Dear AP Calculus BC student, Hello and welcome to the wonderful world of AP Calculus! I am excited that you have elected to take an accelerated mathematics course such as AP Calculus BC and would like to welcome you to what will surely be a challenging yet rewarding year of mathematics. AP Calculus BC is intended for students who have a thorough knowledge of algebra, geometry, trigonometry and basic functions. The functions you must be familiar with include linear, polynomial, exponential, logarithmic, trigonometric, and piecewise-defined. In particular, before studying calculus, you must be familiar with the properties, algebra, and graphs of functions. Knowledge of the basic trigonometric identities is also essential, along with the values of the trigonometric functions of common angles. I would like to help you get a good start in AP Calculus BC. These problems are to be completed over the summer, and it is my expectation that all problems will be thoughtfully attempted, with all work and justification shown. Additionally, there is a significant amount of memorization expected for success on the AP exams. It is expected that students will return to school having mastered those procedural basics to permit more time to deeply understand the critically important conceptual aspects of the course. Also attached are 4 mostly new and skill-based calculus topics students are expected to master prior to the start of the school year. Please plan to devote approximately 2 hours of work to the review questions and at least 1 hour of work to each of the 4 new content topics. I look forward to supporting you in a successful and rewarding year in mathematics! Sincerely, Ms. Gersowsky Anticipated workload: 6 hours Summer Packets are due Thursday, August 24, 2017 Summer Assignment Quiz (including a unit circle quiz) the same day Serving the communities of Antioch, Lake Villa, Lindenhurst, and Old Mill Creek Community High School District 117, being a community of learners with a vision of excellence, is committed to providing an educational experience that encourages all learners to develop to their fullest potential, to engage in lifelong learning, and to be responsible members of society.
Algebra Key Concepts 1. Write the equation of the line that passes through the points ( 3, 1 ) and ( -5, -3 ) in point-slope form then convert it to slope-intercept form. 2. Write an equation of the line that is the perpendicular bisector of the segment connecting an x-intercept of 2 and a y-intercept of 6 then convert it to slope-intercept form. 3. Find the points of intersection of the line y = x + 1 and the parabola y = x 2 11 using algebraic methods. Show work. 4. If f x = $ % and g x = x' + 2x find f g x and g f x. Geometry Key Concepts 5. If the area of a rectangle is x * 3x 4 and the height of the rectangle is x + 1, find an expression for the length of the base. 2
6. If the volume of a sphere is 36π V = 0 ' πr', find the surface area SA = 4πr *. 7. Find the volume of a cone (V = $ ' πr* h) if the ratio of the height to the radius is 3:1 and the cone has a height of 12 cm. 8. Find the area of the larger trapezoid if 9. Find the shaded area if the circles are concentric, the shapes are similar. and the circumference of the larger circle is 58π, and AB, which is tangent to the smaller circle, is 42 units. 4 5 7 6 A C B Advanced Algebra Key Concepts Solve for all values of x, where x is a real number. Do not use a calculator unless directed. Leave answers exact. 10. 1 + 2x 6 7 = 0 11. 27 *% = 9 %;' 3
12. log * x 5 = 3 13. e % = cos x *Use a calculator. Simplify without a calculator: 7 14. ln e * 15. 5a E 7 4a 7 E 16. log F (2x) + 3 log F (y) 17. sec arcsin N $O Rewrite using exponent properties 18. (3x * ) 0 19. %PE Q P7 R T 20. 3x 21. b F 22. '% U 23. % $* % PV 4
Sketch a quick graph of the function labeling key values (calculator allowed) and determine whether the function is even or odd (explain what feature on the graph indicates even v. odd): 24. g x = 3 cos % * 25. f x = x ' x 26. f x = 2 x, x 1 x *, x > 1 a. Graph the function. Label the coordinates of at least three points on each graph. b. Domain: c. Range: d. f 2 = e. f f 7 = Graph each function. Label any vertical and horizontal asymptotes, holes, x and y-intercepts: 27. h x = *%[0 % E ;%;F 28. k x = ln(x + 2) 11 10 9 8 7 6 5 4 3 2 1 y 11 10 9 8 7 6 5 4 3 2 1 y 11 10 9 8 7 6 5 4 3 2 1 1 1 2 3 4 5 6 7 8 9 10 x 2 3 4 5 6 7 8 9 10 11 11 10 9 8 7 6 5 4 3 2 1 1 1 2 3 4 5 6 7 8 9 10 x 2 3 4 5 6 7 8 9 10 11 Domain: Range: Domain: Range: 5
