AP Calculus BC Summer Assignment 018 Name: When you come back to school, I will epect you to have attempted every problem. These skills are all different tools that we will pull out of our toolbo at different points in the year to help us solve calculus problems. We will spend a few days working together on the items students struggled with before moving on. Shortly into the school year, you will turn this in and we will take a prerequisite test over this material. Should you lose this assignment, it can be found on my web page or the math department web page. If you run into any problems on this assignment over the summer, feel free to e-mail me. My e-mail address is NADoolin@bluevalleyk1.org. I know this is a long assignment, so hear me out. We finished calculus A before leaving for summer but limits and derivatives are the foundation of the rest of calculus B and C. We will have a hard time in the fall if we ve forgotten the A topics or if our algebra skills have diminished. The BC part is longer than the AB parts, so plan accordingly. If you spend 0 minutes a day working on this starting August 1 st, you should have no problem finishing by the start of the school year. All assignments should be completed on separate paper or in a notebook. Any graph sketches should include the basic shape of the function and any relevant characteristics (asymptotes, ma/min, etc) should be clearly labeled. Round any approimate answers to three decimal places. Part 1 Selections from the AB Calculus Assignment First, I have taken four sections from the AB calculus assignment. Each tutorial contains some definitions and some eample problems. These are provided since you do not have a summer tetbook. The end of the tutorial will have a homework section. Feel free to start with the homework and reference the tutorial if you need help. All of the material you ve seen before in previous courses, but possibly adapted in different ways (calculus is the course that puts everything together). I have kept the original numbering so you can work together with AB students if you wish. The tutorials are: Tutorial # Selected Problems on Composition of Functions Tutorial #4 Eponential and Logarithmic Functions Tutorial #5 Trigonometric Epressions and Equations Tutorial #6 Comple Algebraic Simplifications Afterwards, part two of this handout is labeled Part BC Assignment. These problems are not broken down by section but you should be able to reference your notes from Accelerated Pre-Calc on these sections (Chapter limits, Chapter derivatives, Chapter 4 applications of derivatives). FYI - The BC section is longer than the AB section. I look forward to seeing you in the fall! 1 -Mr. Doolin
Tutorial # Selected Problems on Composition of Functions In Calculus, we will frequently use tables of values to learn more information about functions. In the table you are given below, you are given si selected -values, and corresponding values of f() and g(). For eample, if you were to look at the column where = 0, you would see that f() = - and g() =. This means that the point (0, -) is on the graph of f() and the point (0, ) is on the graph of g(). It also means that f(0) = - and g(0) =. Use this information to help answer the questions below. Questions like 8-10 are function composition questions not included in the tutorial, but if you ve forgotten, try google searching Composition of Functions. Suppose f() and g() are continuous functions with domains of [0, 5]. 0 1 4 5 f( ) - 4 6 8 11 g ( ) 5-6 1 0-4 6. Compute f(1) g(1). 7. Compute f (1) g() f () g(1). 8. Compute f( g (0)). 9. Compute g( f (1)). 10. Compute f( g (4)). Use the table to answer questions 6 1.
Tutorial #4 Eponential and Logarithmic Functions Concept 1 Eponential Functions In Calculus, we will deal a little bit with eponential functions. An eponential function is defined by the equation, f ( ) a. In this definition, a must be positive and not equal to 1, while must be a real number, not equal to zero. Eample 1: Let f( ) 16. Evaluate f (0), f (), f (1/ ), f ( 1),and f ( / 4). Solutions: 0 f (0) 16 1. (Anything to the zero power is 1.) f () 16 56. 1 f 1 16 16 4. (Remember that anything to the ½ power is the same as finding the square root.) f 1 1 1 16. 1 1 16 16 (Any negative eponent can be made positive by moving the base and the power to the other part of the fraction.) f 1 1 1 1 4 16 16 8 / 4 16 / 4 4 Eample : Let f( ), and (a) Find ( g( f (1)). g 1 ( ). Solution: Since you want to calculate g(f(1)), you need to find f(1) first. Once you find f(1), substitute 1 1 that result into g. f(1). g() 8. g (b) Find ( ). f 1 g 1 1 Solution: ( ). f bases, you may subtract the eponents. Since you are dividing two eponential epressions with like
