Mathematics Calculus

Home , Symbolic integration

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C OUR S E OF S TUDY

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CAL CUL US

M TH 4 1 3

APPROVED BY THE BOARD OF EDUCATION

January 24, 2013

Approved: April 1986 Revised: August 1995 October 1996 MTH 413 – Calculus 2

STA TEM EN T OF P U R P OSE:

This Calculus course has been developed for the highly motivated twelfth grade student who has demonstrated proficiency in Algebra II and Precalculus.

The study of calculus involves three distinct states of mathematics: precalculus mathematics, the limit process, and the new calculus formulations of derivatives and integrals. This course is designed to help students make connections between familiar precalculus concepts and their more powerful calculus versions and to have students use precalculus formulas and techniques as tools to produce calculus formulas and techniques. One goal of this Calculus course is to have students recognize and make connections between the numerical, graphical and analytical interpretations of a problem. This course also promotes mathematical communication by asking students to interpret, describe, discuss, justify and make conjectures.

Separately we assess students to gauge progress and inform instruction. Benchmark assessments for students in grades 9 through 12 are administered in the form of a midterm and final exam for full year courses. *Special Note: Only final exams are administered at the end of quarter courses and semester courses.

Real-life application problems are incorporated throughout this course in order for students to see the applied nature, usefulness, and value of calculus.

This revision was undertaken to align with the New Jersey Student Learning Standards for Mathematics, the New Jersey Student Learning Standards for Technology and the College Board AP Calculus AB/BC course outline.

GOA LS

This course offers students the opportunity to:

1. acquire a broad understanding of trigonometry and its real world applications. 2. expand the experience of graphing various types of functions and relations obtained in Honors Algebra II. 3. predict functions and future results given appropriate data. 4. extend the experience of the conics to include three dimensional models in preparation for Calculus. 5. extend the experience of sequences and series to include the concept of limit. 6. use the concept of limit and continuity to graph functions. 7. acquire a broad understanding of higher degree polynomial equations. 8. continue the use of the graphing calculator to explore functions and their graphs. 9. acquire a broad understanding and apply the concept of differentiation. 10. acquire a broad understanding and apply the concept of integration. 11. use self-assessment to identify their mathematical strengths and weaknesses and to help foster a better understanding of the concepts being taught. MTH 413 – Calculus 3

T H E L I VI NG CURRI CUL UM

Curriculum guides are designed to be working documents. Teachers are encouraged to make notes on the document. Written comments can serve as the basis for future revisions. In addition, the teachers and administrators are invited to discuss elements of the guides as implemented in the class- room and to work collaboratively to develop recommendations for curriculum reforms as needed.

AFFI RMAT I VE ACT I O N

During the development of this course of study, particular attention was paid to material which might discriminate on the basis of sex, race, religion, national origin, or creed. Every effort has been made to uphold both the letter and spirit of affirmative action mandates as applied to the content, the texts and the instruction inherent in this course.

MODIFICATIONS AND ADAPTATIONS

For guidelines on how to modify and adapt curricula to best meet the needs of all students, instructional staff should refer to the Curriculum Modifications and Adaptations included as an Appendix in this curriculum. Instructional staff of students with Individualized Education Plans (IEPs) must adhere to the recommended modifications outlined in each individual plan.

MTH 413 – Calculus 4

PARS I PPANY - TR OY HI LLS TOWN SHI P SC HOOLS

CO URS E PRO FI CI E NCI E S AND G RA DI NG PRO CE DURE S

COURSE #: MTH 413 TITLE: CALCULUS

IN ACCORDANCE WITH DISTRICT POLICY AS MANDATED BY THE NEW JERSEY ADMINISTRATIVE CODE AND THE NEW JERSEY STUDENT LEARNING STANDARDS, THE FOLLOWING ARE PROFICIENCIES REQUIRED FOR THE SUCCESSFUL COMPLETION OF THE ABOVE NAMED COURSE. .

The student will:

1. use the  − definition to verify the limit of a function. ∑ 2. use the definition of continuity to verify that a function is continuous at a given point. 3. evaluate the limit of algebraic functions. 4. evaluate the limit of trigonometric functions. 5. use the techniques of cancellation, rationalization, algebraic manipulation, direct substitution and trigonometric substitution to evaluate a limit. 6. use the Squeeze Theorem to evaluate a limit. 7. use the Intermediate Value Theorem to locate the zeros of a function. 8. determine the continuity of a function given a graph and/or given a function. 9. determine infinite limits. 10. determine vertical asymptotes. 11. use the properties of limits to determine limits. 12. find the derivative of appropriate functions using the delta process. 13. find the derivative of appropriate functions using the product rule. 14. find the derivative of appropriate functions using the quotient rule. 15. find the derivative of appropriate functions using the chain rule. 16. find the derivative of appropriate functions implicitly. 17. find the slope of a curve at a point by using the derivative. 18. find the equation of the tangent line to a curve at a given point. 19. use derivatives to solve related rate problems. 20. recognize the graph of the derivative of a given function. 21. given the position function, evaluate its velocity and acceleration. MTH 413 – Calculus 5

Proficiencies (continued)

22. use derivatives to find the maxima and minima of a given function. 23. use derivatives to find the concavity and point of inflections of a given function. 24. use derivatives to sketch the graph of a given function. 25. use derivatives to solve extrema problems. 26. verify that a function satisfies the Mean Value Theorem. 27. evaluate the limits of a function at both vertical and horizontal asymptotes. 28. evaluate differential equations. 29. apply the Fundamental Theorem Of Calculus to evaluate definite integrals. 30. find the average value of a function on an interval. 31. use the substitution method to evaluate definite integrals. 32. use numerical integration methods such as the Trapezoidal Rule and Simpson’s Rule to evaluate definite integrals. 33. evaluate definite integrals involving trigonometric functions. 34. evaluate definite integrals involving the natural logarithmic function. 35. evaluate definite integrals involving natural exponential function. 36. evaluate definite integrals involving general exponential functions. 37. evaluate definite integrals involving inverse trigonometric functions. 38. evaluate definite integrals involving hyperbolic functions. 39. evaluate the area of a regions using the Fundamental Theorem of Calculus. 40. evaluate the derivative of trigonometric functions. 41. evaluate the derivative of the natural logarithmic function. 42. evaluate the derivative of the natural exponential function. 43. evaluate the derivative of general logarithmic functions. 44. evaluate the derivative of general exponential functions. 45. evaluate the derivative of inverse trigonometric functions. 46. graph the natural logarithmic and natural exponential functions. 47. evaluate volume of a solid using the disk and washer methods. 48. evaluate volume of a solid using the shell method. 49. evaluate definite integrals using integration by parts. 50. evaluate definite integrals using partial fractions. 51. evaluate definite integrals involving powers of trigonometric functions. 52. evaluate definite integrals using trigonometric substitution. MTH 413 – Calculus 6

G RADI NG PRO CE DURE S

Marking Period Grades:

Long- and Short-Term Assessments 90% Publisher prepared tests, quizzes and/or worksheets Teacher prepared tests, quizzes and/or worksheets Authentic Assessments Technology applications Projects Reports Labs

Daily Assessments 10% Homework Do Now / Exit Questions Class participation Journal Writing Notebook - checks and open notebook assessments Explorations

Final Grade – Full Year Course

Full Year Course The midterm assessment will count as 10% of • Each marking period shall count as the final grade, and the final assessment will 20% of the final grade (80% total). count as 10% of the final grade.

