McKinney High School AP Calculus Summer Packet

Transcription

1 McKinne High School AP Calculus Summer Packet (for students entering AP Calculus AB or AP Calculus BC) Name:. This packet is to be handed in to our Calculus teacher the first week of school.. ALL work must be shown in the packet OR on separate paper attached to packet.

2 Comple Fractions When simplifing comple fractions, multipl b a fraction equal to which has a numerator and denominator composed of the common denominator of all the denominators in the comple fraction. Eample: ( ) ( ) + ( ) ( ) 5( )( ) ( ) 5 Simplif each of the following.. 5 a a 5 + a

3 Functions To evaluate a function for a given value, simpl plug the value into the function for. Recall: ( f g)( ) f ( g( )) OR f[ g( )] read f of g of, Means to plug the inside function in this case g() into the outside function in this case f(). Eample: f + g f g g m+ Given ( ) and ( ). Find ( ( )) and ( ). f( g( )) f( ) g( m+ ) ( m+ ) + m ( ) + + ( 8 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 eactl. π. f. π f Let f( ), g( ) + 5, and h( ). Find each.. h[ f ( )] 5. f[ g( )] 6. gh [ ( )]

4 Find f ( + h) f( ) for the given function f. h 7. f ( ) f ( ) 5 Intercepts and Points of Intersection To find the intercepts, let 0 in our equation and solve. To find the intercepts, let 0 in our equation and solve. Eample: Find intercept: (let 0) Find intercept: (let 0) 0 0 (0) 0 ( )( + ) or intercepts: (, 0) and (, 0) intercept: (0, ) Find the and intercepts for each

5 Use substitution or elimination method to solve the sstem of equations. Eample: Elimination Method Substitution Method Add two given equations to obtain Solve one equation for one variable ( + 6 9) 9 0 plug into nd ( )( 5) 0 equation and Plug and 5 into an original equation: The rest is the same as elimination eample ± Points of intersection (, 0), (5, ), and (5, ). Find the point(s) of intersection of the graphs for the given equations Interval Notation 6. Complete the table with the appropriate notation or graph. Inequalit Notation Interval Notation Graph < [, 7)

6 Solve each equation. State our answer in BOTH interval notation and graphicall (number line) < 9. > 5 Domain and Range Find the domain and range of each function. Write our answer in INTERVAL notation. (HINT: Domain and range will be all real numbers ecept values where the function does not eist.) 0. f( ) 5. f( ) +. f ( ) sin. f( ) Inverses To find the inverse of a function, simpl switch the and the and solve for the new value. Eample: f( ) Rewrite f() as Switch and, then solve for ( ) ( ) + Cube both sides + Simplif Subtract to solve for f ( ) Rewrite in inverse notation. Find the inverse for each function.. f ( ) + 5. f() 6

7 Also, recall that to PROVE one function is an inverse of another function, ou need to show that f ( g( )) g( f ( )) Eample: 9 If f( ) and g( ) + 9, show f( ) and g( ) are inverses of each other. 9 (+ 9) 9 f( g( )) + 9 g( f( )) f( g( )) g( f( )) therefore the are inverses of each other. Prove f and g are inverses of each other ( ) g( ) f f ( ) 9, 0 g( ) 9 7

8 Equation of a line Slope intercept form: m+ b Vertical line: c (slope is undefined) Point slope form: m( ) Horizontal line: c (slope is zero) 8. Use slope intercept form to find the equation of the line having a slope of and intercept of Determine the equation of a line passing through the point (5, ) 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, 5) with a slope of /.. Find the equation of the line passing through the point (, 8) and parallel to the line Find the equation of the line perpendicular to the ais passing through the point (, 7).. Find the equation of the line passing through the points (, 6) and (, ). 5. Find the equation of the line with intercept (, 0) and intercept (0, ). 8

9 Logarithms log a is equivalent to a Product propert: logbmn logbm + logb n m Quotient propert: lo gb logbm logbn n Power propert: log b p m plog m log Propert of equalit: If b b, then logb n Change of base formula: log a n logb a Solve each equation. Check our solutions. 6. () b m log n m n + 5 log( ) ln + ln( ) ln 8 5. e 9 Radian and Degree Measure Use 80 to convert to degrees. π radians Use π radians to convert to radians. 80 Convert to degrees: 5. 5π 6 5. π 5 5. π radians Convert to radians:

10 Angles in Standard Position 60. Sketch the angle in standard position. (counter-clockwise from the positive -ais) a. π 6 b. 0 c. 5π d..8 radians Reference Triangles Sketch the angle in standard position. Draw the reference triangle and lable the sides, if possible. (reference triangle is alwas drawn up or down from angle to the -ais) 6. a. π 6 b π a. b. 0 0

