Unit 0 Prerequisites for Net Year (Calculus) The following Study Guide is the required INDEPENDENT review for you to work through for your final unit. You WILL have a test that covers this material after AP eams have concluded. REQUIREMENTS:. Complete this ENTIRE Study Guide. You should complete all work, on separate paper. While you may work on any part at any time, your final submission should be in order. This is not only your homework for this unit, but also your guide as to what topics to review (from Algebra through Precalculus).. You are to spend class days working on these problems. You may work at home as well, but when in class you should take advantage of being able to ask each other questions as well as myself. You will be allowed to use your mobile devices to look up things that you do not remember.. The completed guide is due the day of the test which will be announced later.
SKILLS NEEDED FOR CALCULUS I. Algebra: *A. Eponents (operations with integer, fractional, and negative eponents) *B. Factoring (GCF, trinomials, difference of squares and cubes, sum of cubes, grouping) C. Rationalizing (numerator and denominator) *D. Simplifying rational epressions *E. Solving algebraic equations and inequalities (linear, quadratic, higher order using synthetic division, rational, radical, and absolute value equations) F. Simultaneous equations II. Graphing and Functions *A. Lines (intercepts, slopes, write equations using point-slope and slope intercept, parallel, perpendicular, distance and midpoint formulas) B. Conic Sections (circle, parabola, ellipse, and hyperbola) *C. Functions (definition, notation, domain, range, inverse, composition) *D. Basic shapes and transformations of the following functions (absolute value, rational, radical, higher order curves, log, natural log (ln), eponential, trigonometric, piece-wise, inverse functions) E. Tests for symmetry: odd, even III. Geometry A. Pythagorean Theorem B. Area Formulas (Circle, polygons, surface area of solids) C. Volume formulas D. Similar Triangles * IV. Logarithmic and Eponential Functions *A. Simplify Epressions (Use laws of logarithms and eponents) *B. Solve eponential and logarithmic equations (include ln as well as log) *C. Sketch graphs *D. Inverses * V. Trigonometry **A. Unit Circle (definition of functions, angles in radians and degrees) B. Use of Pythagorean Identities and formulas to simplify epressions and prove identities *C. Solve equations *D. Inverse Trigonometric functions E. Right triangle trigonometry *F. Graphs VI. Limits A. Concept of a limit B. Find limits as approaches a number and as approaches * A solid working foundation in these areas is very important.
AP CALCULUS PRERQUISITE UNIT Name Period Work the following problems on your own paper. Show all necessary work. Part I. Algebra A. Eponents: ) 8 yz 4 yz B. Factor Completely: ) 9 + - y - y (use grouping) ) 64 6 - Hint: Factor as difference of squares first, then as the sum and difference of cubes second. 4) 4 4 + 5-8 5) 5 5 4 Hint: Factor GCF / first. 6) - - - + - Hint: Factor out GCF - C. Rationalize denominator / numerator: 7) 8) + + D. Simplify the rational epression: 9) ( + ) ( - ) + ( + ) ( + ) 4 E. Solve algebraic equations and inequalities 0) Use synthetic division to help factor the following, state all factors and roots. a) p() = + 4 + - 6 b) p() = 6-7 - 6 + 7 ) Eplain why cannot be a root of 45 + c - d + 5 = 0, where c and d are integers. (Hint: You can look at the possible rational roots.) ) Eplain why 4 + 7 + - 5 = 0 must have a root in the interval [0, ], ( 0 ) (Hint: You can use synthetic division and look at the y values.)
