AP CALCULUS B/C GRADE 12. THE EWING PUBLIC SCHOOLS 1331 Lower Ferry Road Ewing, NJ Angelina Aiello

Transcription

1 AP CALCULUS B/C GRADE 12 THE EWING PUBLIC SCHOOLS 1331 Lower Ferry Road Ewing, NJ BOE Approval Date: 3/26/07 Written by: Don Wahlers Angelina Aiello Raymond Broach Superintendent

2 Section 7.4 Lengths of Curves Timeline: One Block Objectives: To derive an expression that can be used to find the length of a given curve To find the length of a curve analytically and with the use of a graphing calculator To solve construction and design problems involving the length of a curve o Activity : [C2] [C3] What is the length of the curve y = sin x from x = 0 to x = 2pi? a. Draw a diagram b. Break the whole into measurable parts c. Approximate each arc with the line segment joining its two ends. Label the y and x increments d. Find the length of the segment in terms of the increments e. Set up a sum that represents the length of the arc f. Compare this sum to a Riemman sum. Explain why it cannot be considered a Riemman sum. g. See what happens when you multiply and divide by delta x h. Write your expression in terms of f(x) Length of a smooth curve What properties of y = six of x did we use to write our final expression? [C2] Smoothness property: Arc Length of a Smooth Curve: Sample Examples and Exercises: Fin the exact length of y = x 1 for 0 x 1 3 Vertical Tangents, Corners, and Cusps 1 2. Find the length of the curve 3 y = x between ( 8, 2) and (8, 2) 3

3 3. Find the length of the curve y = x2 4 x x from x = 4 to x = 4 4. Verify your results by using the ArcLen feature in the TI 89 [C4] 5. Additional Examples and Exercises Section 8.1 L Hopital Rule Timeline: One Block Objectives: To recognize the various indeterminate forms To decide when to apply the L Hopital rule to evaluate limits or when to modify the limit so that the L Hopital rule could be applied Form 0/0 L Hopital Rule Proof Verify the proof graphically [C4] Evaluate the following limits: 1 cos x Lim 0 x 2 x + x Lim 0 x x 1+ x x

4 o Activity : a. Explore the L Hopital s Rule Graphically [C4] [C2] [C3] sin x b. Use the function f ( x) = x c. Find the limit analytically by applying the L Hopital s Rule d. Let y1 = sinx, y2 = x, y3 = y1/y2, y4 = y1 /y2 e. Graph y3 and y4 in the same viewing window. How does it provide support for the L Hopital s Rule? f. Let y5 = y3 (y3 = y1/y2). Graph y4 ( y1 /y2 ) and y5 in the same window g. Make a statement about what L Hopital s Rule does not say [C3] Use the L Hopital s Rule with one sided Limits: a. b. sin x x sin x Lim x 0 2 x + Lim x 0 2 Indeterminate Forms 1,0 0, Additional Examples and Exercises 0 Indeterminate Forms, 0,

5 C1 T he course teaches all topics associated with Functions, Graphs, and Limits; Derivatives; and Integrals and Polynomial Approximations and Series as delineated in the Calculus BC Topic Outline in the AP Calculus Course Description. C4 The course teaches students how to use graphing calculators to help solve problems, experiment, interpret results, and support conclusions. C2 T he course provides students with the opportunity to work with functions represented in a variety of ways g raphically, numerically, analytically, and verbally a nd emphasizes the connections among these representations. C3 T he course teaches students how to communicate mathematics and explain solutions to problems both verbally and in written sentences.

6 Section 8.2 Relative Rates of Growth Timeline: Two Blocks Objectives: To compare rates of growths by evaluating limits at infinity To use and understand Order and Oh notation o [C2] [C3] [C4 Comparing rates of growth Faster, Slower, Same rate of Growth 1. Compare e x and x 2 as x goes to infinity 2. Show that ln grows slower than (a) x and (b) x 2 as x goes to infinity 3. Show that log a x and log b x grow at the same rate as x goes to infinity 4. Support your Results graphically [C4] 5. Explain why e x grows faster than x n as x goes to infinity for any positive integer n [C2] Transitivity of Growing Rates a. Show that two or more functions grow at the same rate by using the transitivity property C1 T he course teaches all topics associated with Functions, Graphs, and Limits; Derivatives; and Integrals and Polynomial Approximations and Series as delineated in the Calculus BC Topic Outline in the AP Calculus Course Description. C4 The course teaches students how to use graphing calculators to help solve problems, experiment, interpret results, and support conclusions. C2 T he course provides students with the opportunity to work with functions represented in a variety of ways g raphically, numerically, analytically, and verbally a nd emphasizes the connections among these representations. C3 T he course teaches students how to communicate mathematics and explain solutions to problems both verbally and in written sentences.

