AP Calculus AB Summer Math Packet

Transcription

1 AP Calculus AB Summer Math Packet This is the summer review and preparation packet for students entering AP Calculus AB Dear Bear Creek Calculus Student, The first page is the answer sheet for the attached problems They are due the first day of school Be sure to keep track of how long it takes you to complete the problem set and which problems you couldn t do Fill in this information on the front page where it is requested MOOC stands for minutes outside of class which in this case is all the time you spend completing the problem set If you need some etra help in August, schedule a time to see me at school the week before school starts You will also find a two-page back-to-back summary of math facts that you will need to know by memory Please review this sheet and make sure that you do spend the time needed to recommit these facts to memory You will have a quiz on the first day of Calculus over the material in this review packet Now, to make sure you have read all of these directions, you are to send me an confirmation message when you have started on the questions in August, bmackenney@tbcsorg This has to be done from an account that I can use to contact you Be sure to sign your with the name you would like me to call you There is etra credit awarded if you include an interesting picture (jpeg) of you that I can use as a picture of the day for your class on the overhead Please make it a picture that you would like to tell about in the class I look forward to teaching you Calculus during the net school year Have a wonderful summer Mr MacKenney Remember: This packet is for your benefit and is intended to help you succeed as you move through more advanced math classes Name:

2 The Bear Creek School Review and Preparation Packet for AP Calculus AB Name: Answers: y 0 y 80 y 80 y y a) b) int( s) : 0,, y int : 0 { : or } or655 7 { : 5 or 9} y even 9 a 6 b yes 5 ( 6, ) 6 check on a graphing calculator in radian mode 7 R(, 9) 8 check on a graphing calculator in radian mode 9 ( 6,8) 0 h h 8 0,cos 7 7 sin neither { : or } (,) (, ) 5 (,) 6 ( ) ( y 6) 5 or ( 5) 7 domain :, range : y 8 verte at (,-) and opens down, check on a graphing calculator cos 5 5,tan discontinuous at = - or = -6 and continuous at all other real number 9 - positive negative non real, comple check your memory , 5 s s s s Check on your graphing calculator 56 y-int: 6 check your graph on your calculator 57 y-int: 6 check your graph on your calculator 58 y-int: 8 check your graph on your calculator y-ais 6 origin 6 -ais 6 origin 65 False 66 True 67 True 68 True 69 True 70 False check on a graphing calculator in radian mode

3 August Practice for students entering AP Calculus AB Name: Do not use a graphing calculator to compute any of your answers unless specifically told on the individual problem Check your answers with a graphing calculator where appropriate Show all your work on each problem clearly and cleanly Use a pencil and an eraser 5 Use the Binomial Theorem to epand ( y ) and then simplify NWW: Number worked / 78 MOOC: Minutes to complete If f ( ) 7 and g ( ), find a) f ( g( )) and b) g ( f ( ))???: Question numbers of the Unanswered questions Find the and y-intercepts of y 9 () Solve: 5 Convert to degrees 6 Solve the equation 5 0sin 8cos, where 0 60 ( 9)( 5) 7 Solve: 0 8 Given f ( ) and g ( ) 5, find ( g f )(5)

4 9 Write the slope-intercept form of the equation of the line passing through the point ( 5, -5) and perpendicular to the line - 5y = -0 0 Is the following function an even function, an odd function, or neither? y 8 Convert 80 to radians a Evaluate using memory angles the eact value of sin( ) b Evaluate using a memory angle the eact value of cos( ) 8 Is the relation { ( -, ), (, ), ( 6, ) } a function? Find the distance between the points P(, -) and Q( -, -5) 5 Find the points of intersection of the graphs of: y 6 y 7 6 Graph y tan( ) over M( -, -) is the midpoint of segment RS If S has coordinates (, 5), find the coordinates of R 8 Graph y cos( ) on the interval 9 Determine the midpoint of the line segment whose endpoints are (, ) and ( -, 5) 0 If g( ) 6, find g( h )