29. Fill out the blank unit circle entirely. Be sure to include both degrees and radians for each angle. 30. cot O^F 31. csc N^' 32. sin '^0 33. sin ;$ ( ' * ) 34. csc ;$ (π) 35. sec ;$ 2 6
THE FOLLOWING PAGES CONTAIN NEW CONTENT TO MASTER BEFORE SCHOOL STARTS J THE TOPICS ON THE FOLLOWING PAGES ARE MOSTLY NEW. HONEST COLLABORATION WITH FRIENDS IS ENCOURAGED. PLEASE CONSULT THE VIDEO LINK PRIOR TO ASKING FOR HELP. WITH THAT SAID, PLEASE ASK FOR HELP. CONTACT MS. GERSOWSKY THROUGH EMAIL: SARAH.GERSOWSKY@CHSD117.ORG DON T PROCRASTINATE! BECAUSE THE CURRICULUM OF AP COURSES IS FAIRLY UNIVERSAL, EXCELLENT RESOURCES AND VIDEOS ARE AVAILABLE ONLINE. I D SUGGEST STARTING HERE: http://patrickjmt.com/#calculus EVERY SINGLE CALCULUS STUDENT WILL NEED HELP AT SOME POINT THIS YEAR LETS GET OFF ON THE RIGHT FOOT AND ASK FOR HELP EARLY AND OFTEN! 7
Learn Concept #1 Limits 1. How to evaluate one-sided limits 2. How to evaluate two-sided limits 3. How to evaluate limits at infinity 4. When a limit equals infinity v. negative infinity v. does not exist (hint: not the same thing) Watch https://www.youtube.com/watch?v=zpcbkrpqhqq, http://patrickjmt.com/topic/limits/page/2/ Try Limit hints try the following approaches in order: 1. Plug it in, plug it in substitute the x value into the function. If the solution is a real number, congrats, you re done. If it is _,, 0 and other confusing forms called indeterminate _ (http://en.wikipedia.org/wiki/indeterminate_form). It does NOT equal 0! It means the value cannot be determined without further algebraic analysis. 2. If substituting the x-value yields an indeterminate solution, try one of the algebraic strategies below depending on the function form: a. Rational functions factor the numerator and denominator, simplify, and substitute again. b. Compound rational expressions work towards common denominators for fractions being added or subtracted then refer to (a). c. Radical expressions multiply by the conjugate of the radical expression (i.e. a + b has a conjugate of ( a b)). 3. Analyze the limit graphically. 0. Sketch a graph of f x = hij % %. a. Find lim % _ f(x) from the graph. b. Find lim % m f(x) from the graph. c. Memorize either the two limits above or the graph of f x above. hij *% d. lim % _ 0% (Graph and make logical conclusion about future hij n% o% limits. Evaluate each of the following limits without a graphing calculator using the hints above: 1. lim % ;* x' 2x * + 1 2. lim % ;* % E [$ '% E ;*%[p 3. lim % _ 6 Eqr ;6 E % 4. lim *%E [' % m p% E [O 5. lim %U[%E % m $*% 7 [$*N '% 6. lim E ;*%;$ % ;m % E ['%;0 8
7. lim %[hij % % ;m %[sth % 8. lim '%E ;*%;$ % $ % E ['%;0 9. lim u p u[0;' u;p 10. lim u m 'u[* pu E [$ + 4 {Use a graph} $ 11. lim % _ % 12. lim % _ $ % E 13. Given the function f x whose graph is shown below (ticks marks in picture have scale 1): a) lim f x = e) f 3 = % ' q b) lim f x = f) f 1 = % ' P d) lim f x = % ;$ P g) lim % ;$ f x = e) lim f x = h) lim % ;$ q % ' f x = 14. Sketch a graph of a function h(x) that has the following properties: a. lim % ;m h x = 0 b. lim h x = % ;* P c. lim % ;* q h x = d. lim % _ h x = 0 e. lim % m h x = 1 15. Sketch a possible graph for a function f that has the stated properties. f(-2) exists, the limit as x approaches -2 exists, f is not continuous at x = -2, and the limit as x approaches 1 does not exist. 9
Concept #2 Definition of Derivative Learn 1. What a definition of derivative represents with respect to rate of change and the function 2. How to apply the standard definition of derivative lim z %[y ;z %;y lim y _ %[y ; %;y, etc. 3. How a derivative compares to an average rate of change z %[y ;z % y _ %[y ;% or other definitions Watch http://patrickjmt.com/understanding-the-definition-of-the-derivative/, http://patrickjmt.com/thedifference-quotient-example-2/ Try Use the definition of derivative f z %[y ;z % x = lim to determine f (x) for each of the functions below. Use f (x) y _ %[y ;% to find the slope of the function at the x-value given, then write a tangent line in point-slope form. 1. f x = 3x 1 (hint: if you understand what a derivative is, you don t even need to use the limit for part a) a. Find f (x). b. Find f (3). c. Find the equation of a tangent line at x = 3. d. Sketch f x and the tangent line at x = 3. 2. f x = x * a. Find f (x). b. Find f ( 2). c. Find the equation of a tangent line at x = -2. d. Sketch f x and the tangent line at x = -2. 10
3. f x = 2x * 5x a. Find f (x). b. Find f (1). c. Find the equation of a tangent line at x = 1. d. Sketch f x and the tangent line at x = 1. 4. f x = 2x ' (hint: a + b ' = a ' + 3a * b + 3ab * + b ' ) a. Find f (x). b. Find f ( 1). c. Find the equation of a tangent line at x = -1. d. Sketch f x and the tangent line at x = -1. 5. Carefully compare the original functions f(x) in each of the problems above to the derivatives f (x). There is a fairly simple and incredibly powerful pattern called the Power Rule. Try to describe it in 1-2 sentences below. 11
