AP CALCULUS AB SUMMER ASSIGNMENT

Transcription

1 AP CALCULUS AB SUMMER ASSIGNMENT Attached is your summer assignment for AP Calculus (AB). It will probably take you - hours to complete depending on how well you know your material. I would not do the packet at the beginning of the summer. Give your brain a break and wait until the middle of July to start the packet and see what information you have been able to retain and what you need to work on over the summer. If there are topics you are unsure of read the notes section carefully. In addition one of us will be available one day in August to help students. We will post that date and time on the Calculus AB blog located at ( ). In addition the opening week of school for teachers we will be available by appointment. Please us at halsteadr@calvertnet.k.md.us or straubj@calvertnet.k.md.us if you have any questions or want to meet with us that opening week of school for teachers and we will respond as soon as possible. There are si sections to the packet, each section highlights some critical concepts which you need to retain from pre-calc. At the end of each section there is an assignment that correspond to that sections notes {You can also use websites, old Pre-Calculus notes, and friends if you get stuck}. The assignments are worth 0 points each for a grand total of 0 process points, this will be your first process grade of the first quarter. The assignments will be checked for effort on the first day of school with points deducted from the twenty for every problem not attempted. We will have one two hundred point quiz on these assignments the first week of school. We cannot emphasize enough the importance of knowing the unit circle, trig functions, trig identities and using trig identities to manipulate a trig function, so please be sure to study these over the summer. The packet does not contain everything from Pre-Calculus just what is critical for success in AP Calculus. Have a great summer and we look forward to seeing you in the fall as our students. Sincerely, Mr. Halstead Mrs. Straub

2 AP CALC (AB) : SUMMER ASSIGNMENT This assignment is due the first day of class for 0 process points. Name: Pd: Date: PART Interval Notation: 4,7 This means the Real numbers from four to seven including four and seven.,5 This means the Real numbers from negative one to five including negative one but not five.,5 This means the Real numbers from three to five not including either endpoint.,5 5,9 This means the Real numbers from three to nine not including the endpoints or five., This means all Real numbers. We will not use inequalities a ton in Calc, but they are a good way to practice interval notation. E : Solve and graph the solution set: (,6) E : Solve and graph the solution set: (remember to flip the inequality symbol) [-, ] E : Solve the following quadratic inequality: (Remember on these find critical # s and make a chart) critical numbers:, (,) Piece-wise Function: A function made up of the restricted domains of multiple other functions: E : Graph 7 0 f ( ) 0 4 4

3 Absolute Value Function: (This is really just a piecewise function) f ( ) is really f( ) 0 0 E : Solve and sketch the solution set to (to solve this you must do both the positive and negative case) Positive Case: 5 (,5] Negative Case: [, ) E : Solve and sketch the solution set to Positive Case: (, ) Negative Case: 5 (, 5)

4 Assignment # :. Graph the piecewise functions given. a. f ( ) 4 b. For #8-9 Write the following as piecewise functions. 8. f ( ) 4 9. f ( )

5 Part : Slope y y Slope of a line: m E : Find the slope of a line segment connecting (,6) and (, 8) m 5 Completing the square will also be used occasionally in Calculus, so here is a quick eample of how to do it (You could then proceed to solve for, but not necessary at this time) Now for a little more comple completing the square problem. Find the center and radius of a circle with the equation, then give a rough sketch: 4 4y 0 6y y 4y y y y 4 (Divided everything by 4) Center : 5, Radius: Parallel lines: Two lines in a plane whose slopes are equal (they don t intersect as a result) Perpendicular Lines: First, the AP will sometimes call these Normal Lines, two lines who intersect and whose slopes are opposite reciprocals of each other. Hence, they meet at 90 degree angles. Point slope form of a line: You use this when you have the slope and a point and it is going to be the dominant way we write the equation of a line in this class. y y m E : Find the equation of a line that has a slope of and passes through the point (, ) Plug into point-slope: y ( ) no need to put in slope-intercept form

6 E : Find the equations of a line parallel and perpendicular(normal) to the line in eample though the point (4,6) Parallel: y 4 ( 6) Perpendicular(normal): y 4 ( 6) Summary of equation of a line types: Vertical lines always in the form: = a where a is a number Horizontal lines always in the form: y= a where a is a number Point-slope form of a line: y y m Slope intercept form of a line: y=m +b General form of a line: A By C 0 E : Find the equation of the lines that pass through (, ) and are: A) Parallel to y 5 in slope intercept form B) Normal to y5in general form. 5 A) y, slope is two-thirds. So now get it to go through (, ) B) Still has a slope of two-thirds but remember it is normal or perpendicular. A) y 4 y 7 y B) y y y General form: y 6 0

7 Assignment #:

8 9. Complete the square and solve for for each equation. a b Write an equation of line, f( ), given f () 5 and f ( ). Find f ( ).

