Harrison High School AB Calculus AP Prerequisite Packet Optional help session for prerequisite material to be held at Harrison High School in Room 409 from 9:00 am until noon on Friday, August. To: From: AP Calculus AB Students and Parents Michelle Bowman and Ann Blume, AP Calculus Instructors 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. BC Calculus, corresponding to Calculus II at most universities, is taught during the second semester. Students are encouraged to take both semesters of AP Calculus. The Prerequisite Packet: Students need a strong foundation 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 Analysis. Students should anticipate working approimately 0 hours to complete it properly. This packet includes A copy of the first chapter of your tetbook. Please return it in good condition at the beginning of the semester. This chapter should be used to complete the problem set from the book and as a reference if you need help on the additional problem set. An acknowledgement of receipt of the prerequisite packet. A Toolkit of Functions ; you should be familiar with each of the graphs. A formula and identities section. These are for your reference and do not need to be memorized at this time. A unit circle template with which to practice your unit circle. A list of skills that you will need for AP Calculus. If you feel you are weak in any of the areas, let us know and we will be glad to help you. Calculus Prerequisite Problems, please show all work. Answers are provided at the end of the packet. The packet will be collected on the first 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 89 in class; we recommend a TI-89, but a TI 83 or TI 84 can also be used.
Assistance with the packet: Chapter can be used for reference and there is a lot of help on the internet. There will also be an optional help session as indicated at the top of this page. This is a help session to work on problems you are having difficulty with, not a session in which to do your packet. Please complete as much of the packet as possible before attending the session. Students who will be on vacation during the help session and who have questions should contact us prior to this help session to determine whether an alternate time is available the week prior to school beginning. To obtain help please contact us by email at or michelle.bowman@cobbk.org or ann.blume@cobbk.org. You can also check our blogs at the Harrison High School website for additional information. If you lose this packet another copy can be downloaded from the blog. General Information: Students will have the opportunity to continue studying BC topics second semester through the BC Calculus course. The AB course is a prerequisite for BC. Students taking only the AB course and making a 3, 4, or 5 on the AB - AP test could receive 5 hours of credit for Calculus. Students taking both the AB and BC courses and passing the BC AP test could earn up to 0 hours of college credit for Calculus and Calculus. Be sure to check with your school of choice to determine their AP credit policy. Your success in this AP program will depend on the effort you put into the course. The work you do in the AB course is for you. 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. We both anticipate a motivating and challenging year. Calculus is a stimulating and eciting field of mathematics and we look forward to sharing our ecitement with you. We will be there to help and support you. If you are downloading this packet and have not provided us with the following information, please email the information to Michelle.Bowman@cobbk.org. Name: Address: Phone: Email: Emailing this information acknowledges receipt of the packet and an understanding that completion of this prerequisite packet is a requirement for the AB Calculus course. The packet is due on the first day of class.
Toolkit of Functions Students should know the basic shape of these functions and be able to graph their transformations without the assistance of a calculator. Constant Cubic f() = a f() = 3 Identity Square Root f() = f() = Absolute Value Greatest Integer f() = f() = [ ] Reciprocal Eponential f() = f() = a Quadratic Logarithmic f() = f() = ln
Trig Functions f( ) = sin f( ) = cos f( ) = tan Polynomial Functions: n n A function P is called a polynomial if P( ) = an + an +... + a + a + a 0 Where n is a nonnegative integer and the numbers a 0, a, a, a n are constants. Even degree Leading coefficient sign Odd degree Leading coefficient sign Positive Negative Positive Negative Number of roots equals the degree of the polynomial. Number of intercepts is less than or equal to the degree. Number of bends is less than or equal to (degree ). Formulas and Identities
Trig Formulas: d Arc Length of a circle: L = rθ or L = 360 Area of a sector of a circle: Area = r θ or Area = d 360 Solving parts of a triangle: π r π r Law of Sines: a b c = = sin A sin B sin C Law of Cosines: a = b + c bc cos A b = a + c ac cos B c = a + b ab cos C Area of a Triangle: Area = bc sina or Area = ac sinb or Area = ab sinc Hero's formula : Area = ss ( a)( s b)( s c), where s = semi perimeter Ambiguous Case: θ is acute θ is obtuse or right Compute: alt = adj sinθ opp adj No triangle opp < alt No triangle opp > adj triangle opp = alt triangle (right) opp>adj triangle alt<opp<adj triangles Does a triangle eist? Yes - when (difference of sides) < (third side) < (Sum of sides) Formulas and Identities, continued