7 Order and Oh Notation 6. Show that ln = O(x) as x goes to infinity 7. Show that x 2 = O(x 3 + 1) as x goes to infinity 8. Show that x + sin x = O(x) as x goes to infinity 9. Show that e x + x 2 = O(e x ) as x goes to infinity

8 Calculus Section 8 3a Integration by Parts Timeline: 2 Blocks Objectives: To use the integration by parts method to convert a non integrable expression into an integrable expression o [C2] [C3] [C4] Use the product rule in integral form to develop a new integration formula Integration by Parts Formula 1. x cos xdx 2. Find the area of the region bounded by the curve y = xe x and the x axis from x=0 to x=3 Repeated use of integration by parts 3. Evaluate x x 2 e dx Solving for the unknown integral Evaluate e x cos xdx C1 T he course teaches all topics associated with Functions, Graphs, and Limits; Derivatives; and Integrals and Polynomial Approximations and Series as delineated in the Calculus BC Topic Outline in the AP Calculus Course Description. Tabular Integration: 2 Evaluate x e x dx Evaluate 3 x sin xdx

9 o Exploration: [C2] [C3] [C4 a. Use integration by part with u = ln x and dv = dx to show that the integral is x ln x x + C. b. Differentiante x lnx x + C to confirm your result c. Use a slope field of the differential equation dy/dx = ln x and the graph of y = x ln x x to support your result [C2] [C4] d. Explain the connection [C3] e. Use the graphs of y = x ln x x and y = x tdt 1 ln to support your result in a. [C2] [C4] f. Explain the connection [C3] 6. Additional Examples and Exercises C1 T he course teaches all topics associated with Functions, Graphs, and Limits; Derivatives; and Integrals and Polynomial Approximations and Series as delineated in the Calculus BC Topic Outline in the AP Calculus Course Description. C4 The course teaches students how to use graphing calculators to help solve problems, experiment, interpret results, and support conclusions. C2 T he course provides students with the opportunity to work with functions represented in a variety of ways g raphically, numerically, analytically, and verbally a nd emphasizes the connections among these representations. C3 T he course teaches students how to communicate mathematics and explain solutions to problems both verbally and in written sentences.

10

11 Section 8 3b Improper Integrals Timeline: 3 blocks Objectives: To solve integrals with infinite limits of integration To evaluate integrals with infinite discontinuities in their domain To determine whether an integral converges or diverges by applying several tests for converge and divergence Infinite Limits of Integration: Activity: [C2] [C3] [C4] a. Ask students to graph y = e x and to make a conjecture about the area under the curve b. and above the x axis. [C2] [C3] [C4] c. Find the area analytically d. Make a conjecture about evaluating integrals with infinite limit of integration [C3] Improper Integrals with Infinite Limits of Integration Convergence Divergence 1. Does the improper integral dx converge or diverge? 2. Evaluate dx 1 x x 3. For what values of p does the integral dx converge? Integrals with Infinite Discontinuities 4. Evaluate 3 0 dx (x 1) p x

12 4 dx 5. Evaluate x 1 2 Tests for Convergence and Divergence 6. Does the integral 1 e x 2 dx converge? Direct Comparison Test Limit Comparison Test Additional Examples and Exercises C1 T he course teaches all topics associated with Functions, Graphs, and Limits; Derivatives; and Integrals and Polynomial Approximations and Series as delineated in the Calculus BC Topic Outline in the AP Calculus Course Description. C4 The course teaches students how to use graphing calculators to help solve problems, experiment, interpret results, and support conclusions. C2 T he course provides students with the opportunity to work with functions represented in a variety of ways g raphically, numerically, analytically, and verbally a nd emphasizes the connections among these representations. C3 T he course teaches students how to communicate mathematics and explain solutions to problems both verbally and in written sentences.

13

14 Objectives: Section 8.4 Partial Fractions and Integral Tables Timeline: 3 Blocks To express a rational function as a sum of basic fractions To use the method of partial fractions to integrate rational functions by integrating the sum of its partial fraction 5x 3 1. Use the method of partial fractions to evaluate 2 x 2x 3 6x Express as a sum of partial fractions 2 ( x + 2) 6x Repeated Linear Factor: Evaluate dx 2 ( x + 2) 3 2 2x 4x x 3 4. Improper Fraction: Evaluate 2 x 2x 3 5. Quadratic Factor: Find the solution to dy/dx = 2xy(y 2 + 1) that satisfies y(0) = Evaluate + dx 2 2 x + 1 x 2x + 5 Trigonometric Substitutions Integrating Powers of Trigonometric Functions Evaluate 4 + x 2 dx n sec axdx = C1 T he course teaches all topics associated with Functions, Graphs, and Limits; Derivatives; and Integrals and Polynomial Approximations and Series as delineated in the Calculus BC Topic Outline in the AP Calculus Course Description. Evaluate x 3 dx 9 x 2