11 6. (, 0) (0, ) (0, ) (, 0) Unit Circle You can determine the sine or cosine of a quadrantal angle b using the unit circle. The -coordinate of the circle is the cosine and the -coordinate is the sine of the angle. Eample: π sin 90 cos 0 a. sin80 b. co s 70 c. sin( 90 ) d. sinπ e. co s60 f. cos( π ) Graphing Trig Functions 5 5 f() sin 5 5 f() cos sin and cos have a period of π and an amplitude of. Use the parent graphs above to help ou sketch a graph of the functions below. For f()asin(b + C), A amplitude, π period, B C B phase shift (positive C/B shift left, negative C/B shift right) and K vertical shift. Graph two complete periods of each function and label aes. 6. f ( ) 5sin 65. f ( ) sin π 66. f( ) cos 67. f ( ) cos

12 Trigonometric Equations: Solve each of the equations. Isolate the variable, sketch a reference triangle, find all eact solutions within the given domain, 0 < π, then check with calculator. Remember to double the domain when solving for a double angle. Use trig identities, if needed, to rewrite the trig functions. (See formula sheet at end of packet) 68. sin 69. cos 70. cos 7. sin 7. sin 7. cos cos 0 7. cos sin + cos cos 0

13 Inverse Trigonometric Functions Recall: Inverse Trig Functions can be written in one of two was: arcsin( ) OR sin ( ) Inverse trig functions are defined onl in the quadrants as indicated below due to their restricted domains: Eample: Epress the value of in radians. arctan Draw a reference triangle to solve. cos ( ) < 0 cos ( ) > 0 sin ( ) > 0 ta n ( ) > 0 sin ( ) < 0 tan ( ) < 0 π This means the reference angle is 0 or 6. π So, 6 so that it falls in the interval from π π π < < Answer: 6 For each of the following, epress the value for in radians. 76. arcsin 77. arccos( ) 78. arctan( ) Eample: Find the value without a calculator. 5 cos arctan 6 Draw the reference triangle in the correct quadrant first. Find the missing side using Pthagorean Theorem. Find the ratio of the cosine of the reference triangle. 6 cosθ 6 For each of the following give the value without a calculator. 79. tan arccos 80. sec sin 8. sin arctan 8. 7 sin sin 5 8

14 Circles and Ellipses For a circle centered at the origin, the equation is + r, where r is the radius of the circle. For a circle centered at (h, k) with radius, r, the equation is ( h) + ( k) r. For an ellipse centered at the origin, the equation is + a b, where a is the distance from center to the ellipse along the ais and b is the distance from center to the ellipse along the ais. For an ellipse ( h) ( k) centered at (h, k), the equation is + a b. If the larger number is under the term, the ellipse is elongated along the ais and called a vertical ellipse. Graph the circles and ellipses below:

15 Limits Finding limits numericall. Complete the table and use the result to estimate the limit lim f() lim f() Finding limits graphicall. Find each limit graphicall. Use our calculator to assist in graphing. 89. limcos lim lim f( ) +, f( ), Evaluating Limits Analticall Solve b direct subsitution whenever possible. If needed, rearrange the epression so that ou can do direct substitution lim( + ) 9. lim + 9. lim limcos π 5

16 96. lim HINT: Factor and simplif. 97. lim lim 0 + HINT: Rationalize the numerator. 99. lim ( + h) lim h 0 h One Sided Limits Find the limit if it eists. First, tr to solve for the overall limit (double sided limit). If an overall limit eists, then the one sided limit will be the same as the overall limit. If not, use the graph and/or a table of values to evaluate one sided limits lim lim 9 0. lim lim

17 Vertical Asmptotes Determine the vertical asmptotes for the function. Set the denominator equal to zero to find the value for which the function is undefined. That will be the vertical asmptote. 05. f() 06. f( ) f( ) ( ) Horizontal Asmptotes Determine the horizontal asmptotes using the three cases below. Case I. Degree of the denominator is greater than the degree of the numerator. Case II. Degree of the denominator is the same as the degree of the numerator. Case III. Degree of the numerator is greater than the degree of the denominator. There is no horizontal asmptote. The function increased without bound. (If the degree of the numerator is eactl more than the degree of the denominator, then there is a linear slant asmptote, which is determined b long or snthetic division.) Determine all Horizontal Asmptotes f( ) f( ) f( ) 7 Limits to Infinit A rational functin does not have a limit if it goes to ±, however, ou can state the direction the limit is headed if both the left and right hand side go in the same direction. Determine each limit if it eists. If the limit approaches or, state which one the limit approaches.. lim + +. lim + +. lim 0 sin 7

18 Formula Sheet Reciprocal Identities: Quotient Identities: Pthagorean Identities: csc sec cot sin cos tan sin cos tan cot cos sin sin + cos tan + sec + cot csc Double Angle Identities: tan sin sin cos cos cos sin tan tan sin cos Logarithms: log a is equivalent to a Product Propert: lo gb mn logb m + logb n m Quotient Propert: logb logbm log n n b p Power Propert: logbm plogbm Propert of Equalit: If logbm log bn, then m n Slope Intercept Form: m+ b Point Slope Form: m( ) Standard Form: A + B + C 0 8