AP Calculus AB 08-09 Summer Assignment Dear Future AP Student, I hope you are ecited for the year of Calculus that we will be pursuing together! I don t know how much you know about Calculus, but it is not like any other branch of math that you have learned so far in your math careers. We will be having a lot of fun and doing a lot of work learning about derivatives (for the first semester) and integrals (for the second semester). You don t need to know what those things are (yet) but I will tell you that Calculus is described as the mathematics of change how fast things change, how to predict change, and how to use information about change to understand the systems themselves. Actually, in some ways, Calculus is taking what you already know a step further. You know how to find the slope of a line, right? You probably don t know how to find the slope of a curve because it s constantly changing but Calculus helps us do that. So traditional math tells us how to find the slope of a line, and Calculus tells us how to find the slope of a curve. Traditional math tells us how to find the length of a rope pulled taut, but Calculus tells us how to find the length of a curved rope. Traditional math tells us how to find the area of a flat, rectangular roof, but Calculus tells us how to find the area of a curved dome-shaped roof. Get the idea? How does Calculus manage to pull this off? Imagine a curve like this: If you were to zoom in a few times, each part of the curve would look kind of like a line, wouldn t it? And if a few times wasn t enough, you could zoom in more. And more. And more. In fact, you could zoom in nearly an infinite number of times until the curve became enough like a line that you could treat it that way. What makes calculus such a fantastic achievement is that it actually zooms in infinitely. In fact, everything you do in calculus involves infinity in one way or another, because if something is constantly changing, it s changing infinitely often from each infinitesimal moment to the net. (taken from http://media.wiley.com/product_data/ecerpt/84/0764549/076454984.pdf) This process doing something an infinite number of times until the problem becomes figure-out-able is the foundation of Calculus. The process is called a limit and it s what we ll be talking about when school begins. In order to give you a head start in the understanding of limits, I ve designed a few things for you to do. I ve tried to make it fairly straightforward so you can still enjoy your summer, but I think this will get us rolling more quickly in September and therefore will put you in a better position to take your AP in May. Please answer the questions that follow and bring this packet with you for the first day of school in September. Feel free to send an email to derek.keating@polk-fl.net if you have any questions. See you in August!
AP Calculus 08-09 Major (hon) Keating Summer Assignment Summerlin Academy HIGH SCHOOL AP CALCULUS SUMMER REVIEW PACKET NAME:
READ THE FOLLOWING DIRECTIONS CAREFULLY!. Before answering any questions read through the given notes and eamples for each topic.. This packet is to be handed in on the first day of school.. All work must be shown in the packet OR on separate paper attached to the packet. 4. Completion of this packet is worth test grade. 5. There will be a 0 point dock for each day the packet is turned in late. 6. ALL work must be organized and neatly written.
FUNCTIONS To evaluate a function for a given value, simply plug the value into the function for. Recall: ( f g)( ) f( g( )) OR f[ g( )] read f of g of Means to plug the inside function (in this case g() ) in for in the outside function (in this case, f()). Eample: Given f ( ) + and g( ) 4 find f(g()). f ( g ( )) f( 4) ( 4) + ( 8+ 6) + 6+ + f( g( )) 6+ Let f ( ) + and g( ). Find each.. f (). g( ). f( t+ ) 4. f [ g( ) ] 5. g[ f( m+ ) ] 6. [ f ] ( ) g( ) Let f ( ) sin( ) Find each eactly. π 7. f 8. 4 π f Let f ( ), g( ) + 5, and h( ). Find each. 9. h[ f( ) ] 0. [ ] f g( ). g h ( ) 4
INTERCEPTS OF A GRAPH To find the -intercepts, let y 0 in your equation and solve. To find the y-intercepts, let 0 in your equation and solve. Eample: Given the function y, find all intercepts. int. ( Let y 0) 0 0 ( )( + ) or intercepts (,0) and (,0) y int. ( Let 0) y 0 (0) y y intercept (0, ) Find the and y intercepts for each.. y 5. y + 4. y 6 5. y 4 5
POINTS OF INTERSECTION Use substitution or elimination method to solve the system of equations. Remember: You are finding a POINT OF INTERSECTION so your answer is an ordered pair. CALCULATOR TIP Remember you can use your calculator to verify your answers below. Graph the two lines then go to CALC ( nd Trace) and hit INTERSECT. Eample: Find all points of intersection of ELIMINATION METHOD Subtract to eliminate y 0 ( )( ) + 0 or Plug in and to find y Points of Intersection: (,) and (, ) y y SUBSTITUTION METHOD Solve one equation for one variable. y y Therefore by substitution 0 From here it is the same as the other eample Find the point(s) of intersection of the graphs for the given equations. 6. + y 8 4 y 7 7. + y 6 + y 4 8. y y 6