MTH 413 – Calculus 7

PROFICIENCIES/OBJECTIVES Teacher (Numbers in parentheses indicate correlation Standards Suggested Activities Evaluation/Assessment with the course proficiencies) Notes Students will: Students will: I. LIMITS AND THEIR PROPERTIES (1-11) 1.1 An Introduction to Limits I.A-D evaluate the function: explain techniques of In general, A. The tangent line problem 8.2.12.C.4 x evaluating limits and give reference to f ( x ) = at several the 8.1 B. An introduction to limits x + 1 −1 examples for each type. Technology C. Limits that fail to exist Standards in- points near x = 0 and use the result D. A formal definition of limit dicates the to estimate the limit: use of the

x graphing cal- lim culator. x →0 + x 1 −1

Open-ended

use the  − ∑ definition of describe three different problems

a limit to prove that: functions whose limit does should be assessed

lim (3x − 2) = 4 not exist. using the x →2 general

HSPA Rubric

found in the appendiz 1.2 Properties of Limits evaluate: explain how to determine I.A-D 2 A. Limits of algebraic lim (4 x + 3) the limit of a composite 8.2.12.C.4 x →2 functions function. B. Limits of trigonometric functions evaluate: use a graphing calculator x − 3 lim cos 3x = x → to graph f (x) 2 and x − 9 estimate the limit (if it exists). Determine the domain of the function and explain any danger inherent in analyzing the graph with only a calculator. MTH 413 – Calculus 8

PROFICIENCIES/OBJECTIVES Teacher (Numbers in parentheses indicate correlation AP Standards Suggested Activities Evaluation/Assessment with the course proficiencies) Notes Students will: Students will: I. LIMITS AND THEIR PROPERTIES (continued) 1.3 Techniques of Evaluating I.A-D find the limit of a function: explain how to determine Limits 8.2.12.C.4 x3 − 8 the limit (if it exists) of:

A. Cancellation a. lim x →2 x − 2 x + 1 + 2 B. Rationalization lim x →3 x − 3 C. The Squeeze Theorem sin x D. Direct substitution b. lim x →0 5x tan x −1 E. Trigonometric substitution find: lim x → sin x − cos x 4

1.4 Continuity and One-Sided I.A-D discuss the continuity of each create two functions and Limits 8.2.12.C.4 function: use a graphing utility to A. Continuity at a point and on 1 graph the functions and an open interval a. f ( x ) = determine the one-sided x B. One-sided limits and limit. x2 −1 continuity on a closed b. g( x ) = interval x −1 describe and give C. The Intermediate Value c. h( x ) = sin x examples of removable Theorem and non-removable find the limit of: discontinuity. f ( x ) = x −1 as x approaches 1 from the right. determine a value for a such that x + 3 x ≤ 2 discuss the continuity of: f (x) = 2 + > f ( x ) = 1 − x ax 6 x 2

is continuous on the entire use the Intermediate Value real line. Theorem to show f ( x ) = x3 + 2 x −1 has a zero in the interval [0,1]. MTH 413 – Calculus 9

PROFICIENCIES/OBJECTIVES Teacher (Numbers in parentheses indicate correlation AP Standards Suggested Activities Evaluation/Assessment with the course proficiencies) Notes Students will: Students will: I. LIMITS AND THEIR PROPERTIES (continued) 1.5 Infinite Limits I.E use graphing calculator to describe how to use the A. Determining infinite limits 8.2.12.C.4 sketch and find the limit of properties of Infinite limits from a graph each function as x → 1 from by finding the limit: B. Vertical asymptotes the left and from the right. x − 3 a. lim C. Properties of infinite limits 1 + a. f ( x ) = x →2 x − 2 x −1 1 2 b. f ( x ) = b. lim 2 + (x −1) x →0 sin x

−1 c. f ( x ) = x −1 explain what a vertical −1 asymptote is and describe = d. f ( x ) 2 how to find the vertical (x −1) asymptote of a function.

determine all vertical asymptotes of the graphs of the functions: 1 a. f ( x ) = 2(x + 1) x2 + 1 b. f (x) = x2 −1 c. f (x) = cot x

determine the limits: a. lim1 x →0 b. lim (x2 −1) x →1− c. lim 3 + x →0 MTH 413 – Calculus 10

Tangent Line Problem 8.1.12.A.1 f ( x ) = 2 x − 3 at the point the equation of the tangent A. The Tangent Line problem (2,1). line to the graph of f (x)

B. The derivative of a function at a given point. Sketch C. Differentiability and find the derivative of: both the graph of f (x) and continuity 3 f ( x ) = x + 2 x the tangent line to verify

their conclusion.

determine whether the function

is differentiable at x=0:

1 3 f ( x ) = x

2.2 Basic Differentiation Rules II.F.2 use the Power Rule to find the use the power rule with a and Rates of Change derivative of: negative exponent and 8.1.12.A.1 3 A. The Constant Rule a. f ( x ) = x show it is valid. B. The Power Rule 3 b. g( x ) = x C. The Constant Multiple Rule 1 use a graph to illustrate D. The Sum and Difference = c. y 2 that the derivative of rules x sin x = cos x and the de-

E. Derivatives of sine and rivative of cos x = − sin x . cosine find the derivative of: F. Rates of change a. y = 2 sin x sin x b. y = 2 c. y = x + cos x MTH 413 – Calculus 11

PROFICIENCIES/OBJECTIVES Teacher (Numbers in parentheses indicate correlation AP Standards Suggested Activities Evaluation/Assessment with the course proficiencies) Notes Students will: Students will: II. DIFFERENTIATIONS (continued) 2.3 The Product and Quotient II.F.3 find the derivative using the explain the relationship be- Rules and Higher Order II.D.1 Product Rule: tween the position func- Derivatives 8.1.12.A.1 a. h( x ) = x sin x tion, the velocity function A. The Product Rule d 2 and the acceleration func- B. The Quotient Rule b. (3x-2x tion and describe how to )(5+4x) C. The derivatives of trigono- dx determine the velocity and metric functions acceleration function if the D. The higher-order derivatives find the derivative using the position function is: Quotient Rule: −3t 2 s( t ) = + 2t + 6 d  5x − 2  2 a.  2  dx x + 1 

solve: Because the moon has no at- mosphere, a falling object on the moon encounters no air re- sistance. In 1971 an astronaut, Davis Scott, demonstrated that a feather and a hammer fell at the same rate on the moon. The Position Function for each of these falling objects was given by s( t ) = −0.8t 2 + 2 , where s( t ) is the height in meters and t is the time in seconds. What is the ratio of Earth’s gravitational force to the moon’s? MTH 413 – Calculus 12

