SUMMER MATH PACKET AP Calculus AB Or AP Calculus BC Required
MATH SUMMER PACKET INSTRUCTIONS Attached you will find a packet of eciting math problems for your enjoyment over the summer. The purpose of the summer packet is to review the topics you have already mastered in math, and to make sure that you are prepared for the class you are about to enter. The packet contains a brief summary and eplanation of the topics so you don t need to worry if you don t have your math book. You will find many sample problems, which would be great practice for you before you try your own problems. The eplanations are divided into sections to match the sample problems so you should be able to reference the eamples easily. This packet will be due the second day of class. All of your hard work will receive credit. The answers are provided in the back of the packet; however, you must show an amount of work appropriate to each problem in order to receive credit. If you are unsure of how much work to show, let the sample problems be your guide. You will have an opportunity to show off your skills during the first week when your class takes a quiz on the material in the packet. This packet is to help you maimize your previous math courses and to make sure that everyone is starting off on an even playing field on the first day of school. If you feel that you need additional help on one or two topics, you may want to try math websites such as: www.mathforum.org or www.askjeeves.com. Math teachers will be available for assistance at the high school the week before school. Check the school website for specific dates and times. Enjoy your summer and don t forget about the packet. August will be here before you know it! If you lose your packet, you will be able to access the packets on-line at the school website, www.oprfhs.org starting May 8 th. Etra copies may be available in the OPRFHS Bookstore. See you in August! The OPRFHS Math Department
SUMMER PACKET For students entering AP Calculus AB or BC NAME: Welcome to AP Calculus! This packet contains the topics that you have learned in your previous courses that are most important to calculus. Please read the information, do the sample problems and be prepared to turn this in when school begins again. * Denotes problems for BC calc students only Enjoy your summer! I. Using your graphing calculator Be able to find the intersection of lines and curves. You will need to find the roots of equations on your calculator Do not be dependent on your calculator for graphs that should be memorized (see Function Section) Do not be dependent on your calculator for simple algebra, adding fractions, etc. II. Interval and set notation You will be epected to be able to read and write using different notation, for eample notice the difference between the following solutions: Problem: < + Solution: Problem: + Solution:, 7, 7 Don t forget about the symbols that join information together: U = Union I = Intersection
III. Cartesian Coordinate System Know the quadrants, aes, and how to graph Know the different forms of the equation of a line Δy Slope (incredibly important in calculus); think Δ Parallel and perpendicular lines Remember that means parallel and means perpendicular Distance Formula: Equation of a circle: ( ) + ( y y ) ( h) + ( y k) = r IV. Functions We use functions almost everyday in calculus. It is critical that you are able to identify and do the following: Identify functions from equations, graphs, sets Use function notation: y = f( ) remember is the independent variable and y is the dependent variable Find the domain and range of a function Identify functions as even, odd or neither. You should also be able to identify what lines of symmetry go with even and odd functions. Be able to graph (without a calculator) or identify: y = a = b (where a and b are constants) f( ) f( ) f( ) = f( ) = f( ) f( ) = ln = a + b * f( ) = or f( ) f( ) = f( ) = = f( ) = e = f( ) = a + b f( ) = = * c + d * Remember the following function operations: f + g = f + g ( )( ) () () ( f g)( ) = f() g() ( fg)( ) = f ( ) g( ) f f( ) ( ) = g g( ) g( f( )) = go f ( )( )
You must be able to graph and write the equations of piecewise functions + You should be able to identify discontinuities, i.e. f( ) = +, if < 0 Be able to graph piecewise functions, i.e. f( ) =, if 0 V. Polynomial Functions and Graphs A. Long Division Remember that one of the main reasons we use long division is to check if an algebraic epression is a factor of a polynomial. P() = ( - r) Q() + R The divisor the quotient the remainder For eample, is (+) a factor of +? + + + + + 0 + = ( + )( + ) Answer: ( ) If the remainder is 0, then it is a factor.
