Troy High School AP Calculus Summer Packet As instructors of AP Calculus, we have etremely high epectations of students taking our courses. We epect a certain level of independence to be demonstrated by the students taking AP Calculus. This summer packet is your first opportunity to demonstrate your capabilities and resourcefulness to us through completion of this packet. It is designed for you to retain, relearn, and review topics of Algebra, Trigonometry, and Pre-Calculus. Without mastering these topics, you will find that you will consistently get problems incorrect even though you may understand the calculus topics. Work needs to be shown, when possible, in a neat and organized manner. Since you can only use a calculator on half of the AP eam, you should not always rely on your calculator. You must be familiar with certain built-in calculator functions such as finding maimum and minimum values, intersection points, and zeros of a function. A list of topical understandings represented in this summer packet is listed below. Please be familiar with all of them and ready to apply them to a higher level. Comple Fractions Simplifying Epressions Functions Intercepts and Points of Intersection Interval Notation Domain and Range Inverses Equations of Lines Radian and Degree Measure Reference Triangles Unit Circle Graphing Trigonometric Functions Graphing Calculator Usage Limits Asymptotes Finally, it is not suggested to wait until the week before school starts to begin the packet. If you spread it out, you will more likely retain the information better. A test over this material will be given within the first two weeks of school. Best of Luck! Troy High AP Calculus Teachers
Summer Review Packet for Students Entering Calculus (all levels) Comple Fractions When simplifying comple fractions, multiply by a fraction equal to which has a numerator and denominator composed of the common denominator of all the denominators in the comple fraction. Eample: 6 6 7 7 + + 7( + ) 6 7 + i + + + + + ( ) ( ) + ( ) + 8+ + 8 i ( ) ( )( ) ( ) 0 Simplify each of the following.. a a + a. +. 0 + + +. + + +. +
Simplifying Epressions Make sure you are very comfortable manipulating eponents, positive/negative, fractional. Also, know the relationship between eponents and radicals; the appropriate radical will undo an eponent. Eamples: () 8 8 0 0 ( ) ( ) 0 ( ) ( ) 9 6 6 9 6 ( ) 6 Simplify each epression. Write answers with positive eponents where applicable. A. F. + h a 0 B. G. y 8y C. H. 8 D. E. ( a )( a )
Functions To evaluate a function for a given value, simply plug the value into the function for. Recall: ( f g)( ) f ( g( )) OR f [ g( )] read f of g of means: plug the inside function (in this case g()) in for in the outside function (in this case, f()). Eample: Given f ( ) + and g( ) find f(g()). f ( g( )) f ( ) + ( ) + + ( 8 6) + + 6 f g + ( ( )) 6 Let f ( ) + and g( ). Find each. 6. f () 7. g( ) 8. f ( t+ ) 9. f [ g( ) ] 0. [ ( ) ] g f m+. f ( + h) f ( ) h Let f ( ) sin Find each eactly.. π f. π f Let f ( ), g( ) +, and h( ). Find each.. h[ f ( ) ]. [ ] f g( ) 6. g h ( )
Find f ( + h) f ( ) h for the given function f. 7. f ( ) 9+ 8. f ( ) Intercepts and Points of Intersection To find the -intercepts, let y 0 in your equation and solve. To find the y-intercepts, let 0 in your equation and solve. Eample: y int. ( Let y 0) 0 0 ( )( + ) or i ntercepts (, 0) and (, 0) y int. ( Let 0) y 0 (0) y y intercept (0, ) Find the and y intercepts for each. 9. y 0. y +. 6. y y
Use substitution or eination method to solve the system of equations. Eample: + y 6+ 9 0 y 9 0 Eination Method 6 0 0 + 8+ 0 ( )( ) 0 and Plug and into one original y 9 0 y 9 0 y 0 6 y y 0 y± Points of Intersection (, ), (, ) and (, 0) Substitution Method Solve one equation for one variable. + 6 9 (st equation solved for y) y ( + 6 9) 9 0 Plug what y is equal to into second equation. 6 + 0 0 ( The rest is the same as previous eample 8 + 0 ) ( )( ) 0 or Find the point(s) of intersection of the graphs for the given equations.. + y 8 y 7. + y 6 + y. y y 6 Interval Notation 6. Complete the table with the appropriate notation or graph. Solution Interval Notation Graph <,7) 8 6
