Falls Church High School

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1 Summer AP Assignment Coversheet Falls Church High School Course: AP Calculus BC Teacher Name/s: Marla Schnall Assignment Title: AP Calculus BC Summer Packet Assignment Summary/Purpose: The material in this packet should be a review of past material that will be necessary in BC Calculus. Completion of this packet will set each student up for continued success in the AP Calculus program. Due date (must be completed before school starts) Assigned during: Summer break Due Date: 9//7 Estimated time needed to complete the assignment: Approimately hrs. Description of how the assignment will be assessed: The material from this packet will be used throughout the course and will be assessed as part of unit tests and quizzes. Grade impact to overall course grade: Content will be integrated into the coursework. Tools/resources needed to complete the assignment: TI-8, 84, or 89 graphing calculator Contacts: Name Marla Schnall marla.schnall@fcps.edu

2 FAIRFAX COUNTY PUBLIC SCHOOLS Falls Church High School Mathematics Department 75 Jaguar Trail Falls Church, Virginia 4 (7) 7-4 Dear 7-8 AP Calculus Student, Congratulations on your decision to eplore the world of Calculus this year. Attached to this letter you will find a packet of materials for you to work on over the summer. This packet is designed to ensure that you can recall and apply many of the wonderful algebra, geometry, and trigonometry skills that you have learned in the past few years. While calculus is an entirely new subject to you, it is strongly based on algebra. Your success in calculus depends strongly on your ability to recall and apply the skills that you have learned in prior math courses. You will have the opportunity to ask questions about anything in it during the first week of class. I do not epect you to spend more than hours to complete this assignment. Your readiness for this course will be determined by how much information you retained from your previous math courses and your ability of applying them in real-life situation to solve problems. Therefore, the completion of this assignment will prepare you for the first weeks of Calculus. I will use this assignment to plan review and remediation as necessary. If you need more review over the summer, there are many resources to be found at the library and on the Internet. The public libraries carry algebra and precalculus tetbooks, as well as online and review books for those courses. Many sites on the Internet have math help available. You may me at marla.schnall@fcps.edu, but I may not answer immediately as we all have different plans for the summer. You will need to get your own graphing calculator for net year. Because Calculus is an elective course, the school does not issue Ti-8 calculators. You should get a Ti-8, Ti-8plus, or Ti-84. The Ti-89 is also allowed and recommended. Calculus is challenging and sometimes fun and definitely different from anything you have studied before. As an AP student, I epect you to work hard and to take the initiative to find out about things that you don t understand. I hope you enjoy your summer and look forward to seeing you in September 7! Mrs. Schnall

3 Falls Church High School Mathematics Department Summer Packet 7 Multivariables Cal. & Linear Algebra AP Calculus BC Trig/Math Analysis or PreCalculus Algebra or Adv.Alg Geometry or Geometry GT Algebra Introduction to Algebra For students entering AP Calculus BC In August 7

4 FUNCTIONS A function is a rule that assigns to each member in its domain a unique member in the range. Domain: The possible -values of a function Range: The possible y-values of a function FUNCTION NOTATION: f() To be written as f() means the represented epression satisfies the definition of a function, i.e. for each -value, there is an unique corresponding y-value EVALUATION OF FUNCTIONS: f(a) means replace each in f() with a to find the function s y-value EXAMPLE: f() = - 5 f() = () - 5 f() = 6-5 f() = ADDITION, SUBTRACTION AND MULTIPLICATION OF FUNCTIONS: Functions can be added, subtracted and multiplied just as any other epressions: EXAMPLE: f( ) = + ( ) ( ) 4 f + g= + + ( ) ( ) f g= + + f g ( ) ( ) = + 5 g ( ) = + COMPOSITION OF FUNCTIONS: Composition of functions means instead of replacing in a function with a number, an entire function is inserted instead. EXAMPLE: f( ) = g ( ) = + The composition of f() and g() is written: f( g ( )) To evaluate, simply plug in g() wherever an appears in f(). Function Problems: f( ) = f( g ( )) = ( + ) = g ( ) = + g f ( ( )) = ( ) + = = 6+. Find the -intercepts and the y-intercepts of f() = +. Find the domain and range of the function. h ( ) = + 4

5 . What is the domain of the function f ( ) = ln( 4)? 4. Identify the domain of the function f ( ) 5 =, and find f ( ).. 5. Given f ( ) = a. Domain: Range: b. f() = f(+5) = c. f() = - then = d. Sketch f() 6. a. Graph the piece-wise function, 4 g( ) =, < < 4, b. Identify the domain and the range of g() c. g(-) = g() = g() = d. Is g() a continuous function? 6. Given the graph of g() on the right a. Estimate g(6) g() = 6 b. The ratio in part (a) is the slope of a line segment joining two points on the graph. Sketch this line segment. 5

6 7. The rate at which water is entering a tank ( t > ) is represented by the given graph. A negative rate means that water is leaving the tank. State the interval(s) on which each of the following holds true: a. The volume of water is constant. b. The volume of water is decreasing. c. The volume of water is increasing. d. The volume of water is increasing fastest. 8. Q( ) = + a. What is the domain of Q()? b. Where is this function discontinuous? c. State the equation of the vertical asymptote = d. State the equation of the horizontal asymptote y = e. Sketch the graph. f. Write the equation of the inverse of Q() (Switch the & y and then rewrite as y =) 6