15 Objectives: Calculus Section 9.1 Infinite Series Timeline: 1 Block To know the difference between series and sequences To analyze the convergence and divergence of geometric series Demonstrate graphically the value of the following infinite sum: [C2] = Infinite Sequence Identify various familiar sequences Infinite Series Compare and Contrast Sequences and Series Partial Sums of Infinite Series Convergence and Divergence of Series Examples: 1. Does the series converge? Does the series converge? n Geometric Series 3. Tell whether each geometric series converges or diverges a. 1 n 1 3( ) n= n b ( )

16 c. 3 ( ) n=0 5 k 2 3 π π π d Determine if the following series converges or diverges: 1 k( k ) Express the number as the ratio of two integers a b Harmonic Series Additional Examples and Exercises C1 T he course teaches all topics associated with Functions, Graphs, and Limits; Derivatives; and Integrals and Polynomial Approximations and Series as delineated in the Calculus BC Topic Outline in the AP Calculus Course Description. C2 T he course provides students with the opportunity to work with functions represented in a variety of ways g raphically, numerically, analytically, and verbally a nd emphasizes the connections among these representations.

17 Objectives: To represent function by series To differentiate and integrate series Calculus Section 9.1b Infinite Series Timeline: 2 Blocks Represent Functions by Series Activity: Find Power Series to represent various series that can be obtained from the basic Geometric Power Series Activity I: [C2] [C3] [C4] Given the expansion of the Geometric Power Series: 1 a. Find a power series to represent 2 1+ x b. Find a power series that represents tan 1 x on ( 1,1) c. Graph the first four partial sums [C4] d. Do the graphs suggest convergence on the open interval ( 1,1)? Explain [C4] e. Do you think that the series tan 1 x on ( 1,1) converges at x = 1? f. Use the graphing calculator to verify that the partial sums converge to π/4 [C2] [C4] Finding a Power Series by Differentiation Finding a Power Series by Integration Activity 2 [C2] [C4] n x x x x Given f(x) = 1+ x ! 3! 4! n! a. Find f (x) b. Find f(0) c. What well known function f you suppose f is? d. Verify your conjecture analytically

18 e. Graph the first three partial sums. What appears to be the interval of convergence? f. Graph the next three partial sums to verify your prediction Additional Examples and Exercises C1 T he course teaches all topics associated with Functions, Graphs, and Limits; Derivatives; and Integrals and Polynomial Approximations and Series as delineated in the Calculus BC Topic Outline in the AP Calculus Course Description. C4 The course teaches students how to use graphing calculators to help solve problems, experiment, interpret results, and support conclusions. C2 T he course provides students with the opportunity to work with functions represented in a variety of ways g raphically, numerically, analytically, and verbally a nd emphasizes the connections among these representations. C3 T he course teaches students how to communicate mathematics and explain solutions to problems both verbally and in written sentences.

19 Section 9.2 Taylor Series Timeline: 2 Blocks Objectives: To construct a power series for continuous differentiable functions To approximate functions around a given center through the use of nth order Taylor Polynomials Discovery a. Find a power series to represent ln (1 x) and use it to evaluate ln 2 b. Suppose that f is any function that can be represented by a power series. Find the coefficients of f applying the following procedure: 1. Find f(c) 2. Find f (c ) 3. Find f (c) 4. Find f (c ) c. What will happen if we continue to differentiate and put x = c? d. Use your results to give an expression for the nth coefficient a n Taylor Series of the function f at c Applications 1. Construct a polynomial that matches the behavior of ln (1 + x) at x = 0 through its first four derivatives 2. Construct the seventh order polynomial and the Taylor series for sin x at x = 0 Activity [C2] [C3] [C4] a. Construct the 6 th order Taylor polynomial and the Taylor series at x = 0 for cos x by matching the first 6 derivatives

20 b. Can you find a pattern? c. Explain how can you use the pattern that developed to construct polynomial of higher order for f(x) == cos x [C2] d. Repeat above steps with f(x) = sin x e. Graph the first nine partial sums together with y = sin x [C2] [C4] f. What do you think will happen as the if we add more terms to the partial sums? [C2] [C3] [C4] Acticity 3: [C2] [C3] [C4] How many terms of the Taylor polynomial for y = sin x are required to approximate sin 10 accurate to the third decimal place? a. Find sin 10 on your calculator and write the answer b. Use the sum(sequence(f(n),n,a,b)) feature of your calculator to approximate sin 10 using different number of terms each time [C2] [C4] c. Continue adding terms until you reach the desired results [C2] [C4] d. How many terms are required to stabilize the result in the thousandths place for x = 10? Explain [C3] MacLaurin and Taylor Series Find the fourth order Taylor polynomial that approximates y = cos 2x near x = 0 Taylor Series Generated at x = a Find the Taylor series generated by f(x) = e x at x = 2