9 Part : Intercepts and Points of intersection. -intercept: The place a graph crosses the -ais, found by setting y = 0 and solving for. y-intercept: The place a graph crosses the y-ais, found by setting =0 and solving for y. E : Find the and y intercepts of the function y 4 -intercepts ( 4) 0 ( )( ) (0,0),(,0),(,0) y-intercepts y y 0 (0,0) How do you find points of intersection? You set the equations equal to each other or solve one equation for a variable and then substitute it into the other equation. E : Find the intersection point of y 4 and 6 y 6 first,6 y 6 y 66 y 8 Now set equal to get y-coordinate plug back into one of original equations. 4 y y y Intersection point is:, 5 5 E (Harder one): Find the intersection point(s) of y y and y y ( y) y y y 0 y y 0 ( y )( y) y or y and, Intersection points:, and,

10 Assignment : 4. Create an equation that has -intercepts at = -, = 4, and = Find the points of intersections of the graphs of the equations and check your results.

11 Part 4: Functions and Relations Function: Every input of is assigned to eactly one y-output Relation: An input of has multiple outputs for y. E : Does the set of inputs and outputs given for a mystery equation represent a function or a relation? A),,,4,, 5, 6,4, 4,7, 5, Yes this is a function, each input has only one output. Remember it is okay for two different inputs like and 6 to have the same output but one input cannot have two different outputs. B) 4,6,, 5,,8, 4,, 6, No this is not a function, the input of 4 has two different outputs. Domain: Acceptable inputs that do not cause a function to be undefined. Range: The y-values or outputs created by a function. E : For the functions given below what are their domains and ranges.

12 Evaluating a function: E : For the function f ( ) 7, find A) f() B) f(a) C) f( ) D) f ( ) (Rela young Jedi) A) 7 C) a B) 7 9a 7 D) ( ) 7 7 E 4: (Little bit harder) If 4 f 0 ( ) 4 0, Find A) f(0) B) f ( 5) C) f( t 4) A) 0 B) 5 0 C) since t is squared the input must be greater than zero t 4 t t 4 There are si algebraic functions I epect you to know inside and out, backwards, sleeping, eating, standing on your head drinking a cup of water you get the idea. ) y = ) y ) y 4) y

13 5) y 6) y I also epect you to be able to transform these functions (shift left/right, up/down, flip, take abs value of

14 Function types: Polynomial function: ( ) n n f a b... c, E: p ( ) Rational Function: f ( ), where q( ) 0 E: q ( ) f 4 ( ) f( ) 6 8 Radical Function: f ( ) n p( ) 5 E: f ( ) Transcendental functions: Functions that are not made out of algebraic powers. (Trig functions, logarithmic functions, eponential functions, inverse trigonometric functions) E: 4 f ( ) e, f ( ) ln( ), f ( ) sin, f ( ) cos( ), f ( ) tan, f ( ) arcsin I will epect you to know the graphs of sin, cos, and tan without a calculator.

15 Remember we can add, subtract, multiply, or divide fucntions E : If f g ( ), and ( ) 4 Find A) f ( ) g( ) or ( f g)( ) B) f ( ) g( ) or ( f g)( ) C) f ( ) g( ) or ( f g)( ) D) f( ) g ( ) or ( f / g)( ) A) B) C) D) , 4 Composite functions: When one function is substituted into another function or substituted into itself Notations: f(g()) or f g E : Using the same functions as out last eample find: A) f(g()) B) g(f()) C) f(f()) D) g(f()) f ( 4) 4 6 B) A) f ( ) 4 D) g f C) g( ) ( ()) g() 4

16 Assignment 4:

17 8. Use a piece of graph paper and sketch the graphs below without a calculator A) f ( ) B) f ( ) 4 f ( ) C) D) f ( ) 4 E) F) G) f( ) f 4 ( ) 4 f ( ) 4 f ( ) H)

18 Part 5: Trig stuff (You need to have the unit circle memorized. You will not be making a unit circle for each quiz or test we take.) E: sin = E: 5 Tan 4 E: 5 Cos Cos 6 6 E: Sin 6 Know these identities: sin cos tan sec cot csc sin sin cos sin tan cos cos cot sin cos cos sin cos sin cos cos sec cos csc sin cos sec sin csc

19 0, E : Solve on sin cos 0 E : Solve on sin 0 cos sin sin 7, 6 6 sin sin 0 sin sin 0 sin sin sin or sin 5,, 6 6 sin cos cos 0 cos sin 0 cos 0 or sin 7,,, 6 6 0, E : Assignment #5:

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21 Part 6: Miscellaneous The assignment below has some other topics you need to review. If you are unsure how to do a problem, please look online, ask friend for help, or look at your notes from PreCalc. A. Factor completely B. Find all vertical and horizontal asymptotes (if they eist). 4. y. y

22 y 4. y 5. y 4. y 9 8 C. Simplify each of the following... y y y y D. Function Transformations Given the graph of y=f(). Sketch the following graphs.. y=f(). y=-f(). y=f(-) 4. y=f(+) 5. y= f() 6. y=f

23 E. If, find. F. Rewrite as a single logarithmic epression.