DOMAIN AND RANGE Domain All values for which a function is defined (input values) Range Possible y or Output values EXAMPLE EXAMPLE Find the domain and range of f ( ) 4 Write answers in interval notation. DOMAIN For f ( ) to be defined 4 0. This is true when, Domain: [ ] RANGE The solution to a square root must always be positive thus f ( ) must be greater than or equal to 0. Range: [ 0, ) Find the domain and range of each function. Write your answer in INTERVAL notation. 9. f( ) 5 0. f( ) +. f ( ) sin. f( ) 7
INVERSES To find the inverse of a function, simply switch the and the y and solve for the new y value. f f Recall ( ) ( ) ( y+ ) is defined as the inverse of ( ) Eample : f( ) + Rewrite f() as y y + Switch and y y + Solve for your new y y+ Simplify Cube both sides y Solve for y f ( ) Rewrite in inverse notation Find the inverse for each function.. f ( ) + 4. f( ) 5. 5 g ( ) 6. y 4 + 7. If the graph of ( ) f has the point (, 7) then what is one point that will be on the graph of f ( )? 8. Eplain how the graphs of ( ) f and f ( ) compare. 8
EQUATION OF A LINE Slope intercept form: y m + b Vertical line: c (slope is undefined) Point-slope form: y y m( ) Horizontal line: y c (slope is 0) * LEARN! We will use this formula frequently! Eample: Write a linear equation that has a slope of ½ and passes through the point (, -6) Slope intercept form y b 6 ( ) + b Plug in the given ordered b 7 Solve for b y 7 + Plug in ½ for m ( ) Point-slope form y+ 6 Plug in all variables y 7 Solve for y 9. Determine the equation of a line passing through the point (5, -) with an undefined slope. 0. Determine the equation of a line passing through the point (-4, ) with a slope of 0.. Use point-slope form to find the equation of the line passing through the point (0, 5) with a slope of /. 5. Use point-slope form to find a line passing through the point (, 8) and parallel to the line y. 6. Use point-slope form to find a line perpendicular to y + 9 passing through the point (4, 7). 4. Find the equation of a line passing through the points (-, 6) and (, ). 5. Find the equation of a line with an -intercept (, 0) and a y-intercept (0, ) 9
UNIT CIRCLE You can determine the sine or the cosine of any standard angle on the unit circle. The -coordinate of the circle is the cosine and the y-coordinate (0,) is the sine of the angle. Recall tangent is defined as sin/cos or the slope of the line. (-,0) Eamples: (0,-) (,0) π π π sin cos 0 tan und - *You must have these memorized OR know how to calculate their values without the use of a calculator. 6. a.) sinπ π b.) cos π c.) sin 5π d.) sin 4 π π e.) cos f.) cos( π ) g)cos 4 h) 5π sin 6 i) π cos j)tan 4 π k) tanπ l) tan π m) 4π cos n) π sin 6 o) 7π tan 4 π p) sin 6 0
TRIGONOMETRIC EQUATIONS Solve each of the equations for 0 < π. 7. sin 8. cos 9. 4sin 40. cos cos 0 *Factor **Recall sin (sin ) **Recall if 5 then ± 5 TRANSFORMATION OF FUNCTIONS h ( ) f( ) + c Vertical shift c units up h ( ) f( c) Horizontal shift c units right h ( ) f( ) c Vertical shift c units down h ( ) f( + c) Horizontal shift c units left h ( ) f( ) Reflection over the -ais 4. Given f ( ) and g ( ) ( ) +. How the does the graph of g() differ from f()? 4. Write an equation for the function that has the shape of reflected over the -ais. f ( ) but moved si units to the left and 4. If the ordered pair (, 4) is on the graph of f ( ), find one ordered pair that will be on the following functions: a) f( ) b) f( ) c) f ( ) d) f( ) + e) f ( )
VERTICAL ASYMPTOTES Determine the vertical asymptotes for the function. Set the denominator equal to zero to find the -value for which the function is undefined. That will be the vertical asymptote given the numerator does not equal 0 also (Remember this is called removable discontinuity). Write a vertical asymptotes as a line in the form Eample: Find the vertical asymptote of y Since when the function is in the form /0 then the vertical line is a vertical asymptote of the function. 44. f( ) 45. f( ) 4 46. f( ) + ( ) 4 47. f( ) 6 48. f( ) + 5 + 0 49. f( ) 6