Eample : Solve the eponential equations: (a). Solution: This eponential equation can be solved easily by equating bases. Since both the left and right hand side of the equations can be epressed as powers of, rewrite as 5. Thus, you can rewrite the 5 equation. Thus, if you try and match both sides of the equation = 5. (b) 4 8 Solution: You can rewrite both sides of the equation as powers of. Rewrite 4 as. Then, you d have. Then =, so = /.. Rewrite 8 as (c) 8 5 1 16 Solution: Again, rewrite both sides as a power of. (5) 1 4, then 6 + 15 = -4, 4 19. 6 The main motivation for logarithmic functions stem from problems similar to what you saw in eample. In eample, we were able to solve all of our problems by simply equating bases. However, what happens in the situation where bases can t be equated? For eample, what happens if we want to solve 5 17? Clearly, we can t rewrite 17 as a power of 5. When mathematicians encountered this problem, they knew that getting the eponent out of the variable was a must. The way they removed the eponent from the variable was by taking either the logarithm or natural logarithm of both sides (it doesn t matter which one you use as long as you use it on both sides of the equation.) Logarithmic functions are the inverse functions of eponential functions. In other words, if y a, then log ( y a ). Eample 4: Rewrite the following eponential equations in logarithmic form. a) b) 5 1 c a b 5 Solution: 1 1 5 5 5 log5. Solution: b a c log b c. a Properties of logarithms: b a ln( a ) bln a ln( a) ln( b) ln( ab) ln( a) ln( b) ln b ln( e) 1 ln(1) 0 f ( ) ln b log b ln( e ) f ( ) loga b ln a log a 4
Eample 5: Rewrite the following epressions using properties of logarithms. e a) ln e 1 b) e e e ln ln ln 1. e e 1 1 Solution: e ln ln( e) Solution: Eample 6: Solve the following equations: a) 5 7 1 1 1 1 ln ln( ) ln( ) ln( ) e e e e Solution: Start by taking the logarithm or natural log of both sides. Then, we will have ln(5 ) ln(7). Now, use a property of logarithms to pull the eponent out front of the lefthand epression. So, then we ll have ( ) ln 5 ln 7. Then, solve for, and you ll get ln 7 ln 7 ln 7 ln 5. ln 5 ln 5 b) ln 4 ln(5) 4 Solution: First use properties of logs to rewrite ln( 4) ln(5) as ln. Then, we 5 4 want to rewrite to eliminate the natural log. Thus, e. If we want to solve this for, 5 we d need to start by multiplying through by the denominator Eample 7: Solve for y: ln y ln. 5 5e 4 5e 4. Solution: This equation is tricky to solve, but we are first going to need to get rid of the negative on the left-hand side of the equation. So, let s divide both sides through by -1. This gives us ln y ln. Then, we need to undo the natural log on the left hand side of the equation. This means we now have the statement y e ln. Net, we are going to make a funny simplification. Since we have added eponents in the right hand side of the equation, we can ln rewrite that side to have multiplied bases. In other words, e e e ln. This is useful because ln ln we know that e. So, e e. This means our overall equation is y e. Net, we need to undo the absolute value bars. We do this by recognizing that the right hand side of the equation may be either positive or negative, so y e. Isolating y gives us y e.
Homework for Tutorial #4: 1. Eplain why f ( ) e is an eponential function while f ( ) is not an eponential function.. Eplain why f ( ) a is not an eponential function in the situations where a = 1 or where = 0. What kind of function would f() be in these situations?. Let f( ) 8. Evaluate f (0), f (), f ( ),and f ( 5/ ). 1 4. Evaluate log. 7 5. Simplify the following epressions using properties of logarithms: 8 11 a (a) ln( e ) ln( e ) (b) log ( ) (c) ln( e) ln(1) (d) 1 ln y e 6. Solve the following equations for y: (a) y ln y ln (b) e e 5 4 e 7. Find f(1) f(0) if f( ). 4 1 8. Evaluate when = 0: 4 e (1 e ) 6
Tutorial #5 Trigonometric Epressions and Equations Eample 1: Use the unit circle to evaluate the following epressions: (a) (b) (c) (d) 5 1 cos. 7 cos 7 6 cot. 6 7 1 1 sin 6 19 1 7 5 1 csc csc csc csc csc / sin(0) 0 tan(1 ) tan(10 ) tan(8 ) tan(6 ) tan(4 ) tan( ) tan(0) 0. cos(0) 1 Eample : Evaluate the following inverse trigonometric functions. (Remember that inverse trigonometric functions use ratio as inputs, and output angles.) (a) sin 1. (Keep in mind that the range of arcsine is from,. ) The angle in the range that has a sine of / is 60 or /. 1 (b) tan. (Keep in mind that the range of arctangent is from,. ) The angle in the range that has a tangent of is the angle that has a sine of / and a cosine of 1/. (c) cos 1 1 (Keep in mind that the range of arccosine is from 0,. ) The angle in this range that has a cosine of -1/ is /. 7
