AP Calculus AB Summer Assignment

Transcription

1 AP Calculus AB Summer Assignment Welcome to AP Calculus AB! You are epected to complete the attached homework assignment during the summer. This is because of class time constraints and the amount of material that must be covered to properly prepare for the AP Calculus AB eam. I will check the assignment, and we will spend a few days reviewing the material at the beginning of the school year. We will then have a test on the material. DUE DATE: No later than :0 pm, Tuesday, August 5, 07, to the senior high school office. Epect to work a minimum of 8 to 0 hours on this assignment plan accordingly! SUMMER HELP SESSIONS: This assignment is challenging. You are being asked to review all of the algebra, geometry, and trigonometry skills that you have ever learned and that s a lot! Don t get too discouraged if you have some trouble; it is to be epected. For help with any of the problems, refer to the eamples on the assignment sheet, your pre-calculus notes, or internet resources. Some helpful websites are: (algebra and trig) (trig) Answers will be posted on my school website by the end of June so that you can verify your work. I will be glad to help you with questions during the summer. You may me with questions on particular problems at cindy.pompelia@glsd.us. I will be at the high school all day Monday, August th, 0:00 am :00 pm and Tuesday, August 5 th, 9:am -:0 pm to help you with any problems or concepts. I can also meet with you at the school earlier in the summer by appointment. Register to receive tet messages and updates about this assignment and the class through remind.com by to 800. GRAPHING CALCULATORS: We will be using the TI-Nspire CX CAS calculators this year. You will be loaned one of these calculators in the fall to use for the school year. However, if you would like to purchase your own, I would recommend ordering online from either Amazon.com or schoolsavers.com for the best price. These calculators are very different from the calculators you have used in the past, primarily because they are symbol manipulators. NOTEBOOK: You are required to use a loose leaf three-ring binder (no spiral notebooks) for taking notes and completing assignments for this course as there are several handouts, and I often collect your assignments. This assignment is worth 00 points. I will randomly pick seven problems from the worksheet to grade on correctness and effort; I will also grade your assignment on overall neatness and thoroughness. Homework is weighted 5% of your total grade in this course. Please complete all work on separate paper, unless noted otherwise, and include the original problem with your work. All answers, ecept graphs and tables, should be boed or circled. Verify your answers with the posted answers on my website before submitting your work. I epect high quality work! I look forward to working with you net year, and remember ~ Practice makes perfect. Good Luck! ~Mrs. Pompelia

2 Functions, Composites, and Difference Quotient 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()). f ( g( )) f ( ) Let f ( ) and g( ). Find each.. f (). g( ). f( t). f g( ) 5. ( ) Let ( ) ( 8 6) 6 f g ( ( )) 6 g f m 6. f ( ), g( ) 5, and h( ). Find each. 7. h f( ) 8. f ( h) f ( ) Find h 0. f ( ) 5 Intercepts of a Graph for the given function f. f g( ) 9. g( h) g( ) h g h ( ) To find the -intercepts, also referred to as the zeros or roots of the function, let y = 0 in your equation and solve. To find the y-intercepts, let = 0 in your equation and solve. Eample: y y int. ( Let 0) y 0 (0) y y intercept (0, ) int. ( Let y 0) 0 0 ( )( ) or i ntercepts (,0) and (,0) Find the and y intercepts for each.. y 5. y. y 6. y

3 Points of Intersection Use substitution or elimination method to solve the system of equations. Remember you are finding the point if intersection of two graphs so your answer is an ordered pair. Eample: y y 9 0 Elimination Method Subtract to eliminate y 85 0 ( )( 5) 0 and 5 Plug = and 5 into one original y y 9 0 y 0 6 y y 0 y Points of Intersection (5, ), (5, ) and (,0) Substitution Method Solve one equation for one variable. y 6 9 (st equation solved for y) ( 6 9) 9 0 Plug what y is equal ( previous eample Find the point(s) of intersection of the graphs for the given equations or functions. to into second equation. The rest is the same as ) ( )( 5) 0 or 5 5. f ( ) g( ) 6. y. 7. y 5 f( ) g( ) 8. y y Domain and Range

