AP CALCULUS AB,...) of Topical Understandings ~
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1 Name: Precalculus Teacher: AP CALCULUS AB ~ (Σer) ( Force Distance) and ( L, L,...) of Topical Understandings ~ As instructors of AP Calculus, we have etremely high epectations of students taking our courses. As stated in the district program planning guide, we epect a certain level of independence to be demonstrated by anyone taking AP Calculus. Your first opportunity to demonstrate your capabilities and resourcefulness to us is through this summer work packet. We epect you stay current on your skills as well as improve upon them. Therefore, this packet is a requirement for Calculus AB. It will be your first major grade of the new school year. SHOW US YOUR BEST WORK. It needs to be completed when handed in on the first day of class. Requirements The following are guidelines for completing the summer work packet There are 50 questions you must complete. SHOW ALL WORK ON THE FOLLOWING PAGES. Please include all the steps used to answer each question. Your work should lead to the correct answer. Please circle or bo each answer!! Be sure all problems neatly organized, and all writing is legible. If you have trouble answering any question, be resourceful. This might include looking back to your notes from previous years or even finding a book or website with formulas or eplanations. In the event that you are unsure how to perform functions on your calculator, you may need to read through your calculator manual to understand the necessary synta or keystrokes. You must be familiar with certain built-in calculator functions such as finding maimum and minimum values, intersection points, and zeros of a function. We epect you to come in with certain understandings that are prerequisite to Calculus. A list of these topical understandings can be found below. Please be familiar with all of these and ready to apply them to a higher level. Topical understandings within summer work Factoring Limits of Functions Prove Trig Identities Graphing Piecewise Functions Graphing, simplifying epressions, and solving equations of the following types: Trigonometric, rational, logarithmic, eponential, Polynomial/Power, Radical.
2 Finally, we suggest not waiting until the last two weeks of summer to begin on this packet. If you spread it out, you will most likely retain the information much better. Once again this is due, completed with quality, on the first day of class, and it counts as a grade. It is your ticket into the class. Best of luck and if you have any questions, feel free to contact us. Mr. Johnston dave_johnston@ipsd.org Calc AB Mr. Koos joe_koos@ipsd.org Calc AB Mrs. Jessie Lavin jessie_lavin@ipsd.org Calc AB First impressions are lasting impressions impress us!! "If I have seen farther than others it is because I have stood on the shoulders of giants"--newton "In most sciences one generation tears down what another has built. In mathematics, each generation builds a new story to the old structure"
3 AP CALCULUS SUMMER PACKET. Use an appropriate procedure to simplify each of the following. No negative eponents allowed in final answer. a) b) t t c) t t + d) 3 + e) f) y z z y g) ( ) ( ) h) ( ) ( ) y y y
4 . Find the domain of the following functions. Give answers in interval notation. Give algebraic justifications for each domain. A) y = B) y = C) y = log( ) D) y = tan E) + 5 f ( ) = +
5 3. Determine the range of: where they occur. 4 = Also, find the ma and min values of f ( ) f ( ), and state Solve the double inequality, < 7. Epress your answer in interval notation. 5. Solve the equation 4 3 = graphically and algebracially: 6. Write the following absolute value epression as a piecewise function. y = 6 + +
6 7. Three sides of a fence and an eisting wall form a rectangular enclosure. The total length of a fence used for the three sides is 40 feet. Let be the length of the two sides perpendicular to the wall as shown.? X X Eisting wall Write the area A of the enclosure as a function of the length of the rectangular area as shown in the above figure. Then find the value(s) of for which the area is 5500 ft. 8. Rewrite the epression log 5 ( + 3) into an equivalent epression using only natural logarithms. 9. Solve by completing the square: 4 = 7 0. Solve the inequality, 8 > 4. Epress your answer in interval notation.
7 +. Let f ( ) = 3, and g( ) =. Compute ( g f )( ), and state its domain in interval notation.. Is this function one-to-one? Justify your answer. y = The following three transformations are applied (in the order given) to the graph of y =. I. A vertical stretch by a factor of 3 II. A horizontal shift right 5 units III. A vertical shift down 6 units Which of the following is an equation for the graph produced as a result of applying these transformations? A. y = 5 B. y = 3( 5) 6 C. y = 3( + 5) 6 D. y = 3 + E. y = 3( 6) 5 4. Let y = f ( ) =. Find a rule for f. In other words find the inverse function.
