I) Simplifying fractions: x x. 1) 1 1 y x. 1 1 x 1. 4 x. 13x. x y xy. x 2. Factoring: 10) 13) 12) III) Solving: x 9 Prime (using only) 11)
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1 AP Calculus Summer Packet Answer Key Reminders:. This is not an assignment.. This will not be collected.. You WILL be assessed on these skills at various times throughout the course.. You are epected to be able to do all the questions in this packet without a calculator ecept for those in Section XXI. I) Simplifying fractions: ) y y y ) ) ) y y 5) h h h II) Factoring: 6) 8) 6 6 7) 9) ) ) 9 Prime (using only) ) 7 9 ) ) 5) III) Solving: 7 5
2 6) 5 0 8, 7) 9 5 8) 5 0, 9) 0 0, 0 0,5,6 ) 600 0) ) y z z y z ) 6 ) 9 5) 9 ( = is etraneous.) 6) ) 8) 8 7 9), 6 0 0) 0 ) 5 0 0, 5 IV) Are the following epressions equal to ln?
3 ) ln yes ) ln 8 ln no ) ln8 ln yes 5) ln ln yes 6) ln ln no 6) ln no V) Equations of lines: 7) Find the equation of the line that has a slope of 5 and passes through the y 5 or y 5 9 point (, -). 8) Find the equation of the line that passes through the points (, ) and (, -). y or y or y 9) Find the equation of the line that passes through the points (-, ) and is parallel to the line y y or y. 0) Find the equation of the line that has a slope of 0 and passes through the point (-5, ). y ) Find the equation of the line that passes through the origin and is perpendicular to the line y 7. y ) Find the equation of the line that has an undefined slope and passes through the point (, -5).
4 VI) Intercepts: ) Find the and y intercepts of y 9. -intercepts: (, 0) and (-, 0). y-intercepts: (0, ) and (0, -) ) Find the equation of the line that has an -intercept of 5 and a y- intercept of. y 0 or y 0 5 or y VII) Equations of graphs: 5) y 6) y 7) 8) y 9) y 6
5 VIII) Given the slope, sketch the following lines: [If you do not trust your sketches, see your instructor.] 50) Sketch a line with a slope of. 5) Sketch a line with a slope of. 5) Sketch a line with a slope of -. 5) Sketch a line with a slope of. IX) Sketch the following graphs: [You can check your answers with a graphing calculator, but please know that you are epected to know what these look like without the use of a graphing calculator.] 5) f ( ) 55) f ( ) 56) f ( ) 57) f ( ) 58) 59) y 60) f ( ) ln 6) f ( ) 6) f( ) 6) f( ) 6) f ( ) 65) f ( ) 66) 68) 70) f 67) f ( ) ( ) f 69) f ( ) / ( ) f ( ) / 7) f ( ) e 7) f( ) 7) y 5 X) Absolute value: 7), 0, 0 75),,
6 XI) Domain and range: 76) f ( ) D:, R : 77) f( ) D: 0, R: 0 78) f( ) D :, R :0 y 79) f ( ) e D:, R: 0 80) f ( ) D:, R: 0 8) f ( ) D:, R: 8) f ( ) ln D: 0, R: 8) D f ( ) :,,,, R: 0 8) f( ) 0 6 D: 6, R: 0 XII) Inverses: 85) f ( ) f ( ) 86) f ( ) f ( ) 87) f( ) f ( ) 88) f ( ) ln f ( ) e XIII) Compositions of functions if: f ( ), g( ), and j( ) f 90) j 89) () 9 f 9) f j 6 9) g g f h h h h 9) 9) f h f h h h
7 XIV) Simultaneous equations: 95a) y 8 y 5 95b) y 9 7 y9 (, ) (-0, 89) and (, ) 96a) The length, l, of a certain rectangle is twice the width, w. Write an equation for the perimeter the rectangle as a function of the width, w. P=6w 96b) If the area of the rectangle described above is 50 square feet, find the length and the width of the rectangle. W = 5 feet and l = 0 feet XV) Intersection of curves: 97) Find the point of intersection between the lines y and y 5. (, ) 98) Find the point of intersection between the lines y 7 and the curve y 5. Also sketch the area between the graphs. (-, 5) and (, 8) XVI) What do the following mean? [If you do not trust your answers, see your instructor.] 99) a graph is in the first quadrant f 5 00) 0) an epression is a function 0) a zero of a function is 0) y is directly proportional to (give an eample) 0) the coefficient of the third term is 5 (give an eample) 05) a function has only one root 06) a function is a polynomial 07) two triangles are similar 08) a function is even 09) a function is odd
