Summer Review Packet for Students Entering AP Calculus BC. Complex Fractions

Transcription

1 Summer Review Packet for Students Entering AP Calculus BC Comple Fractions When simplifying comple fractions, multiply by a fraction equal to 1 which has a numerator and denominator composed of the common denominator of all the denominators in the comple fraction. Eample: ( 4) ( 4) ( ) ( 4) 5( )( 4) 1( ) Simplify each of the following a a 5 a

2 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 to 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()). ( ) 1 ( ) 4 f ( g( )) f ( 4) ( 4) 1 ( 8 16) f g ( ( )) 16 Let f ( ) 1 and g( ) 1. Find each. 6. f () 7. g( ) 8. f( t1) 9. f g( ) 10. ( ) g f m 11. f ( h) f ( ) h Let f ( ) sin Find each eactly. 1. f 1. f Let f ( ), g( ) 5, and h( ) 1. Find each. 14. h f( ) 15. f g( 1) 16. g h ( )

3 Find f ( h) f ( ) h for the given function f. 17. f ( ) f ( ) f ( ) 1 0. 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 ( )( 1) 1or i ntercepts ( 1, 0) and (, 0) y int. ( Let 0) y 0 (0) y y intercept (0, ) Find the and y intercepts for each. 1. y 5. y 4

4 . y 4 Use substitution or eination method to solve the system of equations. Eample: y y 9 0 Eination Method ( )( 5) 0 and 5 Plug = and 5 into one original y y 9 0 y 0 16 y y 0 y 4 Points of Intersection (5, 4), (5, 4) and (, 0) Substitution Method Solve one equation for one variable. y 16 9 (1st equation solved for y) ( 169) 9 0 Plug what y is equal to into second equation ( The rest is the same as previous eample ) ( )( 5) 0 or 5 Find the point(s) of intersection of the graphs for the given equations. 4. y 8 4 y 7 5. y 6 y 4 5

5 Interval Notation 6. Complete the table with the appropriate notation or graph. Solution Interval Notation Graph 4 1,7 8 Solve each equation. State your answer in BOTH interval notation and graphically Domain and Range Find the domain and range of each function. Write your answer in INTERVAL notation. 0. f ( ) 5 1. f ( ). f ( ) sin. f( ) 1 6

6 Inverses To find the inverse of a function, simply switch the and the y and solve for the new y value. Eample: f ( ) 1 Rewrite f() as y y = 1 Switch and y = y 1 Solve for your new y y 1 Cube both sides y1 Simplify y 1 Solve for y 1 f ( ) 1 Rewrite in inverse notation Find the inverse for each function. 4. f ( ) 1 5. f( ) 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( ) 4 9 show f() and g() are inverses of each other g( f ( )) 4 9 f ( g( )) f ( g( )) g( f ( )) therefore they are inverses of each other. 7

7 Prove f and g are inverses of each other. 6. f ( ) g( ) 7. f ( ) 9, 0 g( ) 9 Equation of a line Slope intercept form: y m b Vertical line: = c (slope is undefined) Point-slope form: y y1 m( 1) 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 Determine the equation of a line passing through the point (5, -) with an undefined slope. 40. Determine the equation of a line passing through the point (-4, ) with a slope of Use point-slope form to find the equation of the line passing through the point (0, 5) with a slope of /. 8

8 4. Find the equation of a line passing through the point (, 8) and parallel to the line 5 y Find the equation of a line perpendicular to the y- ais passing through the point (4, 7). 44. Find the equation of a line passing through the points (-, 6) and (1, ). 45. Find the equation of a line with an -intercept (, 0) and a y-intercept (0, ). Radian and Degree Measure 180 Use to get rid of radians and radians convert to degrees. radians Use to get rid of degrees and 180 convert to radians. 46. Convert to degrees: a. 5 6 b. 4 5 c..6 radians 47. Convert to radians: a. 45 b. 17 c. 7 9

9 Unit Circle You can determine the sine or cosine of a quadrantal angle by using the unit circle. The -coordinate of the circle is the cosine and the y-coordinate is the sine of the angle. Eample: sin 90 1 cos 0 (0,1) (-1,0) (1,0) (0,-1) a.) sin180 b.) cos 70 c.) sin( 90 ) d.) sin (-1,0) (0,1) (1,0) e.) cos60 f.) cos( ) (0,-1) Graphing Trig Functions - f = sin f = cos y = sin and y = cos have a period of and an amplitude of 1. Use the parent graphs above to help you sketch a graph of the functions below. For f ( ) Asin( B C) K, A = amplitude, = 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. 49. f ( ) 5sin 10

10 50. f ( ) sin 51. f ( ) cos 4 5. f ( ) cos Trigonometric Equations: Solve each of the equations for 0. Isolate the variable, sketch a reference triangle, find all the solutions within the given domain, 0. 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 the end of the packet.) 5. 1 sin 54. cos cos 56. 4cos 0 11