Solve: You may use your graphing calculator to check solutions. ) ( + ) > 4 4) + 5-0 5) 4 5 0 (Factor first) 6) < 7) - 9 + 0 8) - + 4-6 > 0 9) < 4 0) + < 4 F. Solve the system. Solve the system algebraically and then check the solution by graphing each function and using your calculator to find the points of intersection. ) - y + = 0 ) - 4 + = y y = - 5 - + 6-9 = y Part II. Graphing and Functions: A. Linear graphs: Write the equation of the line described below. ) Passes through the point (, -) and has slope -. 4) Passes through the point (4, - ) and is perpendicular to + y = 4. 5) Passes through ( -, - ) and is parallel to y = 5 -. B. Conic Sections: Write the equation in standard form and identify the conic. 6) = 4y + 8y - 7) 4-6 + y + 4y + 5 = 0 C. Functions: Find the domain and range of the following. Note: Domain restrictions - denominator 0, argument of a log or ln > 0, and radicand of even inde must be 0 Range restrictions- reasoning, if all else fails, use graphing calculator 8) y = - 9) y = log( - ) 0) y = 4 + + ) y = - ) y = - 5 ) domain only:
4) Given f() below, graph over the domain [ -, ], what is the range? if 0 f( ) if - < 0 if < - Find the composition /inverses as indicated below. Let f() = + - g() = 4 - h() = ln w() = - 4 5) g - () 6) h - () 7) w - (), for 4 8) f(g()) 9) h(g(f())) 40) Does y = - 9 have an inverse function? Eplain your answer. Let f() =, g() = -, and h() = 4, find 4) (f o g)() 4) (f o g o h)() 4) Let s () = 4 - and t() =, find the domain and range of (s o t )(). D. Basic Shapes of Curves: Sketch the graph of each. You may use your graphing calculator to verify your graph, but you should be able to graph the following by knowledge of the shape of the curve, by plotting a few points, and by your knowledge of transformations. 44) y = 45) y = ln() 46) y = 47) y = - 48) y = - 49) y = - 4 50) y = - 5) y = sin ( - π 6 ) 5) 5 if < 0 5 f( ) if 0, 5 5 0 if = 5
E. Even, Odd, Tests for Symmetry: Identify as odd, even, or neither: Even: f () = f(-) Odd: f (-) = - f() Show substitutions! 5) f() = + 54) f() = 4-6 + 55) f( ) 56) f() = sin 57) f() = + 58) f() = ( - ) 59) f() = + 60) What type of function results from the product of a) two even functions? b) two odd functions? c) an even and odd function? Test for symmetry. Show substitutions. Symmetric to y ais: replace with - and relation remains the same. Symmetric to ais: replace y with - y and relation remains the same. Origin symmetry: replace with -, y with - y and the relation is equivalent. 6) y = 4 + 6) y = sin() 6) y = cos() 64) = y + 65) y = + Part III LOGARITHMIC AND EXPONENTIAL FUNCTIONS A. Simplify Epressions: 66) log 4 6 67) log - log 8 log 7 68) log 9 7 4 45 69) log 70) logw w 7) ln e 7) ln 7) ln e 5 5 B. Solve equations: 74) log 6 ( + ) + log 6 ( + 4) = 75) log - log 00 = log 76) + = 5
Part IV TRIGONOMETRY A. Unit Circle: Know the unit circle radian and degree measure. Complete the chart above as follows: Place degree measures in the circles. Place radian measure in the squares. Place (cos, sin ) in parenthesis outside the square. Place tan outside the parenthesis. Complete the information below using and/ or y. tan = cot = csc = sec =