B. Synthetic Division Synthetic Division is an alternate method of long division where only the coefficients are used. For eample let s use synthetic division to find the quotient and remainder; ( 7 ) ( ) + + Remember divisor must be (-a) form Therefore: 7 0 9 Note: We must write 0 s for any missing terms + 7 = ( + )( + ) + C. Finding Rational Zeros of Polynomials. You should be able to describe the left and right behavior of a graph. D. Rational Functions: finding vertical, horizontal and oblique asymptotes E. Partial Fractions* Remember that decomposing into partial fractions is basically reversing the process of adding rational epressions. If you have a rational epression P()/Q() that have no common factor other than or - there are two possibilities: The degree of the numerator is larger than the degree of the denominator. You should then divide using long or synthetic division. If the degree of the denominator is larger than the degree of the numerator then do the following: E. + a) Factor the denominator ( + )( ) b) A B Write as a sum of terms: = + ( + )( ) +
Do you remember how to determine the numerators? c) Determine A and B by adding the fractions: A( ) + B( + ) = ( + )( ) ( + )( ) d) Set the two numerators equal to each other: = A( ) + B( + ) Set - = 0, so = + and then substitute and solve for B. Set + = 0 so = -, substitute and solve for A. Set A = and B = - e) The decomposition is: + + or + VI. Logarithms and Eponentials Know the properties of eponents Know the properties of logarithms Remember basic facts such as: ln e ln = and e = Remember to remove logarithms from an equation, eponentiate both sides. To remove eponentials from an equation, take the logarithm of both sides. E. Find y if ln y= t+ Eponentiate both sides: e = e + ln y t t y= e + E. Find k if k e = 0 Take the natural log of both sides: k ln e = ln0 k = ln0 ln0 k =
VII. Sequences and Series* A. Writing the terms of a series Given k = k k 0 = + + + + Replace k with consecutive integers starting with k =, and continue through k = B. Find terms from a recursive formula, a formula that defines each term in terms of one or more preceding terms. a = a = n > 7 n an a = 7 a = a = 7 = a = a a = 7 = C Arithmetic and Geometric Sequences.. nth term of an Arithmetic Sequence : an = a + ( n ) d for every n >. nth term of a Geometric Sequence: n a = ar for every n > n. Sum of an Arithmetic Series: n Sn = ( a + an). Sum of a Geometric Series: n a ar Sn = r r. Sum of an Infinite Geometric Series: a Sn = r < r 7
VIII. Trigonometry The Pre-Calc or College Algebra Course you just completed was your formal trig course for your high school math career. However, we use the trig functions and a few of the identities regularly. The following items are the most important: Calculus is always in Radians - never degrees. Watch the mode on your calculator. You need to know, without a calculator, sinθ, cos θ, tan θ, sec θ, csc θ, and cotθ for the following values: θ =,,,, Be able to identify and graph sine, cosine and tangent Know which trig functions are even or odd or neither The following identities are the most used: sin θ + cos θ = cos θ = cos θ tan θ + = sec θ sin θ = sinθcosθ + cos( θ ) cos θ = cos θ sin θ cos θ = cos( θ ) cos θ = sin θ sin θ = IX. Polar and Rectangular Coordinates * Change coordinates from polar to rectangular and rectangular to polar E. Given:, find (, y) = rcosθ y= rsinθ = cos = sin = = = y = So (, y) = (, ) 8
E. Given: (-, ) find ( r, θ ) ( ) ( ) + = r = r tan θ = b ac θ in QII tan.70 by restrictions in QIV θ = 80.70 =.89 (radians) r, θ = (,.89) so ( ) Change equation to Polar Form Given: + y = ( r θ) ( r θ) cos + sin = r cos θ + r sin θ = r (cos θ + sin θ) = r = r = X. Parametric Equations* Obtain an equation in and y by eliminating the parameter and identify the curve (line, parabola, ellipse, circle and hyperbola) A. = t+ < t < y= t t eliminate t so t = y= ( ) ( ) parabola B. = cosθ < θ < y = sinθ eliminate θ so cosθ = sinθ = cos θ + sin θ = y + = y + = 9 ellipse y 9
C. Find the parametric equation for the conic section y y 0 + 7 = 0 By completing the square in and y we obtain the standard form ( ) ( y ) = So the graph is a hyperbola with center (, ) and transverse ais on the line y =. Since letting sec θ tan =, a parametric representation with parameter θ is obtained by y = secθ and = tan θ therefore we get = secθ + < θ < y = + tanθ θ + k, k is an integer Practice Problems. Complete the problems below, showing work where necessary. Feel free to do your work on separate sheets of paper and remember you will be required to turn this in. An answer key is provided for you but in calculus the work is always more important than the answer! *Problems should be completed by BC Calculus students only. Solve the following and write your answer in interval notation:. 7 7+. + +. + + > 0 + +. + >. Write an equation for each of the lines described, in point-slope form. a) passes through (-, -9) and has slope 0 b) passes through (, -) and is parallel to the line + y = c) passes through (, 0) and is normal to the line - y = 0
. Find the center and radius for the following circles. a) ( ) ( y ) + + = b) y y + 8 + + = 0 Center: Radius: Center: Radius: 7. Use your calculator to sketch the following graphs, labeling their intersection points. f( ) = ( ) and g ( ) = + 8. If a particle starts at A (-, ) and its coordinates change by the increments Δ = and Δ y =, find its new position.
