t. If f(20)=88.2 and f(23)=91.4 a. Find the base b. Find the growth rate

c. Evaluate f(25)

Exponential Functions 

withbase

◦ 

e

Any exponential Growth function can be written, for some a rel="nofollow">1 and k>0, in the form or

And any exponential Decay function can be written, for some 00, as ◦

or

◦ We say that P and Q are growing or decaying at a continous rate of k. (k=0.02 corresponds to a Continous rate of 2%) 

Example. Convert the functions Into the form ◦

and and

Concavity 

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The graph of a function is concave up if it bends upward as we move from left to right; It is concave down if it bends downward.

Exercisespg. 14: 1,2,3,4,5,6,7,8,9,10,11 12,17,23,24,25,26,27,37,39

1.3 NEW FUNCTIONS FROM OLD  Shifts and Stretches Multiplying a function by a constant, c, stretches the graph vertically (if c>1). Or shrinks the graph vertically (if 0
y=-2f(x) y=f(x ) y=3f(x )

y=x²+ 4 y =x²

y =x²

y=(x-2)²

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Composite Functions “A Function of a Function” ◦ Example 1. If f(x)=x² and g(x)=x+1, find: ◦ 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+t³)²⁷

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Odd and Even Functions: Symmetry ◦ The graph of any polynomial involving only even powers of x has symmetry about the x-axis. (Even functions. E.g. f(x)=x²) ◦ Polynomials with only odd powers of x are symmetric about the origin. (Odd functions. E.g. g(x)=x³) Even function

f(x)=x ²

Odd function

For any function f, f is an Even function iff(-x)=f(x) for all x. f is an Odd function if f(-x)=-f(x) for all x.

g(x)=x ³

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Inverse Functions fˉ¹(y)=x means y=f(x)

Ex er ci se s. P g 2 1. [1 ,8 ], 1 4, 2 2, 2

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 y=f(x )

Find the Inverse function. y=x³

y=f(x )

C=f(T)=4T-160 fˉ¹(C)

fˉ¹(y)= x

1.4 LOGARITHMIC FUNCTIONS The logarithm to base 10 of x, written log₁₀ x, is the power of 10 we need to get x. log₁₀ x = c means 10^c = x The natural logarithm of x, written ln x, is the powerof e needed to get x. lnx = c means e^c = x Properties of Logarithms 3. 4. 5. 6. 7.

Log (AB)=log A + log B Log (A/B)=log A – log B Log A^p= p log A Log 10^x= x 10 ^ log x= x

Log x and Lnx are not defined when x is negative or 0. Log 1=0 Ln 1=0

Solving Equations using Logarithms  EX 1. Find t such that

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EX 2. Find when the population of Mexico reaches 200 million by solving

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EX 3. What is the half life of ozone?

(Decaying exponentially at a continuous rateof 0.25% per year)

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EX 4. The population of Kenya was 19.5 million in 1984, and 21.2 million in 1986. Assuming it increases exponentially, find a formula for the population of Kenya as a function of time.

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Give a formula for the inverse of the following function. (Solve for t in terms of P )

Exercises pg 27: 1,7,8,9,11,17,25,28,29,41

1.5 TRIGONOMETRIC FUNCTIONS An angle of 1 radian is defined to be the angle at the center of a unit circle which cuts off an arc of length 1. (measured counterclockwise) Arc length= 1 =1 Radian

The Unit Circle

180° = πradians

1 radian = 180° / π

Equation of the unit circle: x² + y² =1 Fundamental Identity: cos² t + sin² t = 1

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Amplitude, Period, and Phase

For any Periodic function of time Amplitude is half the distance between the maximum and the minimum values. (if it exists) Period is the smallest time needed for the function to execute one complete cycle. Phase is the difference a periodic function is shifted with respect to other.

Amplitude =1

Sine and Cosine graphs are shifted horizontally π/2 cos t = sin(t+ π/2) sin t = cos(t – π/2)

Period = 2π Phase = π/2

The phase difference or phase shift between sin t and cos t is π/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 2π/|B| is the period 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) + C Ex 1. Find and show on the graph the Amplitude and Period of the functions. 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) 3

-6π

3

2

6π

-3

h (t)

f (t)

t

-1

3

-2

t

π

-5π

-3

7π

t

EX 3. The High tide was 9.9 feet at midnight. Later at Low tide, it was 0.1 feet. the next High tide is at exactly 12 noon and the height of the water is given by a sine or cosine curve. Find a formula for the water level as a function of time.