4 Find the domain and range of each function. Write your answer in interval notation. 9. f ( ) 5 0. f ( ). f ( ) sin. f( ). Find the domain of each function. Write your answer in interval notation. f( ) 5. h ( ) 5 Inverses 5. f( ) g( ) ln( ) Find the inverse for each function f ( ) 8. f( ) 9. g ( ) 0. If the graph of f() has the point (,7), then what is one point that will be on the graph of f - ()?. Eplain how the graphs of f() and f - () compare. Also, recall that to PROVE one function is an inverse of another function, you need to show that: f ( g( )) g( f ( )) 9 Eample: If: f ( ) and g( ) 9 show f() and g() are inverses of each other f ( g( )) 9 g( f ( )) f ( g( )) g( f ( )) therefore they are inverses of each other.

5 Prove f and g are inverse functions of each other.. f ( ) g( ). f ( ) 9, 0 g( ) 9 Equation of a Line ***In calculus the point-slope form is almost always used, not the slope-intercept form.***. *Do not use this form! Determine the equation of a line passing through the point (5, -) with an undefined slope. 5. Determine the equation of a line passing through the point (-, ) 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 /. 7. 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 (, 7). 9. Find the equation of a line passing through the points (-, -0) and (, ). Trigonometry 5

6 0. Complete the following: a.) csc b.) sec c.) cot d.) tan e.) sin cos f.) To convert angle measures between degrees and radians, π =. Bonus: Using SOH CAH TOA and a graph of the unit circle, eplain why the sine of the angle is the y- value and the cosine of the angle is the -value.. Without a calculator, determine the eact value of each epression. a) sin0 b) sin c) sin d) cos e) cos f) cos g) 7 tan h) tan 6 i) tan j) sec k) 5 csc l) cot Trigonometric Equations: Solve each of the equations for 0.. sin. cos. cos cos 0 (*factor) 5. cos 0 Inverse Trigonometric Functions: Recall: Inverse Trig Functions can be written in one of ways: arcsin sin Inverse trig functions are defined only in the quadrants as indicated below due to their restricted domains. cos - sin - cos - tan - sin - tan - For each of the following, epress the value for y in radians. 6. y = arcsin (or sin ) 7. y = arctan 8. y arcsin 9. y arccos 6

7 Eample: Find the value without a calculator. 5 cosarctan 6 6 Draw the reference triangle in the correct quadrant first. 5 Find the missing side using Pythagorean Theorem. 6 Find the ratio of the cosine of the reference triangle. cos For each of the following find the value without a calculator. a) tanarccos b) secsin c) sin arctan 5 5. Which of the following epressions are identical? a) cos b) (cos ) c) cos 5. Which of the following epressions are identical? d) sin sin 8 7 a) (sin ) - b) arcsin c) sin - d) sin e) sin - Circles and Ellipses (No problems to do; just some information for you.) ( h) ( y k) a b Minor Ais r ( h) ( y k) 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, 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. 7

8 5. Take a break you deserve it! 5. Tell someone in your family that you love them and note their reaction. Transformation of Functions h( ) f ( ) c Vertical shift c units up h( ) f ( ) c Vertical shift c units down h( ) f( ) Relection over the -ais h( ) f ( c) Horizontal shift c units right h( ) f ( c) Horizontal shift c units left 55. Given f ( ) and g ( ) ( ). How does the graph of g() differ from f()? 56. Write an equation for the function that has the shape of reflected over the -ais. f ( ) but moved si units to the left and 57. If an ordered pair (,) is on the graph of f(), find one ordered pair that will be on the graphs of the following functions: a) f ( ) b) f ( ) c) f ( ) d) f ( ) Vertical Asymptotes Determine all vertical asymptotes. 58. f( ) 59. f( ) 60. f( ) ( ) 6. f( ) 9 8