Solve each equation. State your answer in BOTH interval notation and graphically. 7. 0 8. < 9. > Domain and Range Find the domain and range of each function. Write your answer in INTERVAL notation. 0. f. f ( ) +. f ( ) sin. ( ) f ( ) Inverses To find the inverse of a function, simply switch the and the y and solve for the new y value. Eample: f ( ) + Rewrite f() as y y + Switch and y y+ Solve for your new y ( ) ( y+ ) y Cube both sides y+ Simplify Solve for y f ( ) Rewrite in inverse notation Find the inverse for each function.. f ( ) +. f ( ) 7
Also, recall that to PROVE one function is an inverse of another function, you need to show that: f ( g( )) g( f ( )) Eample: If: 9 f ( ) and g( ) + 9 show f() and g() are inverses of each other. ( ) 9 + 9 9 f ( g( )) + 9 g( f ( )) + 9 9 9+ 9 f ( g( )) g( f ( )) therefore they are inverses of each other. Prove f and g are inverses of each other. 6. f ( ) g( ) 7. f ( ) 9, 0 g( ) 9 8
Equation of a line Slope intercept form: y m+ b Vertical line: c (slope is undefined) Point-slope form: y y m( ) Horizontal line: y c (slope is 0) 8. Use slope-intercept form to find the equation of the line having a slope of and a y-intercept of. 9. Determine the equation of a line passing through the point (, -) with an undefined slope. 0. Determine the equation of a line passing through the point (-, ) with a slope of 0.. Use point-slope form to find the equation of the line passing through the point (0, ) with a slope of /.. Find the equation of a line passing through the point (, 8) and parallel to the line y. 6. Find the equation of a line perpendicular to the y- ais passing through the point (, 7).. Find the equation of a line passing through the points (-, 6) and (, ).. Find the equation of a line with an -intercept (, 0) and a y-intercept (0, ). 9
Radian and Degree Measure To convert radian to degrees, make use of the fact that radian equals one half of a circle, or 80. SO, divide radians by, and you ll get the # of half circle, Multiply by 80 will give you the answer in degrees. 80 To convert degrees to radians, first find the # of half circles in the answer by dividing by 80. Since each half circle equals radians, multiply the # of half circles by get the answer in radian. 80 6. Convert to degrees: a. π 6 b. π c..6 radians 7. Convert to radians: a. b. 7 c. 7 Angles in Standard Position 8. Sketch the angle in standard position. Angle is in standard position if its verte is located at the origin and one ray is on the positive -ais (initial side). Other ray is the terminal side. Angle is measured by the amount of rotation from the initial side to the terminal side. a. π 6 b. 0 c. π d..8 radians 0
Reference Triangles (If you don t remember how to do this, google reference triangles ) 9. Sketch the angle in standard position. Draw the reference triangle and label the sides, if possible. a. π b. c. π d. 0 Unit Circle You can determine the sine or cosine for basic angles by using the unit circle. The -coordinate of the circle is the cosine and the y-coordinate is the sine of the angle. See Unit Circle worksheet for more practice. Eample: π π sin cos 0. a.) sinπ π b.) cos π c.) sin( ) d.) sin 6 π e.) cosπ f.) cos( π )
Graphing Trig Functions f( ) sin( ) f( ) cos( ) - - - - y sin and y cos have a period of π and an amplitude of. Use the parent graphs above to help you sketch a graph of the functions below. For f ( ) Asin( B+ C) + K, A amplitude, π period, B C phase shift (positive C/B shift left, negative C/B shift right) and K vertical shift. B Graph two complete periods of the function.. f ( ) sin. f ( ) sin π. f ( ) cos. f ( ) cos
Using the Graphing Calculator +pend some time getting comfortable with your graphing calculator, if you are not already. For calculus, you should be comfortable () graphing an arbitrary function, () finding zeros (or roots) of a function, () and finding the intersection between functions.. Given: f() - + -7- Find all roots to the nearest 0.00 [You can solve for f()0 by graphing f(), then going to Calc menu and # zero function] 6. Given: f() sin + from [-л, л] Find all roots to the nearest 0.00 [ NOTE: all trig functions are done in radian mode] 7. Given f() 0.7 +. +. Find all roots to the nearest 0.00 8. Given: f() - 8 + Find all roots to the nearest 0.00 9. Given f() - + and g() Find the coordinates of any points of intersection