(d) sin 1 4 (Keep in mind that the range of arcsine is from,. ) The angle in this range that has a sine of / is / 4. Many of you will want to use either 5 7 or as the answer here. However, both of 4 4 these are wrong, because they don t fit into the range. Homework from Tutorial #5: 1. Fill in the blank unit circle label each angle in degrees and radians, then label each ordered pair. You may use this one if you like, or print off one from google. You are required to have the unit circle memorized! 8
. Evaluate the following without the use of a calculator. (a) 7 cos 6 (b) cot (c) csc 4 (d) tan( ) (e) cos 1 (f) 1 tan 1 (g) sin 1 1 (h) cos 1. Eplain the difference between y csc and y 1 sin.. Evaluate the following epressions, if f ( ) sin( ). (a) Find f. (b) Find f. 4 (c) Find f. 18 9
Tutorial #6 Comple Algebraic Simplifications: f ( ) g( ) h( ) j( ) Eample 1: Determine what f ( ) is if f g e h e j ( ) 1, ( ), ( ), ( ). Solution: By substitution, we immediately have After distributing, we have f g h j e e f ( ) ( 1) ( ) ( ) ( ) ( ) ( 1)( ) ( )( ). ( 1) e e e e e Combining like terms and factoring the numerator gives us. e e ( 1) e. ( 1) Eample : Solve: e e 0. Solution: Start by factoring the common factor out of each term. Some of you will notice that the common factor here is e. However, the process is doable even if you only recognize e as the common factor. Factoring out e 0. Setting each term equal to zero, makes 0, which we e gives us can solve by factoring out an, yielding, ( ) 0. So, we have solutions at 0,,. Setting e 0 produces an interesting case, as anything raised to a power won t ever equal zero!! So our only solutions are 0,,. Comment: You will make very heavy use of eponent properties in this section. A review of these properties are listed here. n 1 Negative Eponent Property: a n. a Raising a quantity to a negative eponent is equivalent to writing one over the quantity with a positive eponent. m / n n m n Rational Eponents Property: a a a m Raising a quantity to a rational eponent is the same as raising the quantity to the power of the rational eponent s numerator and then taking the root of that quantity that corresponds to the number of the denominator. Eample : Rewrite 7 5r ( r 1) so that no variables are in the denominator and no radicals are in your answer. / 7 5r Solution: First rewrite the denominator as ( r 1). So, we have. Then to move that quantity / 7 ( r 1) out of the denominator, we need to rewrite it with a negative eponent, and this is our final answer: 5r 5rr1 / 7 / 7 ( r 1) 10
Homework from Tutorial #6: f ( ) g( ) h( ) j( ) 1) Determine what f ( ) is. Simplify your answer as much as possible. [See e 1] a) Let b) Let f h g j ( ) 5, ( ), ( ), ( ). f h j e g e ( ), ( ), ( ), ( ). ) Solve the equation employing techniques you know (factoring, common denominators, etc.): a) e e 0 b) 4 5 5 5 e 4 e = 0 c) 1 0 ) Simplify each epression using the properties of eponents. 7 4 a) 4 5 b) 1 y c) 1a b z 9ab 5 0 4) Rewrite the radical epressions using eponents. Your answers will contain neither radicals nor fractions. Your answers, however, may contain negative eponents. (In fact, some of them should ) a) 5 b) 9 5 7 c) 1 d) 9 1 e) 7 5) Rewrite each radical as something to a rational eponent (rational eponents are just fractional eponents), then use properties of eponents to combine like terms: a) y y b) z z z c) 6) Rewrite the following epressions so that no variables are in the denominator. 5 a) 7 1 b) 4 8 c) ( ) d) e 7) Solve for y: (a) y (b) 8 5 1/ y 5 1/ / 8) Calculate the value of C (without using a calculator) if () 8 C 11
Part BC Assignment Directions: Complete all problems on separate paper! These will be due within the first two weeks of the school year, and will be accompanied by an eam. Part I Non-Calculator Problems 1. Evaluate the following limits: 8sin( ) a) lim b) lim 10 0 4 e) (7 ) 49 lim 0 f) 45 lim 5 1 c) lim sec 5 0 4 5 g) lim 5 1 d) h) 4 lim 0 4 5 lim 4. Find the value(s) of c for which f() is continuous. Use proper notation. f( ). Eplain why the following function will not be continuous for any value of c: c 6 f ( ) 5 6 c 6 4. Use the graph below to compute the following: c 5 6 c c 6 6 Suppose the function f() has a domain of [-, ] and is represented by the graph below. Answer the questions below. (a) Determine where f() is discontinuous. Give the type of discontinuity that occurs at these values. (b) Fill in the chart below: *For eample, in the - column, write down the limit as approaches - from the left, from the right, and overall, and also determine f(-). Left Right Overall f(#) - -1 0 1 1