21 Combination of Taylor Series Table of Maclaurin Series C1 T he course teaches all topics associated with Functions, Graphs, and Limits; Derivatives; and Integrals and Polynomial Approximations and Series as delineated in the Calculus BC Topic Outline in the AP Calculus Course Description. C4 The course teaches students how to use graphing calculators to help solve problems, experiment, interpret results, and support conclusions. C2 T he course provides students with the opportunity to work with functions represented in a variety of ways g raphically, numerically, analytically, and verbally a nd emphasizes the connections among these representations. C3 T he course teaches students how to communicate mathematics and explain solutions to problems both verbally and in written sentences. Section 9.3 Taylor s Theorem Timeline: 2 Blocks Objectives: To use Taylor Polynomial to approximate functions To understand and apply the Taylor s Theorem and Remainder Estimation Theorem To estimate the truncation error resulting from the truncating the series to a finite polynomial Activity: [C2] [C4] Find a Taylor Polynomial that will serve as an adequate substitute for sin x on the interval [ pi, pi]. a. Use the sum(sequence(f(n),n,a,b)) feature of your calculator to approximate sin π by adding one term at a time [C2] [C4]

22 b. Support your result graphically by graphing the absolute error of the approximation in the interval [ π, π] [C2][C4] Practical Uses of Taylor Polynomials: Truncation Error: Activity: [C2] [C3] [C4] a. Find a formula for the truncation error if we use 1 + x 2 + x 4 + x 6 to approximate 1/(1 x 2 ) over the interval ( 1,1) b. Graph the truncation error [C2] [C4] c. Predict the truncation error after 5 terms d. Predict the truncation error after n terms. e. Show with a graph, how these errors get closer to 0 on ( 1,1) as n gets larger [C2] [C4] d. Explain why they still get worse as we approach ( 1,1) [C3] How could we handle the error when we truncate a non geometric series? The Remainder Taylor s Theorem with Remainder Lagrange Form and Lagrange Error Bounds

23 Example: Approximate cos (0.1) using a fourth degree Taylor (McLaurin) polynomial, and find the associated Lagrange Remainder. Use the Remainder Theorem to Define Convergence of a Function Prove that the series ( 1 k 2x+ 1 ) x ( x )! converges to sin x for all real x Additional Examples and Exercises C1 T he course teaches all topics associated with Functions, Graphs, and Limits; Derivatives; and Integrals and Polynomial Approximations and Series as delineated in the Calculus BC Topic Outline in the AP Calculus Course Description. C4 The course teaches students how to use graphing calculators to help solve problems, experiment, interpret results, and support conclusions. C2 T he course provides students with the opportunity to work with functions represented in a variety of ways g raphically, numerically, analytically, and verbally a nd emphasizes the connections among these representations. C3 T he course teaches students how to communicate mathematics and explain solutions to problems both verbally and in written sentences.

24 Section 9.4 Radius of Convergence Timeline: 2 Blocks Objectives: To understand convergence of series To apply the nth Term Test and Ration Test to determine the radius of convergence of a series To understand and find Endpoint Convergence Activity I: [C2] [C4] a. Examine the following geometric series and corresponding function and determine for what values of x the function represents an identity b. Support your result graphically Activity II: Develop an strategy for finding the interval of convergence of an arbitrary power series: a. Does it converge at x = a? b. List series with a restricted domain of convergence c. List series where the interval of convergence is all real numbers The convergence Theorem for Power Series The nth Term Test for Divergence 2. Find the radius of convergence of the series n! x n=0 n The Direct Comparison Test 3. Prove that 2n x n=0 ( n!) 2 converges for all real x

25 Absolute convergence 4. Show that (sin x) =0 n! n n converges for all x Ratio Test 5. Find the radius of convergence of n( x) n =0 10 n n Endpoint Convergence Activity III: [C2] [C3] [C4] For what values of x does n x x x n 1 x x ( 1) converge? n a. Apply the Ratio Test to Determine the radius of convergence b. Substitute the left hand endpoint into the series. Decide whether the resulting series converges or diverges c. Substitute the right endpoint into the series d. sum(sequence(f(n),n,a,b)) feature of your calculator to track the progresss of the partial sums as the number of terms increases [C4] e. Use your results in step above to determine whether the numerical series converges [C2] f. Does the series converge absolutely? Explain [C3]

26 6. Find the sum of 1 = n( n + 1 Additional Examples n 1 ) C1 T he course teaches all topics associated with Functions, Graphs, and Limits; Derivatives; and Integrals and Polynomial Approximations and Series as delineated And Exercises in the Calculus BC Topic Outline in the AP Calculus Course Description. C4 The course teaches students how to use graphing calculators to help solve problems, experiment, interpret results, and support conclusions. C2 T he course provides students with the opportunity to work with functions represented in a variety of ways g raphically, numerically, analytically, and verbally a nd emphasizes the connections among these representations. C3 T he course teaches students how to communicate mathematics and explain solutions to problems both verbally and in written sentences.