7 9. Use these functions: f( ) = + g ( ) = + h ( ) = 5 a) f () = b) g (5) = c) h ( ) = d) g( ) = e) f( ) + h ( ) = f) g ( ) h ( ) = g) f( h ()) = h) hg ( ( )) = i) h( f( )) = j) g( f( )) = 7

8 TRIGONOMETRY What you need to know: Trig functions and inverse trig functions for all special angles (unit circle) Fundamental trig identities (reciprocal, quotient, Pythagorean) Graphs of sin, cos, tan, sec, csc Domain and range of sin, cos, tan, sec, csc Solve trig equations Trigonometric Identities + = + = + = sin ( ) cos ( ) tan ( ) sec ( ) cot ( ) csc ( ) sin( ) tan( ) = cos( ) cos( ) cot( ) = sin( ) sec( ) = cos( ) csc( ) = sin( ) cos( + y) = cos( )cos( y) sin( )sin( y) cos( y) = cos( )cos( y) + sin( )sin( y) sin( + y) = sin( ) cos( y) + sin( y)cos( ) sin( y) = sin( )cos( y) sin( y)cos( ). Evaluate without use of a calculator. (a) tan( π 6 ) = (b)sin(π)= (c) cos( π )= (d) csc( π )=. Find the eact values without use of a calculator. (a) sin - ( ) = (b) arctan(- ) = (c) cos- ( )= (d) sec - (-)= (e) arcsin(- )= (f) sin(cos-.6)= 8

9 . Sketch the graphs of the following without the use of a calculator. Show at least two periods: π (a) y = cosπ (b) y = tan( ) 4 4. Solve the following equations for. a. cos (7 π) = d. sin ( ) cos( ) = + b. sec( ) = tan( ) + cot( ) e. tan( ) + sin = c. sin + cos = f. sin sin = 5. Write the following epression in algebraic form. sin(cos ( )) 9

10 Limits The limit of a function is the y-value that you are getting close to as gets close to some number in the domain. In the limit process you never get to the limit, ecept for the limit of a constant function. We write lim f ( ) a, which is read the limit of f() as approaches a domain value of a. The limit must be the same as approaches a from both the left and the right. To find the limit, substitute in values very close to a on both left and right and see if the y-value is approaching a single value. The limit does eist at a hole in a graph, but does not eist at a vertical asymptote or a jump in the graph. Problems: The graphs Limit of eist some at functions are pictured below. Do No you limit think at that lim f ( ) does eits, state its value. eists? If you think the limit No limit at.. f f f 4 4 State the value of each of the following: 5. lim 8. lim ( + ) 6. lim 9. lim lim 7.. lim

11 . Find the limit lim sin π. Let f ( ) + = and g ( ) =. Find the limits: (a) lim f( ) (b) lim g ( ) 4 (c) lim g( f( )). Find the following limit (if it eists). Write a simpler function that agrees with the given function at all but one point. + 6 lim Use a graphing calculator to complete the table and use the result to estimate the limit. + 8 lim X f()

12 THE EQUATIONS OF A LINE Write the equation of the line described in the form y = m + b. Where and b, the y-intercept. The point slope form which is given by the slope m are known or given. Eample: The line through (. 6) and (-9, -) y y m = is the slope of the line y y= m ( ) is the most used in Calculus. A point (, y ) and Solution: m = 6 9 = ( y y ) = m( ) y 6= y = + 4 ( ). The line through (, 4) and (, 6).The line through (5, -) and (-5, 4). The line through (, ) with slope 4 4. The line with slope 7 and passing through (-, -5) 5. The line with slope 8 and y-intercept 9 6. The line with slope 8 passing through (, 5) 7. The line through (-, -8) and parallel to 8. The line perpendicular to y = the line y = 5 - and passing through (8, -)

13 EXPONENTS Properties of eponents m m n m+ n m n m n mn * * = = ( ) = n m n m n m = = = m a a (ab) = a b ( ) = b b log a(y)=log a + logay log ( ) = log log y log log a a y = ylog log = log b b a a a a a y Completely simplify each epression 8 *4 cd... 4 ( y) 6 ( cd ) 7 5 * ( ) * * ( )

14 SOLVING EQUATIONS SOLVE FOR X.. ( ) ( ) 4 + = 4 4. e = =. e 7e + =. = 4 5. ln ln = 4. = 4. ln 4 = 5 5. cos + = 5. 4t t + 8t 4 = 6. sec 4 = = 7. sin = cos 7. 8= 8. log + log5 = 6 8. ( + ) 4 = 7 9. s + 6s+ = 9. 6 = =.. ln ( ) = 4

15 . Write the epression ln ( ) ln ( 4) + as the logarithm of a single quantity 5 Sequences and Series a n = 4n +. What is the 4 th term of the sequnce?. What is the (n+)st term of the series?. What is the difference between the nth term and the (n+)rd term? a n = 5 n 4. What is the 6 th term of the sequence? 5. What is the sum of the first terms 6. Evaluate the sum n= 5 n

16 Constructing a Rain Gutter A rain gutter is to be constructed of aluminum sheets in wide. After marking off length of 4in from each edge, this length is bent up at an angleθ. a- Show that the area A of the opening or cross-section (as shown on the illustration) of the gutter as a π function of θ is given by A( θ) = 6 sin θ(cosθ + ), < θ < b- In Calculus, you will be asked to find the angle θ that maimizes A by solving the equation cos( θ) + cosθ =. Solve this equation for θ by using the Double-Angle Identity c- Solve the equation for θ by writing the sum of the two () cosines as a product. d- What is the maimum area A of the opening? 6