5. Find the values of a and b that make the function differentiable. a 4 f( ) b continuous and 6. Compute the following limits [The definition of the derivative might be helpful here]: (a) csc h 4 lim h 0 h (b) lim h 0 7 7 4 4 h h 7. Use the table below to answer questions about f(). Assume that f() is continuous over the interval [1, 7] and differentiable over the interval (1, 7). 1 4 5 6 7 f() - -5-1 1 8 f '( ) 5 0 1 4-5 (a) Write an equation of the secant line to f() over [, 6]. (b) What is the minimum number of zeros that f() must have over this interval? Why? (c) Write an equation of the line tangent to f() at = 4. (d) Consider the interval (1, 7). Find the value of c, the value for which the mean value theorem applies. (e) Does there eist a value k such that f "( k) 0? Why or why not? dy 8. Find. d 4 5 a) y 6 sin( ) b) y e c) 5 d) y sec e) y ln(cos ) f) y 5y 5 ( e ) 8 y g) sin y sin 4 h) y 7 4 i) y 1 tan (4 ) j) ln k) y y 5e cot y ( 4) m) l) 7 5 7 5 5 y y y 1
9. Suppose the motion of a particle along a horizontal line is modeled by the equation v( t) t (t ), t 0. a. Determine when the particle is moving to the left, to the right, and changing direction. b. Calculate the average acceleration of the particle over the interval [, 5]. c. Find an epression giving the acceleration of the particle at time t. d. Determine when the particle is speeding up and when its slowing down. 10. Given y y 16, answer the questions below. dy a. Find. d b. Determine all points on the curve where the tangent line is vertical. c. Determine all points on the curve where the tangent line is horizontal. 11. Given the table of values below, and the fact that f and g are inverses, write the equation of the line tangent to g at =. f() f '( ) g() 1-6 4 5 1 7 4 1. Answer questions relating to the table of values below. f() f '( ) g() g'( ) 1-6 9 4 5 1 0 7 4-4 d a. Calculate g at =. d d cos( 1) b. Calculate at = 1. d g( ) c. Estimate the value of f "(.5). d. Use the equation of the line tangent to f at = to estimate the value of f (.1). e. If f ' is decreasing for all values between = and =.1, is your answer to question (d) an overestimate, underestimate, or neither? Eplain your answer. 1. Let f ( ) ln. Determine when f() has a maimum, a minimum, when its decreasing, and when its increasing. Also, list any inflection points, and where the function is concave up and down. 14
14. Find the value of c that satisfies the mean value theorem if Eplain what happens at this value of c. 4 f ( ) 5 over the interval [0, ]. 15. Suppose f '( ) ( ) ( 4). Determine where f() has critical points, where f() is increasing and where f() is decreasing. Also, determine where f() has any inflection points, and where f() is concave up and concave down. 16. Suppose e 0 f( ). 1 0 a. Show that f() is differentiable at = 0. b. Determine all values of for which f '( ). c. Since f is always continuous and always differentiable, the mean value theorem actually applies. Find the value(s) of c that satisfy the mean value theorem over the interval [-1, ]. dy 17. Suppose (5 y). d a. Determine the equation of the line tangent to y at the point (, 1). b. Calculate of y. d y. d Use the second derivative to eplain why the graph below cannot be the graph c. Use the equation you found in part (a) to estimate (.). f Is your estimate an overestimate, underestimate, or neither? 15
Calculator Problems: t 18. A particle s moves along a horizontal ais with its velocity at time t is given by v( t) t sin for 0 t 15. a. Calculate when the particle changes directions. Determine when the particle is moving left and right. b. Determine the average acceleration of the particle over the interval [4, 7]. c. At t = 8, is the particle speeding up or slowing down? Eplain your answer. d. At t =, suppose the particle is 5 units left of the origin. Is the particle moving towards the origin or away from the origin at t =? Eplain your answer. 19. Suppose f 4 '( ) 6 5. a. Determine the -coordinates of any relative etrema for f. Justify your answer. b. Determine the -coordinates of any inflection points for f. Justify your answer. c. Suppose f () 5. Write the equation of the line tangent to f at =. 0. A rectangle has its base on the -ais and its upper vertices on the parabola maimum area that the rectangle can enclose. y 60. Find the 1. You are planning to make an open top rectangular bo from a piece of cardboard that measures 1 inches by 18 inches. The bo will be formed by cutting congruent squares from each of the corners and folding up the sides. What are the dimensions of the bo of largest you can make this way, and what is this volume?. A 5-foot ladder slides down a wall. At the instant when the base of the ladder is 7 feet from the wall, the ladder is sliding down the wall at the rate of 5 feet per second. a) At the same instant, find the rate at which the base of the ladder moves away from the wall. b) Suppose angle is the angle that the floor makes with the ladder. Find the rate at which angle changes at the moment in question.. A cylindrical shaped balloon is filled with air at a rate of 00 ft / min. At the moment when the radius of the balloon is 5 feet, the height of the balloon is 6 feet, and the radius of the balloon is increasing at a rate of 1 foot per minute. Find the rate at which the height of the balloon is changing. 16