8 XVII) What are the following formulas? 0) Quadratic formula b b ac a ) Pythagorean Theorem a b c ) the hypotenuse of a isosceles right triangle with a leg of length. ) the hypotenuse of a right triangle with shortest leg having a length of. ) the volume of a sphere V r 5) the volume of a cylinder V r h 6) the volume of a cone V r h 7) the volume of a bo with a square base V h 8) the surface area of a sphere SA r 9) the surface area of a cylinder with no top SA r rh 0) the area of a triangle A bh ) the area of a trapezoid A hb b ) the cross section through the center of a sphere A r ) the volume of a prism that has an equilateral triangle with side of length and height of length y [Ambiguous question -- don t worry about this one.] ) area of an equilateral triangle in terms of the length of a side s XVIII) Solve using similar triangles: s A 5) A si foot man is standing 0 feet away from a 0 foot lamppost. What is the length of his shadow? 0/7 feet 6) Water is dripping out of a conical figure that has a diameter of 8 inches and a height of inches. When the depth of the water is 8 inches, what is the radius of the water? 8/ inches XIX) Find the equations of the horizontal and vertical asymptotes of each function: 7) y 8) y H. A.: y 0, V. A.: H. A.: y, V. A.:
9 XX) Eponent Rules: Which of the following are true? 9) 0 True 0) True 0) y y NO!!! ) No ) 5 y y True ) 5 No 8 ) 6) 5w 5 t 5 t True 5) 5 9 w True 7) No No false. 8) True 8) Technically, this is because we do not know if is positive or negative. Just be careful. 9) 5 5 NO!!! 0) No ) y y NO!!! ) True ) e ln True ) ln e True ) e ln ln5 True 5) ln ln 5 No 6) Epand using the properties of logarithms: ln ln 7 ln 0 ln ) Condense into a single logarithmic epression using the properties of 7 logarithms: 7 ln ln 5 5 ln 5 5
10 XXI) Using the graphing calculator: 8) Graph y 0. on the -y plane on the right: 9) Find the roots of the equation above (not shown in the graph -- danger of graphing calculator ), -.0,.9 50) Find the point of intersection for the graphs y and y. (.087, 8.7) 5) Find the maimum value for the graph f ( ) ) For the function in #5, find the intervals on which f( ) is increasing. (-, 0.60 ) XXII) What are the following trigonometric identities? 5) 55) 57) 59) sc e cos 5) sin tan cs o 56) cos sin 58) cot csc cc s sin cos cot sn i sec tan XXIII) Evaluate the following epressions: 60) sin 6 6) cos 6 6) 7 tan 6 6) cos 0 6) cos 65) 5 csc 6
11 66) sec 67) cot 0 68) sin 6 69) tan undefined 70) sin 6 5 7) cot 7) sin 7) cot 7) sec 75) tan 76) csc undefined 77) sec XXIV) Sketch one period of the following trigonometric graphs: [Check with your instructor if you do not know how to graph these by hand.] 78) y sin 79) y cos 80) y tan 8) y sec 8) y csc 8) y cot XXV) Solve the following trigonometric equations for the given domain: 8) sin cos on 0, 5,
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