Massey Hill Classical High School AB Calculus AP Prerequisite Packet To: From: AP Calculus AB Students and Parents Carolyn Davis, AP Calculus Instructor The AP Course: AP Calculus AB is a college level course covering material traditionally taught in the first semester of college calculus. The course is taught in one semester consisting of 90 -minute classes. The Prerequisite Packet: Students need a strong foundation and willingness to be ready for the rigorous work required throughout the term. Completing the prerequisite packet should prepare you for the material to be taught in the course. This packet consists of material studied during Algebra II and Pre- Calculus Honors. Students should anticipate working approimately 0 hours to complete it properly. The packet will be collected on the third day of class and your grade will be determined on neatness, completeness of solutions, and accuracy. In preparation for the AP test students need to begin showing all work with logical steps. Do not list only an answer. Work neatly and in an organized fashion. Calculators: Students enrolled in AP Calculus AB will be using a graphing calculator throughout the course. A graphing calculator is required on the AP test. We will be using a TI 84 Plus and TI 89 in class;, I recommend a TI 84 Plus Silver Edition and/or the TI-89. If you have a TI-83 calculator some of our needed programs can be downloaded onto your calculator. Since the AP test is offered only in May, students taking only the AB course should study for the test independently and attend any after school review sessions or practice test sessions that are offered. It is the student s responsibility to find out when these sessions will be held. I anticipate a motivating and challenging year. Calculus is a stimulating and eciting field of mathematics and I look forward to sharing the ecitement with you. I will be there to help and support you.
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, root, higher order curves, 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.
Calculus Prerequisite Problems Work the following problems on your own paper. Show all necessary work. I. Algebra A. Eponents: ) 3 3 ( 8 yz) ( ) 4 ( yz ) 3 3 3 B. Factor Completely: ) 9 + 3-3y - y 3) 64 6-4) 4 4 + 35-8 5) 5 3 5 4 Hint: Factor GCF / first. 6) - -3 - + -3 Hint: Factor out GCF -3 first. C. Rationalize denominator / numerator: 3 7) 8) + + D. Simplify the rational epression: ( + ) 9) 3 ( - ) + 3( + ) ( + ) 4 E. Solve algebraic equations and inequalities 0.. Use synthetic division to help factor the following, state all factors and roots. 0) p() = 3 + 4 + - 6 ) p() = 6 3-7 - 6 + 7 ) Eplain why 3 cannot be a root of f() = 45 + c 3 - d + 5, where c and d are integers. (hint: You can look at the possible rational roots.) Solve: You may use your graphing calculator to check solutions. 3) ( + 3) > 4 4) + 5-3 0 5) 3 3 4 5 0 (Factor first) 6) < 7) - 9 + 0 8) - + 4-6 > 0 0) < 4 ) + < 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 3) - 4 + 3 = y y = - 5 - + 6-9 = y II. Graphing and Functions: A. Linear graphs: Write the equation of the line described below. 4) Passes through the point (, -) and has slope - 3. 5) Passes through the point (4, - 3) and is perpendicular to 3 + y = 4. 6) Passes through ( -, - ) and is parallel to y = 3 5 -. B. Conic Sections: Write the equation in standard form and identify the conic. 7) = 4y + 8y - 3 8) 4-6 + 3y + 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, radicand of even inde must be 0 range restrictions- reasoning, if all else fails, use graphing calculator 9) y = 3-30) y = log( - 3) 3) y = 4 + + + 3) y = - 3 33) y = - 5 34) domain only: y = 35) Given f() below, graph over the domain [ -3, 3], what is the range? if 0 f( ) = if - < 0 if < - Find the composition /inverses as indicated below. Let f() = + 3 - g() = 4-3 h() = ln w() = - 4
36) g - () 37) h - () 38) w - (), for 4 39) f(g()) 40) h(g(f())) 4) Does y = 3-9 have an inverse function? Eplain your answer. Let f() =, g() = -, and h() = 4, find 4) (f o g)() 43) (f o g o h)() 44) Let s () = 4 - and t() =, find the domain and range of (s o t )(). D. Basic Shapes of Curves: Sketch the graphs. 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. 45) y = 46) y = ln 47) y = 48) y = - 49) y = - 50) y = - 4 5) y = - 5) y = 3 sin ( - ϖ 6 ) 53) 5 if < 0 5 f( ) = if 0, 5 5 0 if = 5 E. Even, Odd, Tests for Symmetry: Identify as odd, even, or neither and justify you re answer. To justify your answer you must show substitution using -! It is not enough to simply check a number. Even: f () = f (-) Odd: f (-) = - f () 54) f() = 3 + 3 55) f() = 4-6 + 3 56) f( ) = 3 57) f() = sin 58) f() = + 59) f() = ( - ) 60) f() = +