77) State the domain, range and fundamental period for each function? a) y = sin b) y = cos c) y = tan B. Identities: Simplify: 78) (tan )(csc ) - (csc)(tan )(sin) 79) - cos 80) sec - tan 8) Verify : ( - sin )( + tan ) = C. Solve the Equations 8) cos = cos +, 0 π 8) sin() =, 0 π 84) cos + sin + = 0, 0 D. Inverse Trig Functions: Note: Sin - = Arcsin 85 ) Arcsin 86 ) Arcsin ) Arccos ) sin Arccos 87 88 89) State domain and range for: Arcsin(), Arccos (), Arctan () E. Right Triangle Trig: Find the value of. (Note: Degree measure!) 90) The roller coaster car shown in the diagram above takes.5 sec. to go up the degree incline segment AH and only.8 seconds to go down the drop from H to C. The car covers horizontal distances of 80 feet on the incline and 60 feet on the drop. How high is the roller coaster above point B? Find the distances AH and HC. How fast (in ft/sec) does the car go up the incline? What is the approimate average speed of the car as it goes down the drop? (Assume the car travels along HC. Is your approimate answer too big or too small? ( Advanced Mathematics, Richard G. Brown, Houghton Mifflin,994, pg 6) F. Graphs: Identify the amplitude and period of these functions 9) y = - sin() 9) y cos
Part V. Calculator Skills A. Be able to do the following on your graphing calculator: Be familiar with the CALC commands; value, root, minimum, maimum, intersect. You may need to zoom in on areas of your graph to find the information. Answers should be accurate to decimal places. Sketch graph. 9. 96. Given the following function f() = 4 - - + 0 9) Find all roots/zeros. Note: Window min: -0 ma: 0 scale y min: - 00 y ma: 60 scale 0 94) Find all local maima. A local maimum or local minimum is a point on 95) Find all local minima. the graph where there is a highest or lowest point within an interval such as the verte of a parabola. 96) Find the following values: f (-), f (), f (0), f (.5) 97) Graph the following two functions and find their points of intersection using the intersect command on your calculator. y = + 5-7 + and y = 0. + 0 Window: min: -0 ma: 0 scale y min: - 0 y ma: 50 scale 0 Neatly graph and label the following. Use Graph Paper and label appropriately. 98) y ln 99) y e 00) y 0) y = 0) y = 0) the identity function 04) y 05) y 06) y sin 07) y cos 08) y tan
Answers: (Remember you must show all of your work!). 4 / y 4/ z. ( + )( + y). ( - )(4 + + )( + )(4 - +) 4. 7( + 4)( - ) 5. / ( - 4)(5 + 6) 6. - ( - )( - ) 7. + - 8. 9. - + ( + ) 0a. ( )( + )( + );, -, - 0b. ( + )( 7)( ); -, 7,. not a possible rational root. f(0) = neg and f() = positive. > - or < -5 4. -5 < 5. or 0 5 6. 0 < < or < - 7. [ -, -) U [, ) 8. > 6 or < < 9. < < 0. - 5 8 < < - 8. (, 4), (-, -). (, -), (, 0) 7 7. y 4. y 5. y 6. = 4(y + ) 7 parabola 5 5 y4 7. ellipse 8. D: R: y 0 9. D: > R: all reals 4 0. D: all reals R:. D: > / R: y > 0. D: all reals R: y 0. >- and 4. R: 0 y or - 5 < y < - 5. f ( ) 4 6. h - () = e 7. y = + 4 > 0, 8. f(g()) = 6 9. ln 5 40. no, eplain 4. 4. - 8 4. D: [ -, ], R: [0, ] 45. 47. 49. 5. 5. You must show work on these! 5. odd 54. even 55. odd 56. odd 57. neither 58. odd 59. even 60. even, even 6. symm y ais 6. origin 6. y ais 64. ais 65. y ais 66. - 67. 68. / 69. / 70. 45 7. 7. 0 7. 74. - 75. = 0, -0 76. log 5 log - 77. a) D: all reals, R: - < <, л b) D: all reals, R: - < <, л
c) D: π/ R: all reals, π 78. 79. sin 80. 8. yes 8. π. 5, 7,, 8 84. 85. 6 6 6 6 86. 87. 88. 89. Arcsin() D: [-, ] Range:, 4 6, Arccos () D: [-, ]. R: [0, π] Arctan () D: all reals R:, 90. BH 70., AH 9.5, HC 9.446, speed up 8. ft/sec, speed down.0 ft/ sec. 9. A:, P: π 9. A = π, Per. 4, y = - π cos π ( + π) 9. -.5, 0,, 5 94. rel ma. (.07, 0.) 95. rel. min ( -.89, - 8. 48), (.94, - 88) 96. f( -) = - 8 97. points of intersection - one is ( -5.77, 6.66) 98. Do not use trace to find points. Use CALC commands.