27 Section 9.5 Testing Convergence at Endpoints Timeline: 2 Blocks 1. Integral Test 2. Harmonic Series and p series 3. Comparison Test 4. Limit Comparison Test 5. Alternating Series Test Activity [C2] [C3] [C4] a. Consider n+ ( 1) n= 1 n b. Use the sum (seq (f(n), n, a, b) ) feature in your calculator to find the following partial sums: [C4] [C2] 1 1 S 11= S 12= S 13= S 14= S 15= S 16= c. Verify that any two consecutive differences is always less than the next term in the series: S n S n 1 < n th term What does the Alternating Series Test say about n= [C3] 1 ( 1) n+ 1 1 n and its 100th partial sum? Explain 6. Absolute and Conditional Convergence 7. Intervals of Convergence Closure: Write a flow chart that can be used to decide what test to use to determine the convergence or divergence of a series

28 Additional Examples and exercises

29 Section 10 1 a Parametric Functions Timeline: 1 block Objectives: To understand and graph parametric equations. To convert functions to parametric equations and vice versa Activity I: [C2] [C4] Use a graphing calculator se to parametric mode to graph x = t and y = t, t 0 Use the trace feature to trace the curve and examine the window settings Write the equivalent function you think this graph represents Confirm your results analytically Definition of Parametric Curves Activity II [C2] [C4] [C3] 1. Let x = a cos t and y = a sin t 2. Graph the parametric equation 3 times in the same viewing window using a = 1, 2 or 3 3. How does changing a affect this graph? 4. What is the current t interval? L 5. Let a = 2 and change the intervals to 0 π, 2 NM O L QP, 0,π, 0, 3 π NM O 2 QP, 2π, 4 6. Describe the role of the length in the parameter interval [C3] π 7. Let a = 3 and graph the parametric equation using the intervals: L π 3π, L3 π,2 π,, π,5π NM O QP, π NM O QP, π What are the initial and terminal point in each case?

30 8. Graph x = 2cos ( t ) an y = 2 sin ( t) using the parameter intervals 0, 2π, π,3 π, each case, describe how the graph is traced [C3] L π 3π, 2 2 NM O QṖ In 10. Verify analytically that these equations in fact represent circles 11. Draw and identify the graphs of the parametric curves and verify your results algebraically: a. x = 3t, y = 2 2t, [0,1] b. x = 3cos t, y = 4 sit t, 0, 2π Parametrize a function a. The line segment with endpoints ( 2,1) amd (3,5) b. The left half of the parabola y = x 2 + 2x Find a Cartesian equations. What portion of the graph of the Cartesian equation is traced by the parametrized curve? c. x = 4cost, y = 2 sin t, 0, 2π Additional Examples and Exercises C1 T he course teaches all topics associated with Functions, Graphs, and Limits; Derivatives; and Integrals and Polynomial Approximations and C4 The Series as course delineated teaches in students the Calculus how to use BC graphing Topic Outline calculators in the to AP help Calculus solve problems, Course Description. experiment, interpret results, and support conclusions. C2 T he course provides students with the opportunity to work with functions represented in a variety of ways g raphically, C3 T he course teaches numerically, analytically, and students how to communicate verbally a nd emphasizes the mathematics and explain connections among these solutions to problems both representations. verbally and in written sentences.

31

32 Section 10 1 b Calculus of Parametric Functions Timeline: 1 block Objectives: To find the derivative of parametric curves. To determine a parametric formula for the second derivative To find the length and surface areas of curves written in parametric mode Activity I: Use analytic procedures to verify that the parametric curve x = cos t and y = sin t in [0,2pi] represent a differentiable curve Derivative at a Point of a parametrized curve Use the chain rule to determine the derivative of a parametrized curve Smoothness of a parametrized curve 2 d y Find a parametric formula for 2 dx in terms of t 2 d y Summarize the necessary steps to find 2 dx in terms of t Examples 2 d y Find : 2 dx Length of a Smooth Curve Examples 1. Find the length of x = cos 3 t, y = sin 3 t, in [0,2pi]

33 2. Verify your answer numerically [C2] Activity II: Cycloids Find parametric equations for the path of the point P in the following figure 1. Assume the wheels rolls to the right, P is at the origin when the turn angle is 0 2. Determine x and y in terms of t and theta after the circle has turned t radians 3. Express theta in terms of t 4. Write the final parametric equation Activity III: Investigating Cycloids [C2] [C3] [C4] 1. Let x = a (t sin t), y = a ( 1 cos t), a > 0 2. Graphs the equations for a = 1, 2, and 3 3. Find the x intercepts 4. Show that y 0 for all t 5. Explain why the arches of a cycloid are congruent [C3] 6. What is the maximum value of y? Where is it attained? 7. Describe the graph of a cycloid [C3] Surface Area Examples Additional Examples and Exercises