Trig Identities: Reciprocal Identities: csc A = sin A sec A = cosa cot A = tana Quotient Identities: Pythagorean Identities: sin A tan A = cot A = cos A cosa sin A sin A + cos A = tan A + = sec A + cot A = csc A Sum and Difference Identities: sin(a + B) = sina cosb+cosa sinb sin(a B) = sina cosb - cosa sinb cos(a + B) = cosa cosb sina sinb cos(a B) = cosa cosb + sina sinb tan (A + B) = tan A+ tan B tan Atan B tan (A B) = tan A tan B + tan Atan B Double Angle Identities: sin(a) = sina cosa tan(a) = tana tan A cos(a) = cos A - sin A cos(a) = cos A cos(a) = sin A Half Angle Identities: A cosa sin =± A + cosa cos =± A cosa tan =± + cos A Polar Formulas: + y = r = rcos θ y = rsin θ Geometric Formulas: y tan = θ > 0, tan =θ+π < 0 Area of a trapezoid: A= h( b + b ) Area of a triangle: A= bh 3 Area of an equilateral triangle: A= s 4 Area if a circle: A= π r Circumference of a circle: C = π r or C = dπ y
Unit Circle Degrees and Radians Place degree measures in the circles. Place radian measure in the squares. tanθ = cot θ = Place (cos θ, sin θ ) in parenthesis outside the square. cscθ = Place tan θ outside the parenthesis. secθ =
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 (use grouping) 3) 64 6 - Hint: Factor as difference of squares first, then as the sum and difference of cubes second. 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: ( + ) 3 ( - ) + 3( + ) 9) ( + ) 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.) 3) Eplain why f() = 4 + 7 + - 5 must have a root in the interval [0, ], ( 0 ) Check the graph and use signs of f(0) and f() to justify you re answer. Solve: You may use your graphing calculator to check solutions. 4) ( + 3) > 4 5) + 5-3 0 6) (Factor first) 3 3 4 5 0 7) < 8) - 9 + 0 9) - + 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?
Test for symmetry. Show substitution with variables to justify you re answer. 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 + 63) y = sin() 64) y = cos() 65) = y + 66) y = + IV LOGARITHMIC AND EXPONENTIAL FUNCTIONS A. Simplify Epressions: 3 67) log 4( 6 ) 68) 3log33 - log38+ 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. Right Triangle Trig: Find the value of. (Note: Degree measure!) 90. 50 o 9. 9. 0 9 X 93. H o o 70 70 0 A 80 ft B 60 ft C 93) The roller coaster car shown in the diagram above takes 3.5 sec. to go up the 3 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. Decimals in answer may vary. a. How high is the roller coaster above point B? b. Find the distances AH and HC. c. How fast (in ft/sec) does the car go up the incline? d. What is the approimate average speed of the car as it goes down the drop? e. Assume the car travels along HC. Is your approimate answer too big or too small? ( Advanced Mathematics, Richard G. Brown, Houghton Mifflin,994, pg 336) F. Graphs: Identify the amplitude, period, horizontal, and vertical shifts of these functions. 94) y = - sin() 95) y = π cos π ( + π )
G. 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. 96. 99. Given the following function f() = 4-3 - + 30. 96. Find all roots. Note: Window min: -0 ma: 0 scale y min: - 00 y ma: 60 scale 0 97. Find all local maima. A local maimum or local minimum is a point on 98. Find all local minima. the graph where there is a highest or lowest point within an interval such as the verte of a parabola. 99. Find the following values: f (-), f (), f (0), f (.5) 00. 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 0. 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 f() = -. (d) On what interval is f decreasing? (e) State the domain and range of f. (f) State the domain and range of g. 0. The number N (in thousands) of cellular phone subscribers in Malaysia is shown in the table. (Midyear estimates are given.) t 99 993 995 997 N 3 304 873 46 (a) Use the data to sketch a rough graph of N as a function of t. (b) Use your graph to estimate the number of cell-phone subscribers in Malaysia at midyear in 994 and 996.
03. If f() = 3 -+, find f(), f(-), f(a), f(-a), f(a+), f(a), f(a ), [f(a)], and f(a+h). 04. Find the domain of each function. a) f( ) = 3 b) gu ( ) = u+ 4 u 05. Find an epression for the bottom half of the parabola +(y-) =0. 06. A rectangle has perimeter 0 m. Epress the area of the rectangle as a function of the length of one of its sides. 07. Find an epression for the function whose graph is the given curve. 08. 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. 09. At the surface of the ocean, the water pressure is the same as the air pressure above the water, 5 lb/in. Below the surface, the water pressure increases by 4.34 lb/in for every 0 ft of descent. (a) Epress the water pressure as a function of the depth below the ocean surface. (b) At what depth is the pressure 00 lb/in?
