Welcome to Advanced Placement Calculus!! Summer Math
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1 Welcome to Advanced Placement Calculus!! Summer Math As Advanced placement students, your first assignment for the school year is to come to class the very first day in top mathematical form. Calculus is a world of change. While words like it, continuity, derivative and integral may seem foreign and daunting to you, I assure you that at this time net year at this time they will seem as commonplace to you as graduation and senioritis! In order to begin our journey, you must do some mathercise to keep your mind in shape through the summer months. So, I have prepared four assignments to help you burn your mathematical muscles! Make sure all of your work is presented in a logical order showing ALL of your work indicating your answers clearly. IF I CAN T READ YOUR HANDWRITING, IT IS WRONG!! This packet is a review of some Pre-calculus topics and some Calculus topics. It is to be done NEATLY and on a SEPARATE sheet of paper. Use your discretion as to whether you should use a calculator or not. When in doubt, think about if I would use one that should guide you! Points will be awarded only if the correct work is shown, and that work leads to the correct answer. Have a great summer and I am looking forward to seeing you in September. Sincerely, BHS Math Teachers PS: A good website to check out if you need some help would be
2 AP CALCULUS BC SUMMER ASSIGNMENT Please check the prerequisites for Calculus BC and study the concepts. PREREQUISITES FOR CALCULUS BC Before studying calculus, all students should complete a full four-year preparation of secondary mathematics designed for college-bound students: courses in which they study Algebra, Geometry, Trigonometry, Pre- Calculus (analytic geometry and elementary functions) Students must be familiar with functions that include: - Linear - Polynomial - Rational - Eponential - Logarithmic - Trigonometric - Inverse Trigonometric - Piecewise defined Students must be familiar with - the properties of functions, - the algebra of functions, and - the graphs of functions. Students must also understand the language of functions: - domain and range, - odd and even, - periodic, - symmetry, - zeroes, - intercepts, and - know the eact values of the trigonometric functions of common angles such as 0,,,, 6 4
3 Students must be familiar with pre-calculus topics. 1. Functions, Graphs, and Limits Analysis of graphs using technology and calculus to predict and to eplain the observed local and global behavior of a function.. Limits of functions (including one-sided its) Calculating its from graphs or tables of data or using algebra. Asymptotic and unbounded behavior Understanding asymptotes in terms of graphical behavior, describing asymptotic behavior in terms of its involving infinity, comparing relative magnitudes of functions and their rates of change, and understanding the very significant behavior of functions that the asymptotes reveal. 4. Continuity as a property of functions Understanding continuity in terms of its and geometric understanding of graphs of continuous functions. 5. Parametric, polar, and vector functions. 6. Derivatives The concept of the derivative presented geometrically, numerically, and analytically, and is interpreted as an instantaneous rate of change. Understand the definition of a derivative as the it of the difference quotient. Understand the relationship between differentiability and continuity. 7. Derivative at a point. Understand the slope of a curve at a point, the tangent line to a curve at a point, and local linear approimation. Understand instantaneous rate of change as the it of average rate of change. Approimate rate of change from graphs and tables of values. 8. Derivative as a function Corresponding characteristics of graphs of and Relationship between the increasing and decreasing behavior of f() and the sign of f ().