2.3 The Product and Quotient II.F.3 dy 2 3 explain how to find the find for y = (x + 1) using Rules and Higher Order 8.1.12.A.1 dx derivative of: Derivatives (continued) the Chain Rule. f (x) = sin2 (5x)

dy find by applying the Chain dx Rule to trigonometric functions: a. y = sin zx b. y = cos (x −1) c. y = tan 3x

solve the following Harmonic Motion problem: The displacement from equi- librium of an object in harmonic motion on the end of a spring is 1 1 y = cos12t − sin12t , where y 3 4 is measured in feet and t is time in seconds. Determine the position and velocity of the object when t = 8 . MTH 413 – Calculus 13

evaluate the derivative of use a graphing utility to graph x2 − y3 = 0 at the point

the equation. Find the equa- (1.1). tion of the tangent line to the graph at the indicated point

and sketch its graph: x + y = 3, (4,1)

2.5 Related Rates II.E.6 determine the rate of change determine the rate of II.E.4 of a certain quantity given the change of the radius of a 8.2.12.E.1 rate of change of a related spherical balloon when the 8.2.12.E.3 quantity. radius is 2 inches if air is pumped in at a rate of

in 3 4.5 . min MTH 413 – Calculus 14

PROFICIENCIES/OBJECTIVES Teacher (Numbers in parentheses indicate correlation AP Standards Suggested Activities Evaluation/Assessment with the course proficiencies) Notes Students will: Students will: III. APPLICATIONS OF DIFFERENTIATION (20-28) 3.1 Extrema on an Interval II.B.2 find the extrema of emphasize the difference A. Extrema of a function II.E.1 f ( x ) = 3x4 − 4 x3 on the between absolute and B. Relative extrema and critical II.E.3 interval [−1,2] . relative extrema. numbers 8.1.12.A.1

C. Finding extrema on a closed explain the guidelines for interval analyzing the graph of a

function and illustrate by locating the absolute

extrema of the function 〉2x + 2 0 ≤ x ≤ 1 f (x) ∫ 2 3.2 Rolle’s Theorem and the II.C.3 let f ( x ) = x4 − 2 x2 find all ⌠ 4x 1 < x ≤ 3 Mean Value Theorem II.C.3 values of c in the interval on the interval [0,3]. A. Rolle’s Theorem 8.1.12.A.1 ( − 2,2) such that f ′(c) = 0 . B. The Mean Value Theorem explain how and when to use Rolle’s Theorem and  4  given f ( x ) = 5 −   , find all the Mean Value Theorem.  x  values of c in the open interval explain why the Mean (1,4) such that Value Theorem holds for

1 f (4)-f (1) x f (c)= f (x) = on [0,2] and 4 −1 x + 1 find the appropriate x- value where the average rate of change equals the instantaneous rate of change. MTH 413 – Calculus 15

PROFICIENCIES/OBJECTIVES Evaluation/ Teacher (Numbers in parentheses indicate correlation AP Standards Suggested Activities with the course proficiencies) Assessment Notes Students will: Students will: III. APPLICATIONS OF DIFFERENTIATION (continued) 3.3 Increasing and Decreasing II.C.2 find the open intervals on which find the critical Have students express an- Functions and the First II.C.2 3 numbers, the open f ( x ) = x 3 − x 2 is increasing or swers using Derivative Test 8.1.12.A.1 2 intervals on which interval A. Increasing and decreasing decreasing. f (x) is increasing or notation. functions decreasing and locate B. The first derivative test find the relative extrema of: all relative extrema 2 3 2 2 of f (x)=x − 3x + 5 . f ( x ) = (x − 4) 3 Use graphing calculator to verify their conclusions.

3.4 Concavity and the Second II.D.2 determine the open intervals in sketch the graph of a Derivative Test II.E.1 x2 + 1 function such that: which the graph of f ( x ) = is A. Concavity II.D x2 − 4 a. f ′ is negative and B. Points of inflection 8.1.12.A.1 concave upward or downward. decreasing. C. The second derivative test b. f ′ is everywhere

increasing

(continued on next page) MTH 413 – Calculus 16

PROFICIENCIES/OBJECTIVES Evaluation/ Teacher (Numbers in parentheses indicate correlation AP Standards Suggested Activities with the course proficiencies) Assessment Notes Students will: Students will: III. APPLICATIONS OF DIFFERENTIATION (continued) 3.4 Concavity and the Second II.D use a symbolic differentiation utility Derivative Test (continued) 8.1.12.A.1 to analyze f ( x ) = x2 6 − x2 over   − 6, 6 : a. find the first-order and second- order derivatives of the function. b. find any relative extrema and points of inflection. c. sketch the graph of f , f ′, f ′′ on the same coordinate axes and state the relationship between the behavior of f and the signs of f ′ and f ′′ . MTH 413 – Calculus 17

3.5 Limits and Infinity I.C.1 determine the limits: x 4 + 2 y find: lim A. Limits at infinity I.C.2 2x −1 x →∞ 4 2 a. lim 3x + x B. Horizontal asymptotes 8.1.12.A.1 x →∞ x + 1

3x − 2 explain how to find all b. lim x →∞ 2 the horizontal asymp- 2x + 1 3x − 2 totes of: c. lim 2x3 −1 x →−∞ 2 = 2 x + 1 f (x) 3 x + x

use a symbolic differentiation utility to analyze the graph of f (x) .

Label any extrema and/or asymptotes that exist:

1 f ( x ) = 5 − 2 x

3.6 A Summary of Curve II.E.1 analyze the graphs of: explain how finding Sketching 8.1.12.A.1 x2 − 2 x + 4 the derivatives helps f ( x ) = and x − 2 in curve sketching. f ( x ) = x4 −12 x3 + 48x2 − 64 x by stating the first and second create a function that has a vertical tote of derivatives, x and y intercepts, x = 5 and a horizon- vertical and horizontal totes, tal tote of y = 0 . critical numbers, points of inflection, domain, the intervals in which f (x) is increasing or decreasing and concavity. MTH 413 – Calculus 18

PROFICIENCIES/OBJECTIVES Teacher (Numbers in parentheses indicate correlation AP Standards Suggested Activities Evaluation/Assessment with the course proficiencies) Notes Students will: Students will: III. APPLICATIONS OF DIFFERENTIATION (continued) 3.7 Optimization Problems II.E.3 find the dimension of a box determine the level at

Applied minimum and 8.2.12.C.4 having maximum volume if which the average maximum problems 8.2.12.E.1 the box has a square base and a production cost per unit

8.2.12.E.3 surface area of 108 square will be a minimum if the inches. total cost of operating a

factory is 2 determine how much of a four C = 0.5x + 15x + 500.0

foot length of wire should be where is the number of x used to form one circle and units produced (the

how much should be used to average cost per unit is

form one square in order to given by C ). maximize the area enclosed by x

these two figures.

determine what size order

will produce a minimum

cost if the total cost for

ordering and storing x

2 300,000 units is C = 2 x .