11 Inverse Trigonometric Functions: Recall: Inverse Trig Functions can be written in one of ways: arcsin sin 1 Inverse trig functions are defined only in the quadrants as indicated below due to their restricted domains. cos -1 sin -1 cos -1 tan -1 sin -1 tan -1 Eample: Epress the value of y in radians. 1 y arctan Draw a reference triangle. -1 This means the reference angle is 0 or. So, y = so that it falls in the interval from 6 6 y Answer: y = 6 For each of the following, epress the value for y in radians. 57. y arcsin 58. arccos 1 y 59. y arctan( 1) 1

12 Circles and Ellipses r ( h) ( y k) ( h) ( y k) a b 1 Minor Ais b a CENTER (h, k) FOCUS (h - c, k) c FOCUS (h + c, k) Major Ais For a circle centered at the origin, the equation is y r, where r is the radius of the circle. y For an ellipse centered at the origin, the equation is 1, where a is the distance from the center to the a b ellipse along the -ais and b is the distance from the center to the ellipse along the y-ais. If the larger number is under the y term, the ellipse is elongated along the y-ais. For our purposes in Calculus, you will not need to locate the foci. Graph the circles and ellipses below: 60. y y y y

13 Limits Finding its numerically. Complete the table and use the result to estimate the it. If the it is increasing without bound, write. If the it is decreasing without bound, write f() f() Finding its graphically. Find each it graphically. Use your calculator to assist in graphing. If the it is increasing without bound, write. If the it is decreasing without bound, write. 66. cos f( ), 1 f( ), 1 Evaluating Limits Analytically Solve by direct substitution whenever possible. If needed, rearrange the epression so that you can do direct substitution. (NO CALCULATORS) 69. (4 )

14 cos HINT: Rationalize the numerator ( h) h0 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, investigate the one-sided it using a test value close to (within.5 unit) of the value is approaching. 15

15 Evaluate the it. No Calculator 4 4 (4.5) (4) 1 - Check the it by plugging in = 4: Since we get not 0 when plugging in 4, will be or. To determine the sign of the infinity, plug a number in that is to the right of 4 (greater than 4 - but not greater than 4.5). The sign of this answer will be the sign of the infinity. Since we got a positive answer when we plugged in a number to the right of 4 (greater than 4), the it is 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 f( ) 8. f( ) f( ) (1 ) 16

16 Horizontal Asymptotes To find the horizontal asymptotes of a function, investigate f( ) classified as one of three cases. and f( ). Limits at infinity can be Case I. Degree of the numerator is less than the degree of the denominator (the function is bottom heavy). f( ) 0 and f( ) 0. The horizontal asymptote is y = 0. Case II. Degree of the numerator is the same as the degree of the denominator. The its and the horizontal asymptote are all equal to the ratio of the lead coefficients. Case III. Degree of the numerator is greater than the degree of the denominator (top heavy). The it is either (increases without bound) or (decreases without bound). This can be determined by the eponent of the quotient of the degrees in the numerator and denominator. If the eponent of the quotient of the degrees in the numerator and denominator is odd, then f( ) and f( ). If the eponent of the quotient of the degree is even, then f( ) and f( ). There is no horizontal asymptote. (If the degree of the numerator is eactly 1 more than the degree of the denominator, then there eists a slant asymptote, which is determined by long division or synthetic division.) Determine all Horizontal Asymptotes. Check your answer with the graphing calculator. 85. f( ) f( ) f( ) Determine each it as goes to infinity 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. 17

17 Determine each it if it eists. If the it approaches or, please state which one the it approaches = = 9. = 0 sin Continuity A continuous function can be drawn without lifting your pencil. In order for a function, f ( ), to be continuous at c, these three conditions must be met: 1.) f (c) eists..) f( ) eists. c.) f ( ) f ( c) c Continuity of Function Families: Polynomials continuous everywhere on,. Rational Functions continuous everywhere ecept for values of that make the denominator zero. Piecewise Functions check the continuity at the endpoints of each branch. Trig Functions sine and cosine are continuous everywhere; tangent and secant have discontinuities at all odd multiples of ; cosecant and cotangent have discontinuities at all multiples of. Removable Discontinuities If a function, f ( ), has a discontinuity at at c. c but f( ) c eists, then f ( ) has a removable discontinuity Determine the values of, if any, for which the function is not continuous. Determine if the discontinuity is removable. 94., f( ) 0, f ( ) 96. f( )

18 , f( ), 99. f( ) , f( ) 5, 19

19 Formula Sheet (should be memorized) Reciprocal Identities: 1 csc sin 1 sec cos 1 cot tan Quotient Identities: sin tan cos cos cot sin Pythagorean Identities: sin cos 1 tan 1 sec 1cot csc Double Angle Identities: sin sin cos cos cos sin 1 sin cos 1 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 Change of base formula: Derivative of a Function: log Slope-intercept form: y m b Point-slope form: y y1 m( 1) b a b log n log Standard form: A + By + C = 0 b b b n a b n n, then m = n Slope of a tangent line to a curve or the derivative: 0 f ( h) f ( ) h0 h You should also be able to do the following without hesitation: SOLVE any equation (i.e. linear, quadratic, eponential, logarithmic, trigonometric) FACTOR by all methods (i.e. difference of two squares, sum/difference of cubes, trinomials, perfect square trinomials) DIVIDE polynomials using synthetic or long division USE graphing calculator to graph functions, find intersection points, zeros, and values

20 Graphs You Should Know 1