34 C1 T he course teaches all topics associated with Functions, Graphs, and Limits; Derivatives; and Integrals and Polynomial Approximations and Series as delineated in the Calculus BC Topic Outline in the AP Calculus Course Description. C4 The course teaches students how to use graphing calculators to help solve problems, experiment, interpret results, and support conclusions. C2 T he course provides students with the opportunity to work with functions represented in a variety of ways g raphically, numerically, analytically, and verbally a nd emphasizes the connections among these representations. C3 T he course teaches students how to communicate mathematics and explain solutions to problems both verbally and in written sentences. Section 10 2 Vectors in the Plane Timeline: 1 block Objectives: To understand vectors and equality of vectors To show whether vectors given by the coordinates of their endpoints are equal To find the component form of a vector To perform operations with vectors including inner product (dot product) To find the angle between to vectors To find Vectors Tangent and Normal to a Curve Activity I: 1. List some scalar quantities 2. List some vector quantities 3. Compare and contrast scalar quantities with vector quantities 4. Identify magnitude and direction in the vector quantity Show That Two Vectors are Equal Let A = (0, 0), B = (3, 4), C= ( 4, 2), and D = ( 1, 6). Show that the vectors And CD are equal by determining the magnitude and slope of each vector AB

35 Vectors in Standard Position Component Form of a Vector Activity II: 1. Find the component form and length of at least two vectors defined by their initial and endpoints 2. Write a generalization for the component form of a vector in terms of the coordinate of the endpoints 3. Write a generalization for the length of a vector in terms of the coordinate of the endpoints 4. Write the vertical component and horizontal component of v= <a,b> 5. What is the slope of the vector? ZeroVector Vector Operations Properties of Vector Operations Parallelogram Law of Addition Activity III: Performing Operations on Vectors 1. Let u = <1,3> and v = <4,7>. Find a. 2u + 3v b. u v 1 c. u 2 2. Verify your answers to a and b using the parallelogram law of addition Dot Product or Inner Product

36 Angle Between TwoVectors Find the angle in the triangle ABC determined by the vertices A = (0,0), B = (3,5), and C = (5,2) Finding Vectors Tangent and Normal to a Curve Additional Examples and Exercises C1 T he course teaches all topics associated with Functions, Graphs, and Limits; Derivatives; and Integrals and Polynomial Approximations and Series as delineated in the Calculus BC Topic Outline in the AP Calculus Course Description.

37 Section 10 3 Vector Valued Functions Timeline: 2 blocks Objectives: To understand the Unit Vector To express vectors as the a linear combinations of the unit vectors i and j To understand and graph vector valued functions To find the limit of vector functions To determine whether a vector function is continuous To find points of continuity and discontinuity of a vector function To find derivatives and integrals of vector functions To analyze motion by using vector functions Activity [C2] [C3] [C4] 1. Express vectors as linear combinations of the two standard unit vectors 2. Vector Functions 3. Graph the vector functions r(t) = (t cos t ) i + (t sin t) j, t 0 [C4] 4. Explain how is the above vector function different from the parametric equation r(t) = (t cos t ) i + (t sin t) j, t 0. How is it the same? [C3] [C2] 5. Limit and continuity of a vector function 6. Find the limit of the vector function r(t) = (t cos t ) i + (t sin t) j as t approaches π/4 7. Find the points of continuity and discontinuity of the vector function r(t) = (1/t) i + (sin t) j 8. Find derivatives of vector functions 9. Velocity, speed, acceleration, and direction of the motion of a particle whose vector position function is given 10. The vector r(t) = (3 cos t ) i + (3 sin t) j gives the position of a moving particle at time t, Find: a. The velocity and acceleration vectors b. The velocity, speed, acceleration, and direction of the motion at t = π/4 c. v a. 11. The vector r(t) = (2t 3 3t 2 )I + (t 3 12t)j gives the position of a moving particle at time t a. Write an equation for the line tangent to the path of the particle at t = 1 b. Find the coordinates of each point on the path where the horizontal component of the velocity is 0

38 Differentiation Rules for Vector Functions Indefinite Integrals of Vector Functions Definite Integrals of a Vector Function Find antiderivatives of vector functions Evaluate definite integrals given by vector functions Additional Examples and Exercises C1 T he course teaches all topics associated with Functions, Graphs, and Limits; Derivatives; and Integrals and Polynomial Approximations and Series as delineated in the Calculus BC Topic Outline in the AP Calculus Course Description. C4 The course teaches students how to use graphing calculators to help solve problems, experiment, interpret results, and support conclusions. C2 T he course provides students with the opportunity to work with functions represented in a variety of ways g raphically, numerically, analytically, and verbally a nd emphasizes the connections among these representations. C3 T he course teaches students how to communicate mathematics and explain solutions to problems both verbally and in written sentences.

