AP Calculus BC Prerequisite Knowledge

Transcription

1 AP Calculus BC Prerequisite Knowledge Please review these ideas over the summer as they come up during our class and we will not be reviewing them during class. Also, I feel free to quiz you at any time on these facts and procedures. When you join me in the fall, I will using the TI Nspire CX CAS calculator as our demonstration calculator. While you are welcome to use any graphing calculator, I won t be lecturing about the other calculators in class. You may purchase the TI Nspire CX CAS on Amazon.com for approimately $40. If can wait until the fall, you can order one from me for $30. There is an order form on the back of this page. As you look through these pages, feel free to ask me, other math teachers and your fellow future calculus students for clarifications. todd@toddfadoir.com. Cell: Website: We ll be using Calculus: Graphical, Numerical, Algebraic. Third Edition by Finney, Demana, Waits, & Kennedy. Have a great summer! Sincerely, Mr. Fadoir

2 Order for TI Nspire CX CAS, $30 Cash or Check (payable to Berkley High School) Student s Name: PRINT Cash Check # Name of check if different than student s: Return to Mr. Fadoir by September 9, 04 toddfadoir.com/apcalc

3 Find the equation of a line given a slope (m) and, y a point 0 0 What is the equation of a line with slope m= 5 that goes through the point( 3,4 ) What is the equation of a line that goes through,, y. the points ( y ) and 0 0 What is the equation of a line that goes through 5,8. the points ( 3,4 ) and When graphing polynomials, factor to find the zeroes. If the polynomials have factors with odd powers, the polynomial changes sign at the corresponding roots. If the polynomials have factors with even powers, the polynomials doesn t change sign at the corresponding root. Algebra I know your tendency to want to create the equation in the form y= m+ b, I would like you to start using the form ( ) y= y + m. 0 0 y= Done. Leave it in this form. y y m= 0 0, then use m in the equation y= y + m m= = =, then use in the equation 5 3 y 8 5 y= 4+ 3 works. = + ( ). Also E. 3 y= + Factor to find the zeroes y = There are at = 0 and =. Also, the factor that provide the zeroes at = 0 is to an even power, so sign change at = 0. The factor that provides the zero at = is to an odd power, so there is a sign change at = 3 Lastly, lim ( + ) =, so the function starts positive, touches ais at = 0, then goes through the ais at =. Let s confirm by graphing. See later in the document for review of limits. toddfadoir.com/apcalc 3

4 When we ask lim f, we are asking, a what value does f ( ) take on when we infinitesimally close to a, as compared to the question, what is the value of f ( a ). Limits Limits can be one-sided. The value f =, even though the f lim = lim a f asks the question, what value does f give as we get infinitesimally close to a from the left hand (or negative side). lim + a f asks the question, what value does f give as we get infinitesimally close to a from the right hand (or positive side). When the left-hand limit and the right-hand do not agree, we say the overall limit does not eist. =, lim f =, lim lim f + f does not eist toddfadoir.com/apcalc 4

5 We say the function f is continuous at a if three conditions are met. f a eists... lim f eists. a lim f = f a a 3. Limits For continuous functions, the value of the function is the value of the limit. For some discontinuous functions, the limit s value can be found through algebraic manipulation. When the limit can be found through algebraic manipulation, the graph of these functions appear with removable discontinuities. f is not continuous at lim = 4 = because lim f a Eample : lim = lim = lim = + + ( ) Eample : lim = lim = lim = lim 0 0 ( ) ( ) does not eist. = 5 lim = toddfadoir.com/apcalc 5

6 Some limits can t be found using the algebraic manipulation technique. Limits The Intermediate Value Theorem states that if a function is continuous on the a, b and you choose any value interval [ ] between f ( a ) and then there eists a c in the interval ( a, b ) such that f ( c) = L. f b (let s call it L ), Some limits have unbounded behavior in a particular direction. While we say their limits do not eist, we also say their limit approach infinity. lim does not eist. Eample: Show that 3 f = has at least one zero. Condition check: First we acknowledge that f is continuous on [,] and that f ( ) = and f =. 0 is a value between and. Consequence: Therefore f must take on the value 0 somewhere,. on the interval When a function has a limit that approaches infinity as approaches a finite value, we say the function has a vertical asymptote. lim does not eist and lim + + = lim does not eist and lim = f has a vertical asymptote at = toddfadoir.com/apcalc 6