5 Given tan and sec 0, find sin and cos 0 Is the function y 5 5 an even function, an odd function, or neither Solve and graph: 5 Solve the inequality and write the solution set in interval notation Find the points of intersection of the graphs of: y 0 y 0 6 Write the equation in standard form for the circle with center at ( -, 6) and radius 5 7 Determine the domain and range of y 8 Graph: y ( ) 9 Convert o to radians 0 Given sin and sec 0, find cos and tan Evaluate: sec( 9 ) Evaluate: Arctan( ) Find: lim5 sin( ) Use a calculator on

6 Use the graph to estimate the lim f ( ) 5 lim 6 lim( 7 5) 7 lim sin 0 Use a calculator on 7 8 Determine where ( ) ( )( 6) f is continuous 9 Use the graph to find the limit (if it eists) 05, lim, 0 Find the vertical asymptote(s) if any for 6 ( 8)( 6) f ( ) Find: 6 6 lim 0 Find: lim 6 6 Complete this table of eact values from memory Degrees Radians 0 π/6 π/ π/ π/ π/ π/ 5π/6 π 7π/6 5π/ π/ π/ 5π/ 7π/ π/6 π sin cos tan

7 Complete the table using Descartes' Rule of Signs for all the possible combinations of zeros for a 9 th degree polynomial function P ( ) 5 5 Find the sum and the product of the solutions: = 0 s + s = s s = 6 Find the remainder when 5-00 is divided by + 7 Graph: y 8 Graph: y 5 9 Graph: ylog ( ) 50 Graph: y 5 5 Graph: y 5 Graph: y ( )( ) ( )( ) 5 Graph: y 5 Graph: y 55 Graph: y 5 Make a sketch of the following polynomial functions and name the y-intercepts 56 p( ) ( )( )( ) 57 p( ) ( ) ( ) 58 p( ) ( )( ) y-intercept = y-intercept = y-intercept =

8 59 Find a quadratic equation with 6 5and 6 5 as solutions 60 Find a quadratic equation with 8 i and 8 i as solutions 6 6 Check for symmetry with respect to each ais and the origin 6 y 6 y 6 y 0 6 y True or False? 65 True False If (, -) is a point on a graph that is symmetric with respect to the -ais, then (-, -) is also a point on the graph 66 True False If (, -) is a point on a graph that is symmetric with respect to the y-ais, then (-, -) is also a point on the graph 67 True False If b ac > 0 with a, b, c Real numbers and a 0, then the graph of y = a + b + c has two intercepts 68 True False If b ac = 0 with a, b, c Real numbers and a 0, then the graph of y = a + b + c has only one intercept 69 True False The lines represented by a by c and b ay care perpendicular Assume a0 and b 0 70 True False It is possible for two lines with positive slope to be perpendicular to each other 7 7 Find the distance between the point and line, or between the lines using the formula for the distance between the point (, y) and the line A By C 0 Distance= A By C A B 7 Point: (-, ) Line: y 0 7 Line: y Line: y 5 7 Graph: y sin 7 Graph: ytan( ), for - << 75 Graph: y Tan Given sin, find sin 77 Given sin, find cos Given sin, find tan 7