3.8 Newton’s Method II.E use Newton’s Method to find the point on the graph Students can A. Newton’s Method IIE. approximate the zeros of of f (x) = x2 that is closest use graphing calculator to 8.2.12.C.4 3 2 B. Algebraic solutions of f ( x ) = 2x + x − x + 1 . Con- to (4,−3) . check their polynomial equations tinue the iterations until two solution. successive approximations differ by less than 0.0001. MTH 413 – Calculus 19

PROFICIENCIES/OBJECTIVES Teacher (Numbers in parentheses indicate correlation AP Standards Suggested Activities Evaluation/Assessment with the course proficiencies) Notes Students will: Students will: III. APPLICATIONS OF DIFFERENTIATION (continued) 3.9 Differentials II.E.8 show that the magnitude of the dy Discuss linear find the derivative and A. Linear approximations 8.1.12.A.1 relative error in R caused by a approxima- dx tion using the B. Comparing the derivative change in I is equal in the differential dy if tangent line. and the differential magnitude to the relative error f (x) = x2 . C. Estimating an error in I when the voltage is

D. Calculating differentials constant using Ohm’s Law find and compare ∆y and which states that when a current of I amps passes dy , given: 2 through a resistor of R ohms, y = 1 − 2x , x = 0,∆x = dx = −0.1

the voltage E applied to the resistor is E=IR.

Use differentials to approxi- use differentials to approx- mate the error and percent imate 3 28 and compare error made in calculating the their approximation to that area of a square when the side of a calculator. of the square is measured to be 5cm with an accuracy of ± 0.2cm. MTH 413 – Calculus 20

PROFICIENCIES/OBJECTIVES Teacher (Numbers in parentheses indicate correlation AP Standards Suggested Activities Evaluation/Assessment with the course proficiencies) Notes Students will: Students will: IV. INTEGRATION (29-39) 4.1 Antiderivatives and Indefinite III.D.1 integrate the following func- evaluate: A. Antiderivatives 8.1.12.A.1 tions: 3 ∫ x dx B. Notation for antiderivatives a. ∫ (x + 2)dx C. Basic integration rules 3 4 2 x + sin x dx D. Initial conditions and b. ∫ (3x − 5x + x)dx ∫ ( ) particular solutions x + 1 x2 + 5 c. dx dx ∫ x ∫ x

find the position function given 2 the height (s) as a function of if f ' ( x ) = x + 5x and the time (t) and determine when = the ball hits the ground, given f (0) 7, find f (x). that a ball is thrown upward with an initial velocity of 64 feet per second from an initial height of 80 feet.

MTH 413 – Calculus 21

PROFICIENCIES/OBJECTIVES Teacher (Numbers in parentheses indicate correlation AP Standards Suggested Activities Evaluation/Assessment with the course proficiencies) Notes Students will: Students will: IV. INTEGRATION (continued) 4.2 Area III.A.1 use the five rectangles to find using four subintervals of C. Sigma notation 8.1.12.A.1 two approximations of the area equal length, approximate D. Area of the region lying between the the area under f x = x2 E. The area of a plane region graph of f ( x ) = − x2 + 5 and ( ) F. Upper and lower sums the x-axis between x=0 and on the interval 1,4 using  x=2. upper and lower sums.

use the limit definition to find use the limit process to the area of the region bounded find the exact area under by the graph of f x = x2 on 1, 4 . f ( x ) = x3 , the x-axis and the ( )  vertical lines x=0 and x=1. MTH 413 – Calculus 22

PROFICIENCIES/OBJECTIVES Teacher (Numbers in parentheses indicate correlation AP Standards Suggested Activities Evaluation/Assessment with the course proficiencies) Notes Students will: Students will: IV. INTEGRATION (continued) 4.3 Reimann Sums and Definite III.A-1 sketch the region corres- evaluate: Integrals III.F 5 ponding to each definite x − 2 dx A. Reimann Sums 8.2.12.C.4 integral, then evaluate each ∫1 B. Definite integrals 4 integral using a geometric x − 3 dx C. Properties of definite formula: ∫0 ( ) integrals 3 a. 4dx ∫1 3 b. (x + 2)dx ∫0 2 c. 4 − x2 dx ∫− 2

sketch the region whose area is indicated y the definite integral, then use a geometric formula to evaluate the integral (a>0, r>0) 3 9 − x2 dx ∫−3 MTH 413 – Calculus 23

PROFICIENCIES/OBJECTIVES Teacher (Numbers in parentheses indicate correlation AP Standards Suggested Activities Evaluation/Assessment with the course proficiencies) Notes

Students will: Students will: IV. INTEGRATION (continued) III.C.1 Emphasize 4.4 The Fundamental Theorem of evaluate the function: evaluate: 2 relationship Calculus III.B,E 4 2 2 x −1 dx x dx between ∫ 0 ∫ 1 A. The Fundamental Theorem III.B,E Mean Value of Calculus III.C.1 sin x dx Theorem for B. The Mean Value Theorem 8.1.12.A.1 find the average value of a ∫ 0 Integrals and for integrals function f ( x ) = 3x2 − 2 x on d x Average 1 + t dt Value. C. Average value of a function the interval [1,4]. dx ∫o on an interval D. The Second Fundamental find c such that f(c) equals Theorem of Calculus the average value of f x = x2 + 3 on 0, 4 . ( ) 

find the average value of f x = x2 + 3 on 0, 4 ( ) 

MTH 413 – Calculus 24

PROFICIENCIES/OBJECTIVES Teacher (Numbers in parentheses indicate correlation AP Standards Suggested Activities Evaluation/Assessment with the course proficiencies) Notes

Students will: Students will: IV. INTEGRATION (continued) Discuss 4.5 Integration by Substitution III.D.2 evaluate: 2 x −1 dx evaluate: ∫ changing A. Pattern recognition 8.1.12.A.1 x x2 + 1 dx ∫ limits. B. Change of variables 5 x C. The general Power Rule for evaluate: A = dx sin x ∫1 dx integration 2x −1 ∫ cos x D. Change of variables for 2 definite integrals/integration 2 8 use the fact that x dx = to ∫0 of even and odd functions 3 evaluate the definite integrals

without using the Fundamental

Theorem of Calculus: 0 2 2 2 a. x dx c. − x dx ∫− 2 ∫0 2 0 2 2

2 ∫− 2

4.6 Numerical Integration use the Trapezoidal Rule to Approximate the definite This is a A. The Trapezoidal Rule III.F integral using four sub- calculator approximate sin x dx and topic. B. Error analysis 8.2.12.C.4 ∫0 intervals with (a) the compare the results for n=4 Trapezoidal Rule and (b) and n=8. Simpson’s Rule. 2 1 dx ∫0 3 1 + x