7 We can ask what happens to a function f when gets arbitrarily large, and we use the notation lim f. If the function gets closer to a particular value, we say the function has a horizontal asymptote. Limits lim f = 0. y= 0 is a horizontal asymptote.. Many times, we take the limit of a rational functions; rational functions are functions P in the form where P( ) and Q( ) Q are polynomials. When the order of P is larger than P Q( ), lim =. Q When the order of P is smaller than P Q( ), lim = 0. Q Eample : 3 lim = 3 Eample : 3 lim 0 3 = Eample 3: lim 3 3 = = When the order of P is equal than P Q( ), lim Q is found by looking at the ratio of the coefficients of the highest terms. toddfadoir.com/apcalc 7

8 Some function follow the rule that f a = f a. We call these functions even functions. They have symmetry about the y-ais. Functions f ( ) and f ( ) and f both equal 3. f both equal 0. Some function follow the rule that f a = f a. We call these functions odd functions. They have symmetry about the origin. Symmetry about the origin means that the midpoint between two corresponding points is the origin. If f f f f =, we = and say the f and f are inverse functions. We use inverse functions to undo the other function when solving equations. e and ln are inverse functions. and are inverse functions. sin and sin (sometimes called arcsin ) are inverse functions. toddfadoir.com/apcalc 8 f ( ) = 3 and f( ) = 3= f( ). f = and f ( 3 ) 5 f ( 3 ) 3 5 Eample : e = 0 ( e ) = ln ln 0 = ln 0 Eample : ln = 0 e = e ln 0 = =. 0 = e We used the fact that ln and isolate. e are inverse functions to

9 For every point on function f, the is a corresponding point on f, the inverse function of f. Functions Graphically, the function f is the reflection of the inverse function y=. f over the line Rules for logs ln a + ln b = ln a b a ln( a) ln( b) = ln b ln b ( a ) = b ln( a) ln a logb a= ln b ln e= ln= 0 f = e, f = ln. Because the point graph of f, then the point ( e, ) is on the graph of f. Because the point (,0 ) is on the graph of f, the point ( 0, ). Eample : ln( 3e) = ln( 3) ln( e) = ln( 3) Eample : ( ) ( ),e is on the ln ln + = ln + ln + ln + = ln + + ln + Eample 3: ( ) ( ) = ln = ln ( ) ( + ) + toddfadoir.com/apcalc 9

10 sin Trig (Note: we work eclusively in radians in AP Calculus) The function has zeroes at = n, n is an integer. The function has a maimum value of. The function has a minimum value of -. The function is odd. cos The function has zeroes at = n +, n is an integer. The function has a maimum value of. The function has a minimum value of -. The function is even. tan tan sin = cos The function has zeroes wherever sin = 0. The function is undefined when cos = 0. The function is increasing. The function is odd. toddfadoir.com/apcalc 0

11 sec Trig (Note: we work eclusively in radians in AP Calculus) sec = cos The function has local maimum of -. The function has local minimum of. The function is undefined wherever cos = 0. The function is even, just like cos( ). csc csc = sin The function has local maimum of -. The function has local minimum of. The function is undefined wherever cos = 0. The function is odd, just like sin( ). cot cot cos = sin The function is undefined when sin = 0. The function is always decreasing. The function is odd, just like sin( ). toddfadoir.com/apcalc

12 tan Trig (Note: we work eclusively in radians in AP Calculus) In the normal tangent function, you give an angle, and you get back a ratio. For the inverse tangent function, you give a ratio and get back an angle. tan = because tan =. 4 4 y= is a horizontal asymptote because lim tan =. y= is a horizontal asymptote because lim tan =. The function is odd. You should be able to state the value of any of the big si trig functions for any multiple of from to 6 and any multiple of from to. These are 4 the values you memorized on the unit circle. cos( 0) = ; sin( 0) = 0 ; tan( 0) = 0; 3 sin = ; sin = ; sin = ; cos = ; cos = ; cos = ; tan = ; tan = ; tan = 3 ; cos = 0 ; sin = ; tan is und.; sin = ; sin = ; sin = ; and more 3 Trig Pythagorean Identities sin + cos = tan + = sec + cot = csc toddfadoir.com/apcalc