9 Horizontal Asymptotes Determine all horizontal asymptotes. 6. f( ) 7 6. f( ) f( ) ( 5) f( ) Logarithms and Eponentials y log is equivalent to b b Product property: log mn log m log n b b b Quotient property: m logb logb m logb n n Power property: p log m plog m b Property of equality: If log m log n, then m = n Change of base formula: b log a b b logb n n log a log 0, ln=0, log a b b, ln e b y Because logarithms and eponentials are inverse functions of each other: log ln log ( b ) b, ln( e ), b, e b 9

10 66. Solve each eponential or logarithmic equation. 5 a) 5 5 b) 9 c) 8 d) 7 9 e) 8 f ) log g) log 9 h) log i) log ( 7) log ( ) j) log log( ) 67. Epand each of the following using the properties of logs. 68. Evaluate the following epressions without a calculator. 5 a)log 5 b) ln y a) ln e b) f) log/ 8 g) Even and Odd Functions ( ln ) e c) ln d) ln e e) log (/ ) ln e h) log (000 ) i) (log 000) 69. Determine algebraically whether the following functions are even, odd, or neither. (Bonus: include a sketch of the graph of the function to confirm graphically your answer.) a f b f c f d f ) ( ) ) ( ) ) ( ) ) ( ) 5 e) f ( ) cos f ) f ( ) sin Comple Fractions and Rational Epressions When simplifying comple fractions, there are different ways to simplify, two of which are shown below:. work separately with the numerator and denominator, rewriting each with a common denominator, and then multiplying the numerator by the reciprocal of the denominator; or. multiply the entire comple fraction by a fraction equal to which has a numerator and denominator composed of the common denominator of all the denominators in the comple fraction. 0

11 Eample: 6 7( ) ) = OR: ) ( ) ( ) ( ) = 5( ) ( ) 5 ( )(5 ) 5 5 OR: ( ) ( ) ( ) ( ) 5( )( ) ( ) Simplify each of the following a) 0 a b) a 5 a 5 e) f) c) d) Miscellaneous algebra problems 7. Write as a single fraction. Where applicable, the denominator should be in factored form a) b) 0 c) d) Bonus ( ) ( ) 6 h 7. Simplify. Final answers should have only positive eponents, and numerators and denominators should be in factored form. Do not rationalize the denominators. a) f) j) b) c) / / ( ) ( ) (5 a )( a ) g) sin cos sin k) cot tan 8 h) l) cot sec 5 d) 5 6 i) e) y y 5 ( y) ( y) 7. Solve for z: a) 0yz 0 b) y yz 8z

12 7. Epand: ( y) 75. Solve for. a) b) 0 c) ( 5) 9 d) 5 e) 0 f) 5 0 ***use test intervals (sign chart) g) h) Rationalize the denominator: (a) (b) 77. Factor: ( ) ( ) Word Problems (Include a diagram with your solutions.) 78. The length of a rectangle is equal to three times the square root of its width. Epress the perimeter and area of the rectangle as a function of its width. 79. An open bo has a square base with base dimensions by cm and height y cm. Its volume is 00 cm. Epress the surface area of the bo as a function of. 80. The base of a triangle is 6 cm more than the height. If the area of the triangle is 0 sq cm, what is the length of the base? 8. Find the area enclosed by the graphs of y = 0, = 0, = 5, and y = +. Series 8. Epand and evaluate. a) Limits n b) n0 n n 8. You are probably approaching your limit with this assignment, so we will wait to review this concept in class!. Graphing 8. Print a hard copy of, complete, and attach the last three pages of this assignment to your work for the previous problems. Graph the functions neatly and accurately, indicating - and y- intercepts and scales where appropriate. Note the domain and range for each graph. Also, know how to sketch vertical and horizontal translations of theses graphs, such as y = + -. Reflection 85. Congratulations! You have finished your summer assignment! Please reflect on your work. Some things to include: how long did it take you to complete? Who did you work with and in what capacity? What concepts are you the most comfortable with and what areas are your weakest? What did you learn as a result of doing this assignment?

13 Graphing a) y b) y c) y d) y e) y sin f) y cos

14 g) y tan h) y 5 i) y j) y k) y l) f ( ) (greatest integer function)

15 m) y ln n) y e o) y p) 0 f ( ) 0 q) f( ) r) f( ) 5