MTH 413 – Calculus 25

PROFICIENCIES/OBJECTIVES Teacher (Numbers in parentheses indicate correlation AP Standards Suggested Activities Evaluation/Assessment

with the course proficiencies) Notes Students will: Students will:

V. LOGARITHMIC, EXPONENTIAL AND OTHER TRANSCENDENTAL FUNCTIONS (40-46) 5.1 The Natural Logarithmic II.F.1 evaluate the differentiation of find: Refer back to 2 Second Function and Other II.F.1 logarithmic functions: = y' if y ln x Fundamental II.F.1 Transcendental Functions d Theorem of A. The Natural Logarithmic 8.1.12.A.1 a. [ln(2x)] dx find: Calculus. Function and differentiation d 2 b.  ln(x +1)  2  B. The number e dx  y' if y = ln  x x + 1  C. The derivative of the   d rational logarithmic function c. [x lnx ] dx

d 3 d. (lnx)  dx

find the derivative of: f ( x ) = ln [cos x]

use Implicit differentiation to dy find : dx x2 − 3 ln y + y 2 = 10

MTH 413 – Calculus 26

PROFICIENCIES/OBJECTIVES Teacher (Numbers in parentheses indicate correlation AP Standards Suggested Activities Evaluation/Assessment

with the course proficiencies) Notes Students will: Students will:

V. LOGARITHMIC, EXPONENTIAL AND OTHER TRANSCENDENTAL FUNCTIONS (continued) 5.2 Integration of the Natural III.D find the area of the region evaluate:

Logarithmic Function III.D bounded by the graph of x A. Log rule for integration 8.1.12.A.1 x dx = ∫ 2 y 2 , the x-axis and the x + 2 B. Integrals of trigonometric (x + 1)

functions line x=3. tan x dx ∫

/ 4 2 evaluate: 1 + tan xdx ∫ 0

5.3 Inverse Functions F.BF.a-d find the inverse of: find f −1 ' a if A. Inverse functions II.E.5 f ( x ) = 2x − 3 ( ) ( )

B. Existence of an inverse II.F.1 3 function 8.1.12.A.1 f x = 2x + 4x − 5 and  −1  ( ) evaluate: Dx tan x C. derivative of an inverse  a = -5. function

MTH 413 – Calculus 27

PROFICIENCIES/OBJECTIVES Teacher (Numbers in parentheses indicate correlation AP Standards Suggested Activities Evaluation/Assessment

with the course proficiencies) Notes Students will: Students will:

V. LOGARITHMIC, EXPONENTIAL AND OTHER TRANSCENDENTAL FUNCTIONS (continued) 5.4 Exponential Functions: II.F.1 find: evaluate: x x Differentiation and Integration III.D.2 sin x f ' x if f x = e Dx(e sin e ) A. Derivatives involving natural 8.1.12.A.1 ( ) ( )

exponential function

B. Integrals involving natural evaluate: evaluate: ln x exponential function ' −2 x e dx e ∫0 dx ∫ x

MTH 413 – Calculus 28

PROFICIENCIES/OBJECTIVES Teacher (Numbers in parentheses indicate correlation AP Standards Suggested Activities Evaluation/Assessment Notes with the course proficiencies) Students will: Students will:

V. LOGARITHMIC, EXPONENTIAL AND OTHER TRANSCENDENTAL FUNCTIONS (continued) 5.5 Bases Other Than e and II.F1 evaluate: ∫ 2x dx find: Discuss the III.D2 relationship Application d  3 x  4 + log5 x between C. Bases other than e 8.1.12.A.1 dx   natural solve: A bacterial culture is D. Differentiation and integra- growing according to the logarithms and tion function: evaluate: exponentials. E. Power Rule for real experi- 2 y x 3x dx ments = 1 < y ∫ ( ) t F. Application where y is the weight of the culture in grams, t is the time

in hours, and y (u) = 100. Find the weight of the culture after:

a. 1 hour b. 10 hours

c. What is the limit as t approaches infinity?

5.6 Differential Equations, III.E.2 solve the differential equation: solve: Discuss the

Growth and Decay 8.1.12.A.1 2 x importance y ′ = y' = x y where y 0 = 1 of initial A. Differential equations ( ) y condition. B. Growth and decay models (optional) The number of bacteria grows at a rate proportional to the number present. If there are 100 present at a certain time and 400 present 5 hours later, how many will there be 11 hours after the initial time?

MTH 413 – Calculus 29

PROFICIENCIES/OBJECTIVES Teacher (Numbers in parentheses indicate correlation AP Standards Suggested Activities Evaluation/Assessment with the course proficiencies) Notes

Students will: Students will: V. LOGARITHMIC, EXPONENTIAL AND OTHER TRANSCENDENTAL FUNCTIONS (continued)

5.7 Inverse Trigonometric Func- II.F.1 differentiate: evaluate: II.E.5 2 tions and Differentiation = + −1 x a. y arcsin x x 1 − x cos ( sin )0

trigonometric functions −1 x use a graphing utility to y = tan (1 + e ) C. Review of basic differential graph f ( x ) = sin x and rules = evaluate y′′ in terms of x g( x ) arcsin ( sin x) . a. Why isn’t the graph of g the and y if : −1 = + line y=x? x sin y x y b. Determine the extreme of g.

5.8 Inverse Trigonometric Func- III.D.2 evaluate: describe how they know

tion Integration and III.D.2 x + 2 when to use Completing a. dx Completing the Square 8.1.12.A.1 ∫ 2 the Square to evaluate 4 − x A. Integrals involving inverse integrals. dx trigonometric functions b. ∫ 2 B. Completing the Square 2x − 8x + 10

MTH 413 – Calculus 30

PROFICIENCIES/OBJECTIVES Evaluation/ Teacher (Numbers in parentheses indicate correlation AP Standards Suggested Activities with the course proficiencies) Assessment Notes Students will: Students will: VI. APPLICATIONS OF INTEGRATION (47-52) 6.1 Area of a Region Between III.B find the area of the region between find the area of the Use graphing Two Curves 8.2.12.C.4 the graphs of f ( x ) = 3x3 − x2 −10 x region bounded by calculator to visualize 2 3 2 A. Area of a region between and g( x ) = − x + 2x . y = 3x – x – 10x regions and 2 two curves and y = -x + 2x. find limits. B. Area of a region between use a symbolic integration utility to intersecting curves graph the region bounded by the use a graphing graphs of the functions and find the calculator to find area of the region the area of the 4 region bounded by f ( x ) = x , g( x ) = 3x + 4 f(x) = x2 and g(x) = 3x + 4. MTH 413 – Calculus 31

PROFICIENCIES/OBJECTIVES Teacher (Numbers in parentheses indicate correlation AP Standards Suggested Activities Evaluation/Assessment with the course proficiencies) Notes Students will: Students will: VI. APPLICATIONS OF INTEGRATION (continued) 6.2 Volume: The Disc Method III.B find the volume of the solid find the volume of the Do some A. The Disc Method 8.1.12.A.1 formed by revolving the region solid formed by revolving examples using both B. The Washer Method bounded by the graph of the region bounded by disks and C. Solids with know cross f ( x ) = sin x and the x-axis y = sin x , the x-axis, shells. sections (0 ≤ x ≤ ) about the x-axis. x = 0, x = π about the x-

axis.

find the volume of the solid formed by revolving the region

bounded by the graphs of y = x and y = x2 about the

x-axis.