Limits Finding its numerically. Complete the table and use the result to estimate the it. 60..9.99.999.00.0. f() 6. + -. -.0 -.00 -.999 -.99 -.9 f() Finding its graphically. Find each it graphically. Use your calculator to assist in graphing. 6. cos 0 6. 6. f ( ) +, f ( ), Evaluating Limits Analytically Solve by direct substitution whenever possible. If needed, rearrange the epression so that you can do direct substitution. 6. ( ) + 66. + + + 67. 0 + 68. cos π
69. HINT: Factor and simplify. 70. + 6 + 7. 0 + HINT: Multiply numerator & denominator by the conjugate. 7. 9 7. ( + h) h 0 h One-Sided Limits Find the it if it eists. First, try to solve for the overall it. If an overall it eists, then the one-sided it will be the same as the overall it. If not, use the graph and/or a table of values to evaluate one-sided its. 7. + 7. 9 76. + 0 0 0 77. +
Vertical Asymptotes Determine the vertical asymptotes for the function. Set the denominator equal to zero to find the -value for which the function is undefined. That will be the vertical asymptote. 78. f ( ) 79. f ( ) + 80. f ( ) ( ) Horizontal Asymptotes Determine the horizontal asymptotes using the three cases below. Case I. Degree of the numerator is less than the degree of the denominator. The asymptote is y 0. Case II. Degree of the numerator is the same as the degree of the denominator. The asymptote is the ratio of the lead coefficients. Case III. Degree of the numerator is greater than the degree of the denominator. There is no horizontal asymptote. The function increases without bound. (If the degree of the numerator is eactly more than the degree of the denominator, then there eists a slant asymptote, which is determined by long division.) Determine all Horizontal Asymptotes. 8. f ( ) + + 7 8. f ( ) + 8 + 8. f ( ) 7 Determine each it as goes to infinity. RECALL: This is the same process you used to find Horizontal Asymptotes for a rational function. ** In a nutshell. Find the highest power of.. How many of that type of do you have in the numerator?. How many of that type of do you have in the denominator?. If highest power in numerator highest power in denominator, that ratio is your it! 8. + + 8. + 86. 7+6 ++ 6
Limits to Infinity A rational function does not have a it if it goes to +, however, you can state the direction the it is headed if both the left and right hand side go in the same direction. Determine each it if it eists. If the it approaches approaches. or, please state which one the it 87. + + 88. + + 89. 0 sin 7
Circle with Radius otherwise known as unit circle (,y) (cosθ, sinθ) You can determine the sine or cosine of an angle by using the unit circle. The coordinate of the circle is cosine, and the y-coordinate is the sine. 8
Unit Circle Worksheet ( 0, ) DIRECTIONS: Please fill in the (, y) coordinates above for the Unit Circle. Then proceed by filling in the values of the appropriate sin, cos and tan functions below. sin (0) sin (π/6) cos(0) cos (π/6) sin(π/) tan (π/) cos(π) sin (π/) sin (π/) cos (π/6) tan (0) sin (π) 9
Formula Sheet Reciprocal Identities: csc sin sec cos cot tan Quotient Identities: sin tan cos cos cot sin Pythagorean Identities: + sin cos tan + sec + cot csc Double Angle Identities: sin sin cos tan tan tan cos cos sin sin cos Logarithms: y log a is equivalent to a y Product property: log mn log m+ log n b b b m Quotient property: log log m log n b b b p Power property: log m p log m Property of equality: If log m log n, then m n b b b b n Change of base formula: Derivative of a Function: log Slope-intercept form: y m+ b a log n log b b n a Slope of a tangent line to a curve or the derivative: f ( + h) f ( ) h h Point-slope form: y y m( ) Standard form: A + By + C 0 0