8 5. Find an equation for the parabola whose verte is (, -5) and passes through (4, 7). Epress your answer in the standard form for a quadratic function. 6. Which of the following could represent a complete graph of 3 f ( ) = a, where a is a real number? 7. Find a degree 3 polynomial with leading coefficient 4 and zeros -,, and Let g ( ) be a sinusoidal function with a min at ( 3π,5) and the net ma at (,8) Write an equation for g ( ). State the amplitude and the period of g ( ). 5π.
9 9. Solve the following by factoring and making appropriate sign charts. A) > 0 B) C) sin sin 0 < π 0. MULTIPLE CHOICE: Solve the inequality > 0., 4 3, B. = 4, = 3 3,4 A. ( ) ( ) C. ( ) D. (, 3) ( 4, )
10 + 3. The graph of y = a for a > is best represented by which graph?. Describe the transformations that can be used to transform the graph of log to a graph of f ( ) = 4log( + ) Arturo invests $700 in a savings account that pays 9% interest, compounded quarterly. If there are no other transactions, when will his balance reach $4550? 4. State whether the following relations are functions. Write yes or no. Then for each one that is a function, state the domain and range. a) {(-, ), (3, 0), (-, 0), (3, )} b) {(0, ), (3, 6), (, ), (3, )}
11 5. Show work to determine if the relation is even, odd or neither. = A) f ( ) 4 3 B) f ( ) = + C) f ( ) = e e 6. Simplify (csc tan )sin cos. A. sin cos B. cos sin C. sin + cos D. cos sin 7. Evaluate all si trigonometric functions of the angle θ for the triangle given below. 8 θ 7
12 8. Solve each equation. A) 4 ( ) 5( ) = B) ( + )( 3) = C) 3 + = 6 D) = 9. π Which transformation was not performed on y = sin to obtain y = sin(3 + )? 3 A. π Horizontal shift left by units 9 B. Horizontal stretch by a factor of 3 C. Vertical stretch by a factor of D. Reflection through the -ais
13 B 30. Solve the right triangle ABC for all its unknown parts, if β = 38 and b = 4.5. c β a A α b C 3. Use a half-angle identity to find the eact value of cos(67.5 ). 3. Find the equations of both the vertical asymptote(s) and horizontal asymptotes (if they eist). State your answers as equations. A) y = 3 B) y = C) y =
14 33. Without using a calculator, find the eact value of cos ( cos 7 π ) 5. Justify your answer. 34. Determine sin lim if possible students are 80 feet apart on opposite sides of a telephone pole. The angles of elevation from the students to the top of the pole are 35 and 3. Find the height of the pole. 36. Graph the piecewise function. < f ( ) = = < Using f ( ) from #38 above, at what points, c, in the domain of f ( ) does f ( ) lim eist? c
15 38. Solve the equation sin cos = cos by factoring. 39. Find all the eact solutions of the equation on the interval [ 0, π ). sin + 3sin = For the function f () graphed, evaluate lim f ( ). 3 A. lim f ( ) = 0. B. lim f ( ) = C. lim f ( ) =. 3 D. lim f ( ) does not eist 3
16 4. Solve for : e = Solve (sin )(cos 3) + (cos 3)(cos ) = for over [ 0, π ) Graph the function y = + 3. Find the local maimum/minimum values, and all the -intercepts. Sketch the graph and state the window dimensions. 44. Use a graphing calculator to approimate all of the function s real zeros. Round to 3 decimal places f ( ) =
17 45. Simplify the following. A) + 4 B) 3 4 y 4 3 y C) Completely factor the following a) b) 8 y c) y 0y
18 47. Solve each equation for. 6 + = A) 5 B) = Find an equation for the following lines with the given properties. Epress your answer using slope intercept form. a) -intercept =, y-intercept = - b) parallel to the line y = ; containing the point (-, )
19 49. Use the following functions. f ( ) = g( ) = h( ) = 0 4 m( ) = 9 Find the domain and range of the following. Give answers in interval notation. a) f () b) h () c) m () Find each of the following: f h d) ( f g) () e) ( ) f) ( f g)( ) g) g ( ) h) (5 ) f i) g (0) j) m k) ( f + g)( 3) l) ( f g)( )
20 50. Solve. a. 3 b 5 4 b 3 = b b = c. 7 = 5 d. ( ) + log 3 log = e. (5.07) = 00 Simplify. f. 5 4 log 8 3 g. log 7 + log h. log3 48 log3 4 i. e 3ln
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