Stone Bridge Math Department June 06 Dear Advanced Placement Calculus BC Student, Congratulations on your wisdom in taking the BC course. I know you will find it rewarding and a great way to spend your year. As always, I do not want you to stagnate or ferment those marvelous brain cells over the summer. Therefore, it would be beneficial for you to periodically refresh your memory on some math basics before beginning class in September. With this in mind, your first assignment is attached. This assignment is due no later than Friday, September, 06 with no eceptions. This is a 0-point grade. Your work/process is worth points so you must show all work to earn full credit. To encourage you to complete this during the summer, there will be a free quiz grade of 0 points given for turning this in on the first block day of your class, August 9, or August 0. This due date is fied, so if you will not be in school this day it is your responsibility to mail the assignment in to school with a postmark date matching the due date. Do the work independently first and then you may work together on the tweaking of your responses if you choose. Include any questions or comments you have on the material or the specific problems given. We will use this as a diagnostic instrument to determine where we need to begin our review of the AB content. The curriculum, pace, and rigor of the AP Calculus BC course is determined by College Board guidelines. It is suggested that all students enrolled in this course take the AP eam. This year, the eam is on Tuesday May 9, 07 and will require the use of a graphing calculator. Although a large portion of the course is a review/etension of AB topics, there is about 60% new content to cover. This year the eam will follow a similar format as the AB eam but there are some difference we will talk about. Since you are masters of the AB topics, I am sure that the little etras of the BC curriculum will not only be easy but also eciting as well. I am looking forward to a year of more calculus fun! Calculus BC is a demanding course and will require an average of approimately 90 minutes of homework time for every 90-minute block. Getting ahead during the first week of school will be advantageous. The AP Calculus course requires you to think mathematically, something some of you may never have done before. Do not be discouraged in the beginning if it takes you awhile to think mathematically. By the time May 07 rolls around, you will be ready. Throughout the course, I will Verbally NAG you. Verbally NAG is a way to remember the four ways that the College Board epects you to think mathematically. Verbally through written words and thoughts spoken out loud Numerical being able to understand charts, tables, etc and use them to answer questions Algebraic solving equations and word problems using algebra skills Graphical comprehending and interpreting graphs and using them to answer questions Have a restful summer and be ready to talk the Calc again in September. Please follow the links below to register for your SBHSMATH account and download the Summer Assignment. Sincerely, Mr. Marchio
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. Part I: First, let s whet your appetite with a little Pre-Calc! ) For what value of k are the two lines + ky = and + y = (a) parallel? (b) perpendicular? ) 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. ) Graph the function shown below. Also indicate any key points and state the domain and range. 4, f( ) = +, +, < > 4) Write a piecewise formula for the function shown. Include the domain of each piece! 4 5 6
5) Graph the function y = e and indicate asymptote(s). State its domain, range, and intercepts. 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, 0 t π t 0 π 4 π π 4 π π π y 7) = t 5, y = 4t 7, t t 0 y Part II: Unlimited and Continuous! For #-4 below, find the limits, if they eist. ) lim 4 7 4 4 ) lim 9 9 ) lim 5 + 4) lim + 8 + For #5-7, eplain why each function is discontinuous and determine if the discontinuity is removable or nonremovable. 5), < g ( ) = + 5, (+ ) 6) b ( ) = 5 7) h ( ) = 0+ 5 5
For #8-, determine if the following limits eist, based on the graph below of p(). If the limits eist, state their value. Note that = - and = are vertical asymptotes. y 4.0.0.0.0 6.0 5.0 4.0.0.0.0.0.0.0 4.0 5.0 6.0 7..0.0.0 4.0 50 8) lim p ( ) 9) lim p ( ) 0) lim p ( ) ) lim p ( ) ) lim p ( ) + ) lim p ( ) + k 5 4) Consider the function f ( ) = π, 5sin > 5 In order for the function to be continuous at = 5, the value of k must be 5) Consider the function sin 0 f ( ) =. k = 0 In order for the function to be continuous at = 0, the value of k must be 4