6.3 Volume: The Shell Method III.B find the volume of the solid of find the volume of the A. The Shell Method 8.1.12.A.1 revolution formed by revolving solid formed by revolving B. Comparison of disc and 8.2.12.C.4 the region bounded by the region bounded by the 2 shell methods y = x − x3 and the x-axis graph of y = x + 1, y = 0 , (0 ≤ x ≤ 1) about the y-axis. x = 0 and x=1 about the y- axis. MTH 413 – Calculus 32

PROFICIENCIES/OBJECTIVES Evaluation/ Teacher (Numbers in parentheses indicate correlation AP Standards Suggested Activities Assessment Notes with the course proficiencies) Students will: Students will:

VII. INTEGRATION TECHNIQUES, L’HÔPITAL’S RULE, AND IMPROPER INTEGRALS (42-45) 7.1 Basic Integration Rules III.D evaluate and compare the evaluate: Review only as necessary. A. Basic integration rules 8.1.12.A.1 following integrals: 9 dx B. Fitting integrands into basic 4 ∫ 2 a. dx x + 9 rules 2 ∫ x + 9 9x

4 x 2 dx and b. dx ∫ x + 9 ∫ 2 x + 9 2 9x 2 4 x 2 dx c. ∫ x + 9 2 dx ∫ x + 9

use a symbolic integration utility

to evaluate the integral, then verify the result “by hand”:

1 dx ∫ 2 x + 24x + 13

7.2 Integration by Parts III.D evaluate: evaluate: 2 A. Integration by parts 8.1.12.A.1 x a. ∫ xe dx a. ∫ x ln x dx B. Tabular Method x 3 b. e sin x dx b. ∫ sec dx ∫ c. ∫ arc sin x dx

MTH 413 – Calculus 33

PROFICIENCIES/OBJECTIVES Evaluation/Assessm Teacher (Numbers in parentheses indicate correlation AP Standards Suggested Activities with the course proficiencies) ent Notes Students will: Students will: VII. INTEGRATION TECHNIQUES, L’HÔPITAL’S RULE, AND IMPROPER INTEGRALS (continued) 7.3 Trigonometric Integrals III.D evaluate: evaluate: A. Integrals involving powers 8.1.12.A.1 a. sin3 x cos4 x dx sin2 x cos5 x dx of sine and cosine ∫ ∫ 4 3 4 B. Integrals involving powers b. ∫ sec 3x tan 3x dx ∫ cos x dx of secant and tangent

7.4 Trigonometric Substitution III.D evaluate: evaluate: A. Trigonometric substitution 8.1.12.A.1 dx dx a. 3 B. Applications ∫ 2 ∫ 4 x + 1 2 2 x + 1 dx ( ) b. ; x > 4 ∫ 2 3/2 (x − 4 x) 2 2 x − 3 dx ∫ 3 x

evaluate: 7.5 Partial Fractions III.D write the partial fraction decomposi-

A. Partial fractions 8.1.12.A.1 1 dx 2 B. Linear factors tion for 2 . ∫ + x − 5x + 6 x − 5x 6 C. Quadratic factors x2 dx 3 ∫ 2 2x − 4x − 8 2 evaluate: dx x + 1 ∫ (x2 − x)(x2 + 4) ( ) MTH 413 – Calculus 34

PROFICIENCIES/OBJECTIVES Teacher (Numbers in parentheses indicate correlation with the AP Standards Suggested Activities Evaluation/Assessment course proficiencies) Notes Students will: Students will: VII. INTEGRATION TECHNIQUES, L’HÔPITAL’S RULE, AND IMPROPER INTEGRALS (continued) 7.6 L’Hôpital’s Rule II.E.9 discuss the use of L’Hôpital’s evaluate: Discuss 2 x sin x 8.1.12.A.1 Rule in evaluating lim e −1 lim x x→0 0 ∞ ∞ o x → 0 x , , 0 ∞, 1 and 0 . 0 ∞ ⋅ evaluate: lim e− x x x → ∞

MTH 413 – Calculus 35

BI BLI OGR A P HY TEXTBOOKS:

Larson, Hostetler, Edwards. Calculus. 5th ed. Lexington: D.C. Heath, 1994.

Larson, Hostetler, Edwards. Calculus. 8th ed. Lexington: D.C. Heath, 2006. (Pending approval, CCPC 4/19/07)

RESOURCES:

Barnes-Robinson, Linda, Sue Jeweler and Mary Cay-Ricci. “Potential: Winged Possibilities to Dreams Realized.” Parenting for High Potential June 2004: 20. (Self-Assessment Rubric for Work Habits)

Foerster, P. Calculus, Concepts and Applications. Berkley: Key Curriculum Press, 1998.

Finney, R., F. Demana, B. Waits and D. Kennedy. Calculus: Graphical, Numerical, Algebraic. Upper Saddle River: Prentice Hall, 2003.

Cade, S., R. Caldwell and J. Lucia. Fast track to a 5: Preparing for the AP Calculus AB and Calculus BC Examinations. New York: McDougal Littell, 2006.

Ruby, T., J. Sellers, L. Kurt, J. VanHorn and M. Munn. Kaplan AP Calculus AB and BC. New York: Simon and Schuster, 2006.