HORIZONTAL ASYMPTOTES Determine the horizontal asymptotes using the three cases below. Case I. Degree of the numerator is less than the degree of the denominator. The asymptote is y 0. Eample: y (As becomes very large or very negative the value of this function will approach 0). Thus there is a horizontal asymptote at y 0. Case II. Degree of the numerator is the same as the degree of the denominator. The asymptote is the ratio of the lead coefficients. + Emaple: y (As becomes very large or very negative the value of this function will + 4 approach /). Thus there is a horizontal asymptote at y. Case III. Degree of the numerator is greater than the degree of the denominator. There is no horizontal asymptote. The function increases without bound. (If the degree of the numerator is eactly more than the degree of the denominator, then there eists a slant asymptote, which is determined by long division.) + Eample: y (As becomes very large the value of the function will continue to increase and as becomes very negative the value of the function will also become more negative). Determine all Horizontal Asymptotes. 50. f( ) + + 7 5. 5 + 8 f( ) 4 + 5 4 5. f( ) 7 5. ( 5) f( ) 54. f( ) + + * Remember ± *This is very important in the use of limits.*
Eample: Solve for + 4 ( ) ( ) + Get a common base + + Simplify + + Set eponents equal 0 Solve for EXPONENTIAL FUNCTIONS Solve for : 55. + 5 + 9 56. 7 9 + 4 57. 6 6 LOGARITHMS The statement y b can be written as log b y. They mean the same thing. REMEMBER: A LOGARITHM IS AN EXPONENT Recall ln log e The value of e is.78888 or lim + Evaluate the following logarithms 58. log7 7 59. log 7 Eample: Evaluate the following logarithms log 8? 60. log 6. log5 5? In eponential for this is 8 Therefore? Thus log 8 6. log9 6. log4 8 4 64. ln e 65. ln e
PROPERTIES OF LOGARITHMS y logb y logb + logb y logb logb logb y logb ylogb y Eamples: log b b Epand log4 6 Condense ln y ln R Epand log 7 5 log 6 + log ln y ln R log 7 + log 4 4 log4 + ln y R log 7 + 5log 5 Use the properties of logarithms to evaluate the following 66. 5 log 67. ln e 68. 5 log 8 69. log 9 log0 70. 7. ln8 e 7. 9lne 7. log 9 9 74. log0 5 + log0 4 75. log40 log5 76. log ( ) 5 5
EVEN AND ODD FUNCTIONS Recall: Even functions are functions that are symmetric over the y-ais. To determine algebraically we find out if f ( ) f( ) (*Think about it what happens to the coordinate (, f( ) ) when reflected across the y-ais*) Odd functions are functions that are symmetric about the origin. To determind algebraically we find out if f ( ) f( ) (*Think about it what happens to the coordinate (, f( ) ) when reflected over the origin*) State whether the following graphs are even, odd or neither, show ALL work. 77. 78. y 79. 80. 4 f ( ) 5 5 g ( ) + 8. 8. h ( ) 5+ j ( ) cos 8. 84. k ( ) sin+ 4 l ( ) cos 6
LIMITS! First of all, watch some eplanations of limits online. Go to the following website http://www.calculushelp.com/tutorials and watch lessons -5. I ve included some questions below that go with each lesson. Then answer the corresponding questions after watching each lesson. Lesson : What is a limit?. How would you describe a limit?. Some graphs are straightforward, like f ( ). What s lim? + 4 + 4. Some graphs are more mysterious, like f( ). What s lim? (Either find the limit, if you know how, or describe how to find it based on the eplanation you saw online) Lesson : When does a limit eist? 4. How is a limit like two friends meeting at a diner? 5. Look at the following graphs. Which one(s) have a limit that eists at, and which one(s) don t have a limit that eists at? 6. What is meant by a right hand limit and a left hand limit? 7
7. For the following graph, find: lim f ( ) + lim f ( ) 8. For a limit to eist, what has to be true for the left hand and the right hand limits? Lesson : How do you evaluate limits? 9. What are the methods for evaluating limits? 0. When can you use the substitution method?. When can you use the factoring method?. When can you use the conjugate method?. Figure out which method to use for the following limits, and evaluate them: a. lim 9 9 b. 4 + 5 lim 6 c. lim + 4 8
Lesson 4: Limits and Infinity 4. How do you know if a function has a vertical asymptote? 5. When you take the limit of a function at its vertical asymptote, the limit will be or. 6. To determine if a function has a horizontal asymptote, look at the 7. If the degrees of the numerator and denominator are equal, how do you find the horizontal asymptote? 8. If the degree of the denominator is greater than that of the numerator, what s the horizontal asymptote? 9. If the degree of the denominator is less than that of the numerator, what s the horizontal asymptote? 0. If we say that the limit of a function EQUALS INFINITY, this really means that. Lesson 5: Continuity. What does it mean for a function to be continuous?. What are the types of discontinuity? Draw an eample of a graph of each kind below:. In order to be continuous, things must be true: There must be no There must be no The limit must be equal to the 4. An easy way to tell if a function is continuous is this: 9