Concept #3 Power Rule Learn 1. How to quickly determine the derivative of a polynomial function. 2. How to apply the Power Rule for simple rational and radical functions. Watch https://www.youtube.com/watch?v=2cvcfkgsunc, http://youtu.be/wrw5wvgz3rs Try 1. Once you ve identified the Power Rule, try to write it in the box below, the find f (x) for each function below by applying the Power Rule. Power Rule If f(x) = x, then f (x) = a. f x = 4x 2 b. f x = 4x * 7x + 2 c. f x = 4x * 7x + 200000000 d. f x = πx $_ e. f x = x p + x 0 + x ' + x * + x + 1 f. f x = $ % (hint: express $ % in the form xn first) g. f x = p % 7 h. f x = x 7 i. f x = x * j. f x = $ 7 % 2. f x represents the second derivative of the function f(x) the derivative of the derivative. Similarly, f (x) represents the third derivative, and for the fourth derivative and higher (since the apostrophes would get a bit ridiculous) derivatives are expressed as f 0 (x), etc. Find f x, f x, f 0 x, f p x, and f F (x) for the function in (e) above. 12
For the three functions below, find the average rate of change (ARoC = z o ;z n ) and instantaneous rate of change o;n (f x using the Power Rule), then use f (x) to write a tangent line and find where the slope has a particular value. 3. b x = 5x * 2x Average RoC over [ 0, 2 ]? Instantaneous RoC expression: X-value where slope is -12? Tangent line @ x = 0? 4. e x = x ' 5x Average RoC over [ -3, 3 ]? Instantaneous RoC expression: X-value where slope is 7? Tangent line @ x = 1? 5. f x = 0 % + 3 Average RoC over [ 1, 2 ]? Instantaneous RoC expression: X-value where slope is -1? Tangent line @ x = 4? 6. Find the point(s) on the curve y = 2x ' 3x * 12x where the tangent line is horizontal. 13
Concept #4 Continuity and Differentiability Learn 1. How to determine where a function is continuous and differentiable 2. How to solve for a missing value in a piecewise function given that the function is continuous and differentiable 3. What types of discontinuities exist and what causes functions to not be differentiable Watch http://youtu.be/youixplqdr0, http://youtu.be/vuem6vwjve4, http://youtu.be/ei9p_h0ml_w Try Definition of Continuity: The function f is continuous at some point c of its domain if the limit of f(x) as x approaches c through the domain of f exists and is equal to f(c). In mathematical notation, this is written as In detail this means three conditions: first, f has to be defined at c. Second, the limit of that equation has to exist. Third, the value of this limit must equal f(c). The function f is said to be continuous if it is continuous at every point of its domain. Primary discontinuities are point (hole), jump, infinite, and oscillating discontinuities. Definition of Differentiability: In calculus, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. As a result, the graph of a differentiable function must have a non-vertical tangent line at each point in its domain, be relatively smooth, and cannot contain any discontinuities, corners, vertical tangents, or cusps. 1. Find the value of A so that the function is continuous. (Hint: The two parts need to have the same y-value at the x- value where they would meet.) g x = 5 Ax*, x 1 4 + 3x, x > 1 2. Determine the location(s) where h(x) is not differentiable and what feature makes it not differentiable. h x = 1, x < 1 x *, 1 x 0 x + 1, x > 0 14
4. Draw a sketch of a graph that meets the following requirements. The limit does not exist as x approaches -2. The functional value at x = 3 is equal to the limit of f as x approaches 3 but the function is not differentiable at x = 3. The function increases without bound as x decreases without bound. 5. Determine the values of A and B that make the function continuous and differentiable at x = -2: g x = x* + 8x 7, Ax + B, x 2 x > 2 6. Let f x = x + 1. Which of the following statements are true about f? I. f is continuous at x = 1 II. f is differentiable at x = 1 III. f has a corner at x = 1 (A) I only (B) II only (C) III only (D) I and III only (E) I and II only 7. Which of the following is true about the graph of f x = x */p at x = 0? (A) It has a corner. (B) It has a cusp. (C) It has a vertical tangent. (D) It has a discontinuity. (E) f 0 does not exist 8. Sketch a graph of a continuous function b(x) y if b 3 = 1 and: b x = 1, x < 2 1, 2 < x < 1 2, 1 < x 6 5 4 3 2 1 6 5 4 3 2 1 1 1 2 3 4 5 6 x 2 3 4 5 6 15