0. 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 +. Match each equation with its graph. Eplain your choices. (Don t use a computer or graphing calculator). 5 8 (a) y = ( b) y = ( c) y =. Suppose the graph of f is given. Write equations for the graphs that are obtained from the graph of f as follows. (a) Shift 3 units upward. (b) Shift 3 units downward. (c) Shift 3 units to the right. (d) Shift 3 units to the left. (e) Reflect about the -ais. (f) Reflect about the y-ais. (g) Stretch vertically by a factor of 3. (h) Shrink vertically by a factor of 3. 3. The graph of y = f() is given. Match each equation with its graph and give reasons for your choices. (a) y = f(-4) (b) y = f()+3 (c) y = /3 f() (d) y = -f(+4) (e) y = f(+6)
4. The graph of f is given. Use it to graph the following functions. (a) y = f() (b)=f( ½ ) (c) y = f(-) (d) y = -f(-) 5. Graph the following, not by plotting points, but by starting with the graph of one of the standard functions and applying the appropriate transformations. π y = sin 3 6 6. Find f+g, f g, fg, and f/g and state their domains. 3 f( ) = +, g( ) = 3 7. Find the functions f og, go f, f o f, and go g and their domains. f ( ) = sin, g( ) = 8. Epress the function in the form f o g. F( ) = ( + ) 0 9. Use a graphing calculator to determine which of the given viewing rectangles 3 produces the most appropriate graph of the function f ( ) = 0 + 5. (a) [-4,4} by [-4,4] (b) [-0,0] by [-0,0] (c) [-0,0] by [-00, 00] (d) [-00, 00] by [-00,00]
0. Use the given graphs of f and g to evaluate each epression, or eplain why it is undefined. (a) f(g()) (b) g(f(0)) (c) ( f og)(0) ( d) ( go f)(6) ( e) ( gog)( ) ( f) ( f o f )(4) and, Determine an appropriate viewing rectangle for the given function and use it to draw each graph.. f ( ) = 5+ 0. f = 4 ( ) 8 4 3. Graph the ellipse 4 + y = by graphing the functions whose graphs are the upper and lower halves of the ellipse. 4. Find all solutions of the equation correct to three decimal places. 3 9 4= 0 5. Graph the given functions on a common screen. How are these graphs related? y =, y = e, y = 5, y = 0 6. Starting with the graph of y = e, write the equation of the graph that results from (a) shifting units downward (b) shifting units to the right (c) reflecting about the -ais (d) reflecting about the y-ais (e) reflecting about the -ais and then about the y-ais
7. Find the eponential function f() = Ca whose graph is given. 8. 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. 9. Determine whether this function is one-to-one. 3 4 5 6 f().5.0 3.6 5.3.8.0 30. Determine whether this function is one-to-one. 3. Here is a verbal description of a function. Determine whether this function is oneto-one. F(t) is the height of a football t seconds after kickoff. For #3-34, find a formula for the inverse of the function. 3. f ( ) = 0 3 33. f ( ) = e 3 34. y = ln(+3)
For #35-36, find the eact value of each epression (non-calculator). 35. (a) log64 ( b ) log6 36 36. (a) log0.5 + log080 ( b) log50 + log5 0 3log5 37. Epress the given quantity as a single logarithm. ln 4 ln 38. 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.
Answers: (Remember you must show all of your work!). 4 /3 y 4/3 z. (3 + )(3 - y) 3. ( - )(4 + + )( + )(4 - +) 4. 7(3 + 4)( - ) 5. / (3-4)(5 + 6) 6. -3 ( - )( - ) 7. + - 8. + 9. - + ( + ) 0. ( )( + )( + 3);, -, -3. ( + )( 7)(3 ); -, 7, 3. not a possible rational root 3. f(0) = neg and f() = positive 4. > - or < -5 5. -5 < 3 6. or 0 5 7. 0 < < or < - 8. [ -3, -) U [3, ) 9. > 6 or < < 3 5 3 0. < <. - 8 < < - 8. (3, 4), (-, -) 3. (, -), ( 3, 0) 7 3 7 4. y = 5. y = 6. y = 7. = 4(y + ) 7 parabola 3 3 3 3 5 5 ( ) ( y+ 4) 8. + = ellipse 9. D: R: y 0 30. D: > 3 R:all reals 3 4 3. D: all reals R: 3. D: > 3/ R: y > 0 33. D: all reals R: y 0 + 3 34. >- and 35. R: - 5 y < - 3 or 0 y 3 36. g ( ) = 37. h - () = e 4 38. y = + 4 > 0, 39. f(g()) = 6 40. ln 5 4. no, not to 4. 43. - 8 44. D: R: 0 y 45. 47. 49. 5. 53. You must show work on these! 54. odd 55. even 56. odd 57. odd 58. neither 59. odd 60. even 6. even, even 6. symm y ais 63. origin 64. y ais 65. ais 66. y ais 67. - 68. 69. 3/ 70. /3 7. 45 7. 73. 0 74. 75. - 76. = 0, -0 77. log 5 log 3-78. a) D: all reals, R: - < <, л π b ) D: all reals, R: - < <, л c) D: ± π n, wheren Z R: all reals, π 79. 80. sin