9. Prove that the triangle formed by the points A (0,0), B (, ), and C (, 0) is equilateral. 0. Find the domain and range of the following functions: a) f( ) 9 = b) g ( ) = c + + Domain: Domain: Domain: Range: Range: Range:. Which of the following functions are even, odd, neither? a) f( ) = b) f( ) = c) f( ) = + d) f( ) = + e) f( ) = + f) f( ) = g) f( ) = h) f( ) = i) f( ) = + j) f( ) = + Even: Odd: Neither:
. If f ( ) = + and g( ) =, find the following: a) f(g()) d) f(f(-)) b) g(f()) e) g(g()) c) f(g(0)) f) f(+h) - f() h g) Given: f( ) =, find + f( + h) f( ) h h) Given: g ( ) = +, find g ( + h) g ( ) h. Graph the following function., f( ) =, > 0 8-0 -8 - - - 8 0 - - - -8
. Find a formula for the graphed function f()=? - - 7 - -. In the following problems, a polynomial P() and a divisor d() are given. Find the quotient Q() and the remainder R() when P() is divided by d() and epress P() in the form P() = d()q() + R() a) P ( ) = + 0 b) P ( ) = + 9 + 0 d ( ) = d( ) = +. Use synthetic division to determine whether the numbers given are roots of the polynomials. a) P () ( 7 ) = + + ; = -, = b) P ( ) = + + 8; =, = -
7. Find all zeros eactly for behavior of the graph. P ( ) = + + 0 + +. Describe the left and right 8. Find the landmarks of the following functions. Include vertical, horizontal and oblique asymptotes as well as the and y intercepts. Sketch each function and label the landmarks. a) f( ) = b) f( ) = 7 0 c) g ( ) = + 0 8-0 -8 - - - 8 0 - - - -8-0 0 8-0 -8 - - - 8 0 - - - -8-0 0 8-0 -8 - - - 8 0 - - - -8-0
9.* Decompose the following into partial fractions a) + 7 ( )( + ) b) + + ( + ) 0. Write in epanded form without summation notation. k = k + k.* Write the first terms of a = ( n+ ) n n n.* Let a, a, a,... be an arithmetic sequence. Given: a = 7 and a 9 =. Find the indicated quantities: a =? and S =?.* Let a, a, a,... be a geometric sequence. Find the indicated quantity. a = 0 a = r =? S =?
.* Find the sum of the infinite geometric series + K 7. Simplify the following: a) ln e = b) ln e = c) ln e = d) ln e + =. Solve for y: ln( y ) ln = + ln 7. Complete the following tables. (Without your calculator!) sin θ cos θ tan θ sin θ 7 cos θ tan θ 7
8. Graph the following. Identify the period of each. a) f( ) = sin( ) b) g ( ) = cos( + ) -.89 -.9.9.89 - - - - -.89 -.9.9.89 - - - - c) h ( ) = tan( ) -.89 -.9.9.89 - - - - 8
9.* Change from polar to rectangular. (note: θ in radians) a) (, ) b) ( 7,.8 ) 0.* Change from rectangular to polar a), b) ( 8, ).* Change this rectangular equation to polar form. Identify the graph as a line, circle, etc. a) y + + 8 = 0 b) = y.* Convert from polar form to an equation in rectangular form r = sinθ.* Obtain an equation in and y by eliminating the parameter, and identify the curve a) = t b) = + sin y = t y = + cos θ θ 9
Answer Key:. (,.,. (, ) (, ). ( ),0 0,,. a) y+ 9= 0( + ) b) y+ = ( ) c) y 0 = ( ). a) Center: (, -) Radius: b) Center: (, -) Radius: 7. Intersection: (.0,0. ) and (.8,0.) 8. (, -) 9. Prove by distance formula, all sides equal units. 0. a) Domain: [,] b) Domain: [ 0, ] c) Domain: (, ) [, ) Range: [ 0, ] Range: ( 0, ] Range: [ 0,) (, ). Even: A, C, G, J Odd: B, E, H Neither: D, F, I. a) + b) + 0+ c) d) e) + f) g). Graph ( + )( + h+ ) h) ( + h) ( + h) + + + 0
. f() = +. a) b) P ( ) = ( )( + 9 + ) + 8 P ( ) = ( + )( + 00) + 0. a) neither b) = no -= yes 7. Zeros are at =, (multiplicity ) and = ± i The left and right behavior of the graph is approaching positive infinity. 8. a) Vertical at = Horizontal at y = 0 No oblique asymptotes 0 8 (0,-0.) -0-8 - - - 8 0 - - - -8-0 b) Vertical at = & - Horizontal at y = No oblique asymptotes 0 8 (0,0) -0-8 - - - 8 0 - - - -8-0 c) Vertical at = No Horizontal Oblique at y = 0 8-0 -8 - - - 8 0 (0,-/) - (,0) - - -8-0
9. a) + + b) + + + 0. + + +., 9,,. a = and S =. r = and S = 7.. 7 8. a) b). y= e + c) 8 d) e 7. sin θ 0 cos θ 0 tan θ 0 Undefined 7 sin θ 0 - - - cos θ - 0 tan θ 0 Undefined - -
8. a) P = b) P = c) P = 9. a) (, 0) b) (.,.) 0. a), b) ( 7,.08) radians. a) r + 8cosθ = 0 circle b) r = tanθsecθ parabola. y y + = circle. a) y = b) ( ) ( y ) parabola + = circle