39 Section 10 5 Polar Coordinates and Polar Graphs Timeline: 1 block Objectives: To understand the definition of polar coordinates To graph a set of points whose polar coordinates satisfy a given condition To graph using the polar mode of the graphing calculator To apply the symmetry tests for polar graphs To convert a Cartesian equation to a Polar equation To convert a Polar equation to a Cartesian equation Definition of Polar Coordinates Find at least five polar points equivalent to the point (2, π/6) Express the point (3, π/3) using a negative r Graph several set of points whose polar coordinates satisfy the given condition Draw the graph r = 1 cosθ. What is the shortest length a θ interval can have and still produce the graph? Symmetry Tests for Polar Graphs Activity I [C2] [C3] [C4] 1. Let r 2 = 4cosθ 2. Solve for r 3. Graph the solutions r1 and r2 4. Explain why the origin is part of the graph. Verify analytically [C2] [C3] 5. What is the shortest length a θ interval can have and still produce this graph? 6. Show algebraically that the graph is symmetric about the x axis, the y axis, and the origin Activity II 1. Draw a Cartesian plane 2. Place the origin of polar coordinates in this plane

40 3. What ray would be the x axis? 4. What ray would be the y axis? 5. Use this relationship to express any (x, y) Cartesian point as a polar point 6. Express r in terms of x and y 7. Express θ in terms of x and y Converting Cartesian to Polar Use your findings in steps 6 and 7 to convert the given Cartesian equations to polar equations Converting Polar to Cartesian Use your findings in steps 6 and 7 to convert the given polar equations to the equivalent Cartesian equations Additional Examples and Exercises C1 T he course teaches all topics associated with Functions, Graphs, and Limits; Derivatives; and Integrals and Polynomial Approximations and Series as delineated in the Calculus BC Topic Outline in the AP Calculus Course Description. C2 T he course provides students with the opportunity to work with functions represented in a variety of ways g raphically, numerically, analytically, and verbally a nd emphasizes the connections among these representations. C4 The course teaches students how to use graphing calculators to help solve problems, experiment, interpret results, and support conclusions. C3 T he course teaches students how to communicate mathematics and explain solutions to problems both verbally and in written sentences.

41

42 Section 10 6 Calculus of Polar Curves Timeline: 2 blocks Objectives: To find the slope of a tangent line to a polar curve To find the area of the region enclosed by a polar curve To find the area of the region between two polar curves To find the length of a polar curve To find the area of a surface of revolution generated by revolving a polar curve about the x axis and the y axis Activity I [C2] [C3] [C4] 1. Graph r = 1 + cos θ 2. Identify all the point that appear to have a vertical tangent line 3. Find dr/dθ 4. Can you verify your answer to question 2 analytically by using dr/dθ? 5. Explain why dr/dθ does not give the slope of the tangent line to a polar curve [C3] 6. Find dy/dx (slope of a polar curve) by converting x and y to polar form: x = r cos θ and y = r sin θ Activity II 1. Use the results in step 6 above to find the horizontal and vertical tangents to a cardioid in [0, 2π] 2. Explore tangent lines at the origin: if r = f (θ) = 0, What does the formula dy = f ( x, y) dy dx f '( θ )sin θ + f ( θ )cosθ = say about the slope of the curve at (0,θ)? f '( θ )cos θ f ( θ )sinθ 3. Write the simplified special case of dy/dx when f(θ) = 0 4. Use your results to find the lines tangent to the rose curve r = cos (2θ) at the pole dx Activity III [C2] [C4] 1. Use the sector of a circle to derive a formula for the area of the region enclosed by a polar curve 2. Find the area of the region between the origin and the cardiod r = 2(1+cos θ) 3. Verify your results numerically by using FNINT in your graphing calculator

43 4. Find the area inside the smaller loop of the limacon r = 2 cos θ Verify your results numerically by using FNINT in your graphing calculator Area Between Polar Curves Length of a Polar Curve Area of a Surface of Revolution generated by revolving a polar curve about the x axis and the y axis Examples and Exercises C1 T he course teaches all topics associated with Functions, Graphs, and Limits; Derivatives; and Integrals and Polynomial Approximations and Series as delineated in the Calculus BC Topic Outline in the AP Calculus Course Description. C2 T he course provides students with the opportunity to work with functions represented in a variety of ways g raphically, numerically, analytically, and verbally a nd emphasizes the connections among these representations. C4 The course teaches students how to use graphing calculators to help solve problems, experiment, interpret results, and support conclusions. C3 T he course teaches students how to communicate mathematics and explain solutions to problems both verbally and in written sentences.