Use the graph of f(), shown below, to answer #6-8. 6) For what value of a is noneistent? 7) 8) Part III: Designated Deriving! ) ) 5
For #-8, find the derivative. ) y = ln( + e ) 4) y = csc( + ) 5) y e = (tan )( π ) 6) y = 4 7 7) f ( ) = ( + ) e 8) e f ( ) = 9) Consider the function f ( ) =. On what intervals are the hypotheses of the Mean Value Theorem satisfied? dy 0) If y y = 5, then = d 9 ) The distance of a particle from its initial position is given by s ( t) = t 5 +, where s ( t + ) is feet and t is minutes. Find the velocity at t = minute in appropriate units. Use the table below for #-. X f () g () f () g ( ) 4 5 7-4 - d d f ) The value of ( f g) at = is ) The value of d d g at = is 6
In #4-5, use the table below to find the value of the first derivative of the given functions for the given value of. X f () g () f () g ( ) 0 ¾ 7-4 - 4) [ f ( )] at = is 5) f ( g( )) at = is 6) Let f be the function defined by + sin π π f ( ) = for < <. cos (a) State whether f is an even function or an odd function. Justify your answer.. (b) Find f (). (c) Write an equation for the line tangent to the graph of f at the point ( 0, f (0)). Part IV: Derived and Applied! For #-, find all critical values, intervals of increasing and decreasing, any local etrema, points of inflection, and all intervals where the graph is concave up and concave down. ) 5 4 + 4 f ( ) = ) y = + 6 ) f ( ) = 5 5 + 7 4) 5 The graph of the function y = + sin changes concavity at = 5) Find the equation of the line tangent to the function 6) For what value of is the slope of the tangent line to y 4 7 = at 6 7 =. y = + undefined? 7
7) The balloon shown is in the shape of a cylinder with hemispherical ends of the same radius as that of the cylinder. The balloon is being inflated at the rate of 6 π cubic centimeters per minute. At the instant the radius of the cylinder is centimeters, the volume of the balloon is 44 π cubic centimeters and the radius of the cylinder is increasing at the rate of centimeters per minute. (The volume of a cylinder with radius 4 r and height h is π rh, and the volume of a sphere with radius r is π r.) (a) (b) At this instant, what is the height of the cylinder? At this instant, how fast is the height of the cylinder increasing? 8) Y O X A ladder 5 feet long is leaning against a building so that end X is on level ground and end Y is on the wall as shown in the figure. X is moved away from the building at a constant rate of ½ foot per second. (a) (b) Find the rate in feet per second at which the length OY is changing when X is 9 feet from the building. Find the rate of change in square feet per second of the area of triangle XOY when X is 9 feet from the building. Part V: Integral to Your Success! 8
) 8 d π / 6 ) sec π / 6 d d 4 d ) t dt 4) d e t dt d 0 sin( 4) 5) + 4 d csc 6) d cot 7) tan sec d 5 8) What are all the values of k for which d = 0? 4 9) What is the average value of y = + 9 on the interval [0, ]? b 0) If g( ) d = 4a + b, then [ g ( ) + 7] d = a b a k ) The function f is continuous on the closed interval [, 9] and has the values given in the table. Using the subintervals [, ], [, 6], and [6, 9], what is the value of the trapezoidal 9 approimation of f ( ) d? 6 9 f() 5 5 40 0 ) The table below provides data points for the continuous function y = h(). 9
0 4 6 8 0 h() 9 5 0 6 5 Use a right Riemann sum with 5 subdivisions to approimate the area under the curve of y = h() on the interval [0, 0]. ) A particle moves along the -ais so that, at any time t 0, its acceleration is given by a ( t) = 6t + 6. At time t = 0, the velocity of the particle is -9, and its position is -7. (a) Find v (t), the velocity of the particle at any time t 0. (b) For what values of t 0 is the particle moving to the right? (c) Find (t), the position of the particle at any time t 0. Part VI: Apply Those Integrals! For #-, find the general solution to the given differential equation. ) dy y = d + dy ) ysin d = ) Find the particular solution to the differential equation du dv sin = uv v if (0) u =. 0
4) The shaded regions, R and R shown above are enclosed by the graphs of g( ) =. f ( ) = and (a) (b) (c) Find the - and y-coordinates of the three points of intersection of the graphs of f and g. Without using absolute value, set up an epression involving one or more integrals that gives the total area enclosed by the graphs of f and g. Do not evaluate. Without using absolute value, set up an epression involving one or more integrals that gives the volume of the solid generated by revolving the region R about the line y = 5. Do not evaluate. R R 5) Let R be the region in the first quadrant under the graph of 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.