ADDITIONAL RESOURCES:

AP Calculus AB. (n.d.) Retrieved from https://apstudent.collegeboard.org/apcourse/ap-calculus-ab/exam-practice Released AP free response problems and scoring guides

Course and Exam Description AP Calculus AB and AP Calculus BC. (n.d.). Retrieved from https://secure- media.collegeboard.org/digitalServices/pdf/ap/ap-calculus-ab-and-bc-course-and-exam-description.pdf Topic outline and curriculum notes

Dawkins, P. (n.d.). Pauls Online Math Notes. Retrieved from http://tutorial.math.lamar.edu/ Notes and examples

JMT, P. (n.d.). PatrickJMT. Retrieved June 07, 2016, from http://patrickjmt.com/#calculus Instructional videos

MTH 413 – Calculus 36

Larson, R., Hostetler, R. P., & Edwards, B. H. (2006). Calculus. Boston: Houghton Mifflin. Course textbook

Mathispower4u Calculus I Videos (Diff). (n.d.). Retrieved from http://www.mathispower4u.com/calculus.php Link to instructional videos

WEBSITES:

www.classzone.com www.domath.org www.enc.org www.forumswarthmore.edu www.illuminations.nctm.org www.mathforum.com www.mathgoodies.com www.nctm.org www.apcentral.collegeboard.com www.mathdemos.gcsu.edu

MTH 413 – Calculus

APPE NDI X A

SA M P LE A U THEN TI C A SSESSM E NT

MTH 413 – Calculus

SA M P LE A U THEN TI C A SSESSM EN T

OPTIMIZATION PROJECT

You have just been hired by the U CAN company, a company that provides can designs to food manufacturers. With your extensive calculus knowledge you decide that your first project will be to determine the efficiency of current can designs. You need to determine if cans of food items are built to optimal size. Create a written proposal complete with all the calculus needed to support it that you would use to approach companies and convince them to employ U CAN to design their cans.

TASK:

You are to go through your food pantry or a supermarket and chose at least three cans to examine. Find the volume of the cans in milliliters. Show how to minimize the cost of the can without altering the volume. (Conversions to remember: 1 fl. oz = 29.6 ml. and 1 ml = 1 cm3.)

While some cans are built to minimize cost, it’s interesting to note that many cans are not.

DIRECTIONS:

Your proposal must include:

a table showing the kind, the volume, the radius, the height, the optimal surface area and the actual surface area for each can. a formula for the surface area in terms of the radius and a graph, on appropriate axes, to go with this equation. Show all of the calculus you use in your analysis. an explanation of why some of your cans were not built to optimal surface area. One way to do this is to suppose that you were the manager of the company and decide what you would do with this new information in order to save money on can production.

MTH 413 – Calculus

S CO RI NG G UI DE F OR WR I TTEN A U THEN TI C A SSESSM EN T

GRADE CONTENT CLARITY

100/A All parts of assignment addressed Few, if any, unclear portions Description is complete Correct spelling and/or grammar Graphs and calculations present

93/A- Characteristics of A, but an element may be Few unclear portions missing Few spelling and/or grammatical errors

85/B Most of assignment complete Mostly clear, some confusing passages Some details missing Graph, calculation or diagram contains some errors

77/C Needs improvement in descriptions, graphs Some passages are confusing and/or explanations Spelling or grammar errors may add to the confusion Parts of assignment missing

60/D Many parts of assignments are missing Writing is incoherent in many places The purpose or procedure is not clear Meaning is difficult to determine

0/F Not handed in, or no effort Not handed in

MTH 413 – Calculus

APPE NDI X C

RUB RI C FO R S CO RI NG O PE N - E NDE D PRO B L E MS

MTH 413 – Calculus

RUB RI C FO R S CO RI NG O PE N - E NDE D PRO B L E MS

Your response showed a complete understanding of the problem’s essential mathematical concepts. You executed procedures completely and gave relevant responses to all parts of the task. Your response contained a 3-POINT RESPONSE few minor errors, if any. Your response was clear and effective, detailing how the problem was solved so that the reader did not need to infer how and why decisions were made.

Your response showed a nearly complete understanding of the problem’s essential mathematical concepts. You executed nearly all procedures and gave relevant responses to most parts of the task. Your response may have 2 POINT RESPONSE had minor errors. Your explanation detailing how the problem was solved, may not have been clear, causing the reader to make some inferences.

Your response showed a limited understanding of the problem’s essential mathematical concepts. Your response and procedures may have been incomplete and/or may have contained major errors. An incomplete 1 POINT RESPONSE explanation of how the problem was solved may have contributed to questions as to how and why decisions were made.

Your response showed an insufficient understanding of the problem’s essential mathematical concepts. Your procedures, if any, contained major errors. There may have been no explanations of the solution or the reader 0 POINT RESPONSE may not have been able to understand the explanation. The reader may not have been able to understand how and why decisions were made.

SOURCE: NEW JERSEY HSPA

MTH 413 – Calculus

APPE NDI X D

SELF - A SSESSM EN T

MTH 413 – Calculus

SELF-ASSESSMENT SHEET FOR STUDENT WORKFOLDER

Name Course

What I did well... How I could improve… Date Assessment Grade Things I need to work on...

Marking Period

MTH 413 – Calculus

NAME:

MI D - YE AR RE FL E CT I O N

After looking over your workfolder with all of your assessments from this year, what are your strengths in math? What are your weaknesses?

How can you continue to use your strengths to be successful in math? Be specific and explain.

How can you improve your areas of weakness? Give yourself at least one goal in order to help you improve.

Which assessment(s) are you most proud of? Explain why.

Which assessment(s) do you think you could have done better on? Explain why and how.

We are now half-way through the school year. What will you continue to strive for in math? How do you plan on doing this?

MTH 413 – Calculus

Workfolder Reflection

Look through the various items in your workfolder and take a moment to think about this school year. Answer the following questions in the form of a paragraph to reflect on your mathematical progress so far this year.

. What were some of your goals in the beginning of this school year? Have you made progress towards achieving them?

. What are some goals you have for the rest of this school year?

. In what areas did you have the most success? Be specific by indicating the topics in which you feel most confident.

. In what areas did you have difficulty? What are some ways you can improve in those areas?

. What can you do to prepare yourself for the final exam?

. Now that more than half of the year has passed, what are some things that you have learned that will help you next year? (e.g., study skills, putting more effort in homework, etc.)

. What are some things that you enjoy about this class? What are some things you don’t like? Do you have any suggestions as to what would make the class better?

MTH 413 – Calculus

APPE NDI X E

ADVANCE D PL ACE ME NT CAL CUL US B C T O PI C O UT L I NE

MTH 413 – Calculus

ADVANCE D PL ACE ME NT CAL CUL US B C T O PI C O UT L I NE

Topic Outline The topic outline for Calculus BC includes all Calculus AB topics. Additional topics are found in paragraphs that are marked with a plus sign (+) or an asterisk (*). The additional topics can be taught anywhere in the course that the instructor wishes. Some topics will naturally fit immediately after their Calculus AB counterparts. Other topics may fit best after the completion of the Calculus AB topic outline. Although the examination is based on the topics listed here, teachers may wish to enrich their courses with additional topics.

I. Functions, Graphs, and Limits

II. Derivatives

III. Integrals

IV. *Polynomial Approximations and Series I. Functions, Graphs, and Limits

A. Analysis of Graphs With the aid of technology, graphs of functions are often easy to produce. The emphasis is on the interplay between the geometric and analytic information and on the use of calculus both to predict and to explain the observed local and global behavior of a function.