Ex 4. The interval between high tides actually averages 12 hours 24 minutes. Give a more accurate formula. Ex 5. Using the info from Ex 4. Write a formula for the water level, when the high tide is at 2 pm.

Exercises pg 35. 13,14,15,16,17,19,20,24,25,38

The tangent function tan t=sin t / cos t

The inverse trigonometric functions arcsine y=x means sin x=y with -π/2 ≤x≤ π/2 (sinˉ¹) arctan y=x means tan x=y with -π/2 <x< π/2 (tanˉ¹) arccos y=x means cos x=y with -π/2 ≤x≤ π/2 (cosˉ¹)

1.6 POWERS, POLYNOMIALS, AND RATIONAL FUNCTIONS A power function has the form Where k and p are constant. Ex: the volume, V, of a sphere of radius r is given by V= g(r)=4/3 πr³ Ex2: Newton’s Law of Gravitation F=k/r²

or

F=krˉ²

Polynomials are the sums of power functions with nonnegative integer exponents

n is a nonnegative integer called the degree of the polynomial. degree of the function=_____ The shape of the graph of a polynomial depends on its degree. 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**

n= 2

n= 3

n= 4

n= 5

EX1: Find possible formulas for the polynomials. 4

-2

f (x)

g (x)

2

x

-3

1 -12

h (x)

2

x

-3

2

x



Ex er ci se s p g 4 2: 5, 7, 8, 9, 1 0,

Rational functions are ratios of polynomials, p and q: y=0 is a Horizontal Asymptote or y→0 as x→∞ and y→0 as x→-∞

y

x

x=K is a Vertical Asymptote if y→∞ or y→-∞ as x →K y

The graphs in Rational functions may have vertical asymptotes where the denominator is zero. Rational functions have horizontal asymptotes if f(x) approaches a finite number as x→∞ or x→-∞.

K

x

1.7 INTRODUCTION TO CONTINUITY A continuous function has a graph which can be drawn without lifting the pencil from the paper. A function is said to be continuous on an interval if its graph has no breaks, jumps or holes in that interval To be certain that a function has a zero in an interval on which it changes sign, we need to know that the function is defined and continuous in that interval. f(x)=3x²-x²+2x-1

f(x)=1/x

5

-1

1

x

-1 1

-5

Zero for 0≤x≤1 F(0)=-1 and f(1) =3 have opposite signs

No zero for -1≤x≤1 although f(-1) and f(1) have opposite signs

x

A continuous function cannot skip values 1

f(x)=cos x -2x²

The function f(x)=cos x -2x² must have a zero because its graph cannot skip over the x-axis.

x 0.4 0.6 0.8 1

f(x) has at least one zero in the interval 0.6≤x≤0.8 since f(x) changes from positive to negative on that

-1

The 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 least

EX: Investigate the continuity of f(x)=x² at x=2

The values of f(x)=x² approach f(2)=4 as x approaches 2. Thus f appears to be continuous at x=2

Continuity The function f is continuous at x=c if f is defined at x=c and if

Exercises pg 47: 15, 17, 15. An electrical circuit switches instantaneously from a 6 volt battery to a 12 volt battery 7 seconds after being turned on. Graph the battery voltage against time. Give formulas for the function represented by your graph. What can you say about the continuity of this function? f (t)

t

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

1.8 LIMITS Notation:

if the values of f(x) approach L as x approaches c.

Ex er ci se s p g 5 5:

general limit

right-hand limit

left-hand limit

When Limits Do Not Exist Whenever there is no number L such that

≠

∞ -∞

UNDERSTANDING LIMITS

Given the graph of below, evaluate the following limits.

   

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Evaluating Limits 

1st Direct Substitution ◦ If it fails… (0/0 Indeterminate form)



2nd Factoring ◦ If it fails…



3rd The Conjugate Method

Algebraic Limits: (a)

(b)

(c)

(d)

(e)

(f)

(g)

Chapter1 REVIEW EXERCISES AND PROBLEMS

1st Period Exam Review 

Concepts are key to AP Exams •A derivative is



•Continuity

Functions ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦

Linear functions Exponential functions New from old functions Logarithmic functions Trigonometric functions Powers, Polynomials, and Rational functions Continuity Limits

lim f(x)=f(x)