44 Section 6.5 Logistic Differential Equations Timeline: 1 Block Objectives: To use exponential model, logistic growth model, and logistic regression to determine population growth Exponential model Activity I [C2] [C4] 1. Find an initial value problem model for the world population described by a table containing data of a population growth following an exponential pattern. [C2] 2. Use your result to predict the population in the year Verify your solution by superimposing the model on a scatter plot of the data [C4] Logistic Growth Model Activity II 1. Use the basic logistic differential equation dp = kp( C P) where C represents the dt current capacity and k the reproduction rate to derive a differential equation that produces dp = kp if P is small and dp dt dt < 0 if P > C C 2. Solve this differential equation to obtain P = + 1 Ce kt Activity III A national park is known to be capable of supporting no more than 100 grizzly bears. Ten bears are in the park at present. We model the population with a logistic differential equation with k=0.1 a. Draw and describe a slope field for the differential equation b. Find a logistic growth model for the population and draw its graph

45 c. When will the population reach 50? d. Verify that the limit as t approaches infinity is the current capacity Activity IV [C2] [C2] [C4] Logistic Regression a. Find the logistic regression equation for the U.S. population data in table 6.8 (page 345) b. Superimpose the graph on a scatter plot of the data c. Determine when the rate of growth predicted by the regression equation changes from increasing to decreasing. d. Estimate the population at this time Additional Examples and Exercises C1 T he course teaches all topics associated with Functions, Graphs, and Limits; Derivatives; and Integrals and Polynomial Approximations and Series as delineated in the Calculus BC Topic Outline in the AP Calculus Course Description. C2 T he course provides students with the opportunity to work with functions represented in a variety of ways g raphically, numerically, analytically, and verbally a nd emphasizes the connections among these representations. C4 The course teaches students how to use graphing calculators to help solve problems, experiment, interpret results, and support conclusions. C3 T he course teaches students how to communicate mathematics and explain solutions to problems both verbally and in written sentences.

46

47 Section 6 6 Euler s Method Timeline: 1 Block Objectives: To solve a differential equation numerically by using Euler s Method Activity I Given dy dx = f ( x, y) and P(x o, y o ), a point in the solution curve 1. Take a small step or increment (dx) 2. Find the linearization by using the given point and the slope at that point: y y o = y (x o, y o )(x x o ) 3. Take another step by picking a point in the linearization (x 1, y 1 ) 4. Find the linearization: y y 1 = y (x 1, y 1 )(x x 1 ) 5. Take another step and repeat the process 6. Use the grid below to illustrate your results

48 Apply the Euler s Method: Find the first three approximations y1, y2, y3 using Euler s method for the initial value problem: y = 1 + y; y(0) = 1 Using a Calculator Program [C2] [C4] Use Euler s method to solve y = 1 + y; y(0) = 1 on [0,1] starting at x = 0: a) Taking dx = 0.1 b) Taking dx = 0.05 c) Compare the approximations to the exact solution y = 2e x 1 Graphical Solutions: [C2] [C4] a. Find a graphical solution by plotting the data pairs obtained in previous exercises with a graphing calculator b. Superimpose the graph of the exact solution on the scatter plot of the data points to support your results Additional Examples C1 T he course teaches all topics associated with Functions, Graphs, and Limits; Derivatives; and Integrals and Polynomial Approximations and Series as delineated in the Calculus BC Topic Outline in the AP Calculus Course Description. C2 T he course provides students with the opportunity to work with functions represented in a variety of ways g raphically, numerically, analytically, and verbally a nd emphasizes the connections among these representations. C4 The course teaches students how to use graphing calculators to help solve problems, experiment, interpret results, and support conclusions.

49

50 Resources: Primary textbook: Calculus Graphical, Numerical, Algebraic Second Edition Finney, Demana, Waits and Kennedy Addison Wesley, Scott Foresman Secondary textbook: Calculus Second Edition Howard Anton Wiley Teacher Resources: Complete Set of Resources from Finney, Demana, Waits, and Kennedy textbook Technology Resources All of my students are assigned a TI 83 plus and a TI 89 graphing calculator for their everyday use Computer Lab with the Geometer s Sketchpad installed in all computers As a supplement to the software we use Exploring Calculus with the Geometer s Sketchpad by Cindy Clements, Ralph Pantozzi, and Scott Steketee Warm Up Resources: We start every block with a Sample AP Question related to the topic of the day. Must of these questions are from previous AP Examinations posted on AP central Additional Teacher Resources: Mathematica Teacher s Edition by Wolfram Research Master the AP Calculus AB and BC Test by Michael Kelley AP Central Teaching Strategies: Each class starts with a sample AP question that may consist of multiple parts addressing the topic of the day. Students are encouraged to discuss and share their solutions, then selected students put final solutions on the board.

51 The class proceeds with a homework review, which is also a student centered activity. I only go over those problems no one in the class could solve After the homework review each student is handed in a packed that presents a detailed outline of the lesson including the activities presented in this syllabus among other activities, class examples and exercises, journal questions, and the daily homework assignment. Students keep a portfolio of these packets and activities for review and evaluation purposes. The two weeks previous to the AP Examination are devoted to the review and solutions of previous AP Examination material Student Evaluation: Marking period grades are computed using homework, journal, quizzes, and chapter tests. Each marking represents 20% of the semester and the Final Exam represents another 20%. The final exam uses multiple choice and free response questions that follow the format of the AP Exam.