6) What type of function (even or odd) results from the product of two even functions? odd functions? IV LOGARITHMIC AND EXPONENTIAL FUNCTIONS A. Simplify Epressions: 3 67) log 4 ( 6 ) 68) 3log3 3 - log3 8+ log 3 ( 7 ) 69) log 9 7 4 3 ( ) 45 70) log 7) log w 7) ln e 73) ln 5 5 w 74) ln e B. Solve equations: 75) log 6 ( + 3) + log 6 ( + 4) = 76) log - log 00 = log 77) 3 + = 5 V TRIGONOMETRY A. Unit Circle: Know the unit circle radian and degree measure. Be prepared for a quiz. 78) State the domain, range and fundamental period for each function? a) y = sin b) y = cos c) y = tan B. Identities: Simplify: 79) (tan )(csc ) - (csc)(tan )(sin) 80) - cos 8) sec - tan 8) Verify : ( - sin )( + tan ) = C. Solve the Equations 83) cos = cos +, 0 ϖ 84) sin() = 3, 0 ϖ
85) cos + sin + = 0, 0 π D. Inverse Trig Functions: Note: Sin - = Arcsin 3 3 86) Arcsin 87) Arcsin 88) Arccos 89) sin Arccos 90. State domain and range for: Arcsin(), Arccos (), Arctan () E.. Graphs: functions. Identify the amplitude, period, horizontal, and vertical shifts of these 9) y = - sin() 9) y = π cos π ( + π ) F. 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 3 decimal places. Sketch graph. 93. 95. Given the following function f() = 4-3 - + 30. 93. Find all roots. Note: Window min: -0 ma: 0 scale y min: - 00 y ma: 60 scale 0 94. Find all local maima. 95. Find all local minima. 96. Graph the following two functions and find their points of intersection using the intersect command on your calculator. y = 3 + 5-7 + and y =. + 0 Window: min : -0 ma: 0 scale y min: - 0 y- ma: 50 scale 0 VI. Functions and Models
97. The graphs of f and g are given. (a) State the values of f(-4) and g(3). (b) For what values of if f()=g()? (c) Estimate the solution of the equation block =-. (d) On what interval is f decreasing? (e) State the domain and range of f. (f) State the domain and range of g. f() each 98. Find the domain of each function. a) f ( ) = 3 b) g( u) = u + 4 u 99. Biologists have noticed that the chirping rate of crickets of a certain species is related to temperature, and the relationship appears to be very nearly linear. A cricket produces 3 chirps per minute at 70 F and 73 chirps per minute at 80 F. (a) Find a linear equation that models the temperature T as a function of the number of chirps per minute N. (b) What is the slope of the graph? What does it represent? (c) If the crickets are chirping at 50 chirps per minute, estimate the temperature. 00. Classify each function as a power function, root function, polynomial (state its degree), rational function, algebraic function, or logarithmic function. 5 9 4 ( a) f ( ) ( b) g( ) ( c) h( ) ( d) r( ) = = = + = + 3 + 0. Find f+g, f g, fg, and f/g and state their domains. f g 3 ( ) = +, ( ) = 3
0. Find the functions f o g, go f, f o f, and go g and their domains. f ( ) = sin, g( ) = 03 and 04, Determine an appropriate viewing rectangle for the given function and use it to draw each graph. 03. f ( ) = 5 + 0 04. f = 4 ( ) 8 4 05. Find all solutions of the equation correct to three decimal places. 3 9 4 = 0 06. Under ideal conditions a certain bacteria population is known to double every three hours. Suppose that there are initially 00 bacteria. (a) What is the size of the population after 5 hours? (b) What is the size of the population after t hours. (c) Estimate the size of the population after 0 hours. (d) Graph the population function and estimate the time for the population to reach 50,000. For #07-09, find a formula for the inverse of the function. 07. f ( ) = 0 3 08. f ( ) = e 3 09. y = ln(+3) For #35-36, find the eact value of each epression (non-calculator). 0. (a) log 64 ( b ) log6 36. (a) log0.5 + log080 ( b) log50 + log5 0 3log5. Epress the given quantity as a single logarithm. ln 4 ln
3. The graph shown gives a salesman s distance from his home as a function of time on a certain day. Describe in words what the graph indicates about his travels on this day.