8. 8. yes 83. π 84. π, π 7, π 4, π 3. π π 85 86. 6 3 6 3 π π π π 87. 88. 89. 90. Arcsin() D: [-, ] Range:, 4 6, π π Arc cos () D: [-, ]. R: [0, π] Arctan () D: all reals R:, 9. 0 sin 50 7.66 9. hint: draw altitude, 4.69 93. a. HB 76.405 ft. b. AH 95.5448 ft., HC 97.48 c. speed up appro. 8.3 ft./sec d. speed down appro 34.696 ft./sec e. Too small, since the distance traveled is longer the distance per foot would be greater. (Note: find average speed for the entire trip!.9 ft/sec) π 94. A:, P: π 95. A = π, Per. 4, v.s. = 0, h. s. = - y = π cos ( ) + 96. -.5, 0,, 5 97. rel ma. (.07, 0.) 98. rel. min ( -.89, - 8. 48), (3.94, - 88) 99. f( -) = - 8 00. 3 points of intersection - one is ( -5.77, 6.66) 00. Do not use trace to find points. Use CALC commands. 0. (a) -,4 (b) - and (c) = -3 and 4 (d) [0,4] (e) d: [-4,4] r:[-,3] (f) d: [-4,3] r:[0.5,4] 0. (b) 540 In 994 and 450 In 996 03. f()=, f(-) = 6, f(a) = a^ -a +, f(-a)=3a^+a+, f(a+)=3a^+5a+4, f(a)=6a^-a+4, f(a)=a^-a+, f(a^)=3a^4 -a^ +, [f(a)]^=9a^4-6a^3+3a^-4a+4, f(a+h)=3a^ +6ah+3h^ -a-h+ 04. (a) (-, /3) U (/3, ) (b) [0,4] 05. f ( ) = (domain: 0) + if 06. A(L) = 0L L^ domain 5<L<0 07. f( ) = 3 + 6 if < 4 08. (a) T=/6 N + 307/6 (b) /6 means that for each increase of 6 cricket chirps per minute corresponds to an increase of degree F. (c) 76 degrees F 09. (a) P=0.434d + 5 (b) appro. 96 feet 0. (a) root fct. (b) algebraic because it Is the root of a polynomial (c) polynomial of degree 9 (d) rational function. (a) matches with h (b) matches with f and (c) matches with g.. (a) y = f()+3 (b) y = f() -3 (c) y = f(-3) (d) y = f(+3) (e) y = -f() (f) y = f(-) (g) y = 3f() (h) y = /3 f() 3. (a) graph 3 (b) graph (c)graph 4 (d) graph 5 (e) graph 4. (a) shrink horizontally by a factor of (b) stretch horizontally by a factor of (c) reflect the graph of f about the y-ais (d) reflect the graph of f about the y-ais, then about the -ais 5.
6. (f+g)() =^3+5^- d: all real numbers (f-g)()= ^3-^+ d: all reals (fg)()=3^5+6^4-^3-s^ d: all reals (f/g)()=(^3 +^)/(3^ -) d: cannot Equal ± 7. ( f og)( ) = sin( ) d:[0, ) ( go f)( ) = sin d:[0, π],[ π,3 π] etc 3 ( f o f)( ) = sin(sin ) d: (, ) ( gog)( ) = d :[0, ) 8. g()=^ + and F()=^0 9. c 0.(a) 4 (b) 3 (c)0 (d) not defined (e) 4 (f) -. [-0,30] by [-50,50]. [-4,4] by [-,4] 3. graph y = +sqr((-4^)/) and -sqr((-4^)/) 4. about 9.05 5. All of these graphs approach 0 as approaches negative infinity, all of them pass Through the point (0,) and all of them are increasing and approach infinity as Approaches infinity. 6. (a) y = e^ - (b) y = e^(-) (c) y = -e^ (d) y = e^(-) (e) y = - e^(-) 7. f() = 3(^) 8. (a) 300 (b) y = 00(^(t/3)) (c) 0,59 (d) 6.9 hrs. 9. No, does not pass horizontal line test. 30. No 3. A football will attain every height up to its maimum height twice: once on the way up, and again on the way down. Hence, not to. 3. f ( ) = + 0 3 d: [0, ) 33. f ( ) = ln 34. f ( ) = e 3 3 3 35 (a) 6 (b) - 36. (a) (b) 37. ln8 38. The salesman travels away from home from 8 am To 9 am and is then stationary until 0:00. The salesman travels farther away from 0 until noon. There is no change in his distance from home until :00, at which time the distance from home decreases until 3:00. Then the distance starts increasing again, reaching the maimum distance away from home at 5:00. There is no chance from 5 until 6, and then the distance decreases rapidly until 7:00 pm, at which time the salesman reaches home.