B. Limits of Functions (incl. one-sided limits)

. An intuitive understanding of the limiting process.

. Calculating limits using algebra.

. Estimating limits from graphs or tables of data.

C. Asymptotic and Unbounded Behavior

. Understanding asymptotes in terms of graphical behavior.

. Describing asymptotic behavior in terms of limits involving infinity.

. Comparing relative magnitudes of functions and their rates of change. (For example, contrasting exponential growth, polynomial growth, and logarithmic growth.)

D. Continuity as a Property of Functions

. An intuitive understanding of continuity. (The function values can be made as close as desired by taking sufficiently close values of the domain.)

. Understanding continuity in terms of limits.

MTH 413 – Calculus

. Geometric understanding of graphs of continuous functions (Intermediate Value Theorem and Extreme Value Theorem).

E. *Parametric, Polar, and Vector Functions The analysis of planar curves includes those given in parametric form, polar form, and vector form.

II. Derivatives

A. Concept of the Derivative . Derivative presented graphically, numerically, and analytically.

. Derivative interpreted as an instantaneous rate of change.

. Derivative defined as the limit of the difference quotient.

. Relationship between differentiability and continuity.

B. Derivative at a Point . Slope of a curve at a point. Examples are emphasized, including points at which there are vertical tangents and points at which there are no tangents.

. Tangent line to a curve at a point and local linear approximation.

. Instantaneous rate of change as the limit of average rate of change.

. Approximate rate of change from graphs and tables of values.

C. Derivative as a Function . Corresponding characteristics of graphs of 'f and f '.

. Relationship between the increasing and decreasing behavior of f and the sign of f '.

. The Mean Value Theorem and its geometric consequences.

. Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa.

D. Second Derivatives . Corresponding characteristics of the graphs of f, f ', and f ".

. Relationship between the concavity of f and the sign of f ".

. Points of inflection as places where concavity changes.

E. Applications of Derivatives . Analysis of curves, including the notions of monotonicity and concavity.

. + Analysis of planar curves given in parametric form, polar form, and vector form, including velocity and acceleration.

MTH 413 – Calculus

. Optimization, both absolute (global) and relative (local) extrema.

. Modeling rates of change, including related rates problems.

. Use of implicit differentiation to find the derivative of an inverse function.

. Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration.

. Geometric interpretation of differential equations via slope fields and the relationship between slope fields and solution curves for differential equations.

. + Numerical solution of differential equations using Euler's method.

. + L'Hôpital's Rule, including its use in determining limits and convergence of improper integrals and series.

F. Computation of Derivatives

. Knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonometric functions.

. Basic rules for the derivative of sums, products, and quotients of functions.

. Chain rule and implicit differentiation.

. + Derivatives of parametric, polar, and vector functions. III. Integrals

A. Interpretations and Properties of Definite Integrals

. Definite integral as a limit of Riemann sums.

. Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval:

. Basic properties of definite integrals. (Examples include additivity and linearity.)

B. *Applications of Integrals Appropriate integrals are used in a variety of applications to model physical, social, or economic situations. Although only a sampling of applications can be included in any specific course, students should be able to adapt their knowledge and techniques to solve other similar application problems. Whatever applications are chosen, the emphasis is on using the method of setting up an approximating Riemann sum and representing its limit as a definite integral. To provide a common foundation, specific applications should include finding the area of a region (including a region bounded by polar curves), the volume of a solid with known cross sections, the average value of a function, the distance traveled by a particle along a line, and for BC only the length of a curve (including a curve given in parametric form).

MTH 413 – Calculus

C. Fundamental Theorem of Calculus

. Use of the Fundamental Theorem to evaluate definite integrals.

. Use of the Fundamental Theorem to represent a particular antiderivative, and the analytical and graphical analysis of functions so defined.

D. Techniques of Antidifferentiation

. Antiderivatives following directly from derivatives of basic functions.

. + Antiderivatives by substitution of variables (including change of limits for definite integrals), parts, and simple partial fractions (nonrepeating linear factors only).

. + Improper integrals (as limits of definite integrals).

E. Applications of Antidifferentiation

. Finding specific antiderivatives using initial conditions, including applications to motion along a line.

. Solving separable differential equations and using them in modeling. In particular, studying the equation y ' = ky and exponential growth.

. + Solving logistic differential equations and using them in modeling.

F. Numerical Approximations to Definite Integrals Use of Riemann sums (using left, right, and midpoint evaluation points) and trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically, and by tables of values.

IV. *Polynomial Approximations and Series

A. *Concept of Series A series is defined as a sequence of partial sums, and convergence is defined in terms of the limit of the sequence of partial sums. Technology can be used to explore convergence or divergence.

B. *Series of constants . + Motivating examples, including decimal expansion.

. + Geometric series with applications.

. + The harmonic series.

. + Alternating series with error bound.

. + Terms of series as areas of rectangles and their relationship to improper integrals, including the integral test and its use in testing the convergence of p-series.

. + The ratio test for convergence and divergence.

. + Comparing series to test for convergence or divergence. MTH 413 – Calculus

C. *Taylor Series . + Taylor polynomial approximation with graphical demonstration of convergence. (For example, viewing graphs of various Taylor polynomials of the sine function approximating the sine curve.)

. + Maclaurin series and the general Taylor series centered at x = a.

. + Maclaurin series for the functions ex, sin x, cos x, and 1/(1-x).

. + Formal manipulation of Taylor series and shortcuts to computing Taylor series, including substitution, differentiation, antidifferentiation, and the formation of new series from known series.

. + Functions defined by power series.

. + Radius and interval of convergence of power series.

. + Lagrange error bound for Taylor polynomials.

MTH 413 – Calculus

APPE NDI X F

NEW JERSEY STUDENT LEARNING

STANDARDS

MTH 413 – Calculus

NEW JERSEY STUDENT LEARNING STANDARDS

4 - Mathematics 8 - Technology 9 - 21st Century Life and Careers

MTH 413 – Calculus

A P P E N D I X G

S H O W C A S E P O R T F O L I O G U I D E L I N E S

MTH 413 – Calculus

Parsippany – Troy Hills Secondary Math Departments

Showcase Portfolio Guidelines

All secondary math courses showcase portfolios will contain evidence of the following NJ Core Curriculum Content Standards in Mathematics:

1. Problem-Solving 2. Reasoning 3. Tools and Technology 4. Patterns, Relationships and Functions

Specifically the student’s showcase portfolio for each subject will display evidence of the following local standards:

Calculus:

1. Transference and extension of pre-calculus concepts. 2. Understanding and applications of major concepts. 3. Problem-solving.

MTH 413 – Calculus

APPENDIX H CURRICULUM MODIFICATIONS & ADAPTATIONS

MTH 413 – Calculus