4 Part I: First, let s whet your appetite with a little Pre-calc! 1) For what value of k are the two lines ky and y 1 (a) parallel? (b) perpendicular? ) Find the value of y for which the line through A and B has the given slope m: A(, ), B(4, y), m = / ) Consider the circle of radius 5 centered at (0, 0). Find an equation of the line tangent to the circle at the point (, 4) in slope intercept form. 4) Graph the function shown below. Also indicate any key points and state the domain and range. 4, f ( ),, 1 1 5) Write a piecewise formula for the function shown. Include the domain of each piece! 4
5 For #6-7, parametric equations are given. Complete the table and sketch the curve represented by the parametric equations (label the initial and terminal points as well as indicate the direction of the curve). 6) 4sin t, y cos t, 0t t y 7) t5, y 4t7, t t y 8) Graph the function ye and indicate asymptote(s). State its domain, range, and intercepts. 9) Write a formula that epresses the first variable as a function of the second: the area of a circle as a function of its diameter 10) Consider the function y 1. (a) (b) State the domain. State the range. (c) Draw its graph. 11) Algebraically determine whether the function is even, odd, or neither by stating and using their respective definitions: y 1 5
6 1) Draw the graph of the function labeling key points. Then find its (b) domain and (c) range. 4, f ( ) (/ ) /,, 1 1 1) If f ( ) 5, g ( ) find each of the following and circle your answers. (a) f g( ) (b) g f ( ) (c) f g (0) (d) g f (0) (e) gg( ) (f) f f () 14) Begin with a circular piece of paper with a 4-in. radius as shown in (a). Cut out a sector with an arc length of. Join the two edges of the remaining portion to form a cone with radius r and height h, as shown in (b). a) Eplain why the circumference of the base of the cone is 8. b) Epress the radius r as a function of. c) Epress the height h as a function of. d) Epress the volume V of the cone as a function of. 6
7 15) The half-life of phosphorus- is about 14 days. There are 6.6 grams present initially. a) Epress the amount of phosphorus- remaining as a function of time t. b) When will there be 1 gram remaining? Solve algebraically. 16) Determine how much time is required for an investment to triple in value if interest is earned at the rate 17) Solve the equation algebraically. Support your solution graphically. a) b) e e t 18) Solve for y algebraically. a) ln y t 4 ln y 1 ln ln b) 19) Without a calculator, sketch one cycle of the curve y 4sin and identify and amplitude and period and label your aes. 0) Algebraically, solve each equation in the interval 0. a) tan 1 b) csc c) 1 cos 1) Evaluate the epression in eact form: sin cos ) The height of a cylinder is twice the diameter. Epress the total surface area A as a function of the height h. 7
8 ) A light m above the ground causes a boy 1.8 m tall to cast a shadow s meters long measured along the ground, as shown. Epress s as a function of d, the boy s distance in meters from the light. 4) A bo with a square base has a surface area (including the top) of m. Epress the volume V of the bo as a function of the width w of the base.water is flowing at a rate of 5 m /s into the conical tank shown at the right.find the volume V of the water as a function of the water level h. 5) As shown, rectangle ABCD has vertices C and D on the -ais and vertices A and B on the part of the parabola y9 that is above the -ais. a) Epress the perimeter P of the rectangle as a function of the -coordinate of A. b) What is the domain of the perimeter function? 8
9 Part II: United and Continuous! For #1-4 below, find the its, if they eist. 1) ) 9 9 ) ) 8 For #5-7, eplain why each function is discontinuous and determine if the discontinuity is removable or nonremovable. 5), g ( ) 5, (1) 6) b ( ) 5 7) h ( ) For #8-1, determine if the following its eist, based on the graph below of p(). If the its eist, state their value. Note that = - and = 1 are vertical asymptotes. y 8) p ( ) 1 9) p ( ) 10) p ( ) 11) p ( ) 1) p ( ) 1) p ( ) 1 9
10 k 5 14) Consider the function f ( ), 5sin 5 In order for the function to be continuous at = 5, the value of k must be 15) Consider the function sin 0 f ( ). k 0 In order for the function to be continuous at = 0, the value of k must be Use the graph of f(), shown below, to answer # ) For what value of a is a f ( ) noneistent? 17) f ( ) 18) f ( ) 10
11 Part III: Designated Deriving! 1) h 0 tan 1 (1 h) 4 h ) h 0 sec( h) sec( ) h For #-8, find the derivative. ) y ln(1 e ) 4) y csc( 1 ) 5) y e 6) (tan )( ) y 4 7 7) f ( ) ( 1) e 8) e f ( ) 9) Consider the function f ( ). On what intervals are the hypotheses of the Mean Value Theorem satisfied? dy 10) If y y 5, then d 9 11) The distance of a particle from its initial position is given by s ( t) t 5, where s is feet and t is ( t 1) minutes. Find the velocity at t = 1 minute in appropriate units. 11
12 Use the table below for #1-1. f () g() X ½ 7-4 f () g( ) -1 d d f 1) The value of ( f g) at = is 1) The value of d d g at = 1 is In #14-15, use the table below to find the value of the first derivative of the given functions for the given value of. X 1 0 ¾ f () g() 7-4 f () 1 g( ) -1 14) [ f ( )] at = is 15) f ( g( )) at = 1 is Free Response Let R be the region in the first quadrant under the graph of 1 y for 4 9. a. Find the area of R. b. If the line = k divides the region R into two regions of equal area, what is the value of k? c. Find the volume of the solid whose base is the region R and whose cross sections cut by planes perpendicular to the -ais are squares. 1
13 . The figure above shows the graph of real numbers such that f ', the derivative of a function f. The domain of f is the set of all a. For what values of is b. For what values of is f f increasing? decreasing? c. For what values of does the graph of f have a horizontal tangent? d. For what values of in the interval 10,10 does f have a relative maimum? Justify yor answer. 1
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