9 You are to be able to recite these following facts from memory: Calculus Preparation To find -intercepts, set y = 0 and solve the equation for To find y-intercepts, set = 0 and solve the equation for y Tests for symmetry a The graph of an equation in and y is symmetric with respect to the y-ais if replacing by - yields an equivalent equation b The graph of an equation in and y is symmetric with respect to the -ais if replacing y by -y yields an equivalent equation c The graph of an equation in and y is symmetric with respect to the origin if replacing by - and y by -y yields an equivalent equation y yy The slope m of a non-vertical line passing through the points (,y ) and (,y ) is m, 5 Summary of Equations of Lines a General form: A + By + C = 0 b Vertical line: = a c Horizontal line: y = b d Point-slope form: y - y = m( - ) e Slope-intercept form: y = m + b 6 Functions can be specified in a variety of ways: a Implicitly: + y = b Eplicitly: (y by itself) y = ( - )/ c Function notation: f() = ( - )/ 7 A function is one-to-one if to each y-value in the range there corresponds eactly one -value in the domain 8 Basic Types of Transformations (c > 0) Original graph: y = f() a Horizontal shift c units to the right: y = f( - c) b Horizontal shift c units to the left: y = f( + c) c Vertical shift c units downward: y = f() - c d Vertical shift c units upward: y = f() + c e Reflection about the -ais: y = - f() f Reflection about the y-ais: y = f(-) g Reflection about the origin: y = - f(-) 9 Terms: polynomial, degree of a polynomial, coefficients, leading coefficient, constant term, rational function, algebraic function, transcendental function 0 Definition of a Composite Function: Let f and g be functions The function given by (fog)() = f(g()) is called the composite of f with g The domain of fog is the set of all in the domain of g such that g() is in the domain of f Even and odd functions a A function is even if its graph is symmetric with respect to the y-ais The function y = f() is even if f(-) = f() b A function is odd if its graph is symmetric with respect to the origin The function y = f() is odd if f(-) = - f() Definition of a Limit: Let f be a function defined on an open interval containing c (ecept possibly at c) and let L be a real number The statement lim f ( ) L means that for each > 0 there eists a > 0 such that if 0 < - c <, then f() - L < Some Basic Limits: Let b and c be real numbers and let n be a positive integer a limb b b lim c n n c lim c Properties of Limits: Let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits lim f ( ) L and lim g( ) K a Scalar multiple: lim[ bf ( )] bl b Sum or difference: lim[ f ( ) g( )] L K c Product: lim[ f ( ) g( )] LK d Quotient: e Power: f ( ) lim L, provided K 0 g( ) K n n lim[ f ( )] L

10 5 Limits of Polynomial and Rational Functions a If p is a polynomial function and c is a real number, then lim p( ) p( c) b If r is a rational function given by r() = p()/q() and c is a real number such that q(c) 0, then 6 The Limit of a Function Involving a Radical: Let n be a positive integer p( c) limr( ) r( c) q( c) n n The following limit is valid for all c if n is odd, and is valid for c> 0 if n is even lim c 7 The Limit of a Composite Function If f and g are functions such that lim g( ) L and lim f ( ) f ( L), then lim f ( g( )) f ( L) L 8 Limits of Trigonometric Functions Let c be a real number in the domain of the given trigonometric function a limsin sinc d limcot cotc 9 Functions that agree at All But One Point b limcos cosc e limsec secc c limtan tanc f limcsc cscc Let c be a real number and let f() = g() for all c in an open interval containing c If the limit of g() as approaches c eists, then the limit of f() also eists and lim f ( ) lim g( ) 0 Intervals on the real line a Bounded open interval: b b Bounded closed interval: b c Unbounded open interval:,a a, : a b} a, : a b} : a} d Unbounded closed interval: [ a, ) { : a} etc The distance between points (,y ) and (,y ) in a plane is given by d ( ) ( y y) y y Midpoint formula for points (,y ) and (,y ) in a plane is given by (, ) Standard Form of the Equation of a Circle The point (,y) lies on the circle of radius r and center (h,k) if and only if ( - h) + (y - k) = r General Form of the Equation of a Circle A + Ay + D + Ey + F = 0, A 0 5 Definitions of the Si Trigonometric Functions a For an acute angle in a right triangle: SohCahToa b Circular functions where is any angle r y sin = y/r cos = /r tan = y/ csc = r/y sec = r/ cot = /y 6 Pythagorean Identities: sin + cos = tan + = sec cot + = csc 7 Reduction Formulas: sin( ) sin cos( ) cos tan( ) tan sin( ) sin cos( ) cos tan( ) tan 8 Sum of Difference of Two Angles: sin( ) sin cos cos sin cos( ) cos cos sin sin tan tan tan Atan B tan( ) A AB B 9 Double Angles: sin = sincos cos = cos sin cos = cos cos = sin 0 Reciprocal Identities: csc = /sin sec = /cos cot = /tan Quotient Identities: tan = sin /cos cot = cos /sin ( ) ( ) Derivative of a function (slope function): ( ) lim f h f f h0 h