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Calculus 1 By Ted Cann

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Westchester Community College Calculus 1

Course Requirements Summer Session 1- 2012

Professor Ted Cann Email: tcann290@gmail.com

Office Hours: before class by appt. TEXTBOOK: Calculus of a Single Variable, Ninth Edition, by Larson & Edwards. PREREQUISITES: Pre-Calculus (Including Trigonometry) or Equivalent. CALCULATOR: TI-83/84+ graphing calculator (Not a TI-85,86,89,92) ATTENDANCE: Attendance will be taken each class. Any student who is absent TWO TIMES OR LESS will have their lowest test grade dropped. CLASS PARTICIPATION: Please come to each class prepared with your textbook, notebook (preferably a graph notebook), worksheets and calculator. Preparedness is essential to participation. Please be prepared to participate in class at all times. Your grade will reflect your preparedness and participation. LATENESS: Please arrive to class on time and be ready to learn. Lateness is a distraction to the class as a whole. Three times late equals one time absent. HOMEWORK ASSIGNMENTS: Homework will be assigned at every class session. Be prepared to hand in your work for a grade at any time. This may or may not be announced in advance. TESTS: Tests will be scheduled with a reasonable amount of notice. Therefore, there will be no make-ups allowed. A missed test will count as a drop unless you do not meet the attendance requirement. In that case, a 0 will be factored into your grade. GRADING POLICY: Final Exam 20% Tests 40% Quizzes/Tech Assignments 30% Attendance/Participations 10% COURSE WITHDRAWAL: Any student may withdraw from this course with a grade of W at any time until Monday, June 11, 2012. After then, a student who does not complete the course will receive a grade of F. The grades of WP, WF, and Incomplete are awarded only in exceptional circumstances beyond the student’s control. This is in full compliance with WCC policy.

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WESTCHESTER COMMUNITY COLLEGE CALCULUS 1

COURSE SYLLABUS

Topics of Study Section in Text Graphs and Models P.1 Linear Models and Rates of Change P.2 Functions and Their Graphs P.3 Fitting Models to Data P.4 A Preview of Calculus 1.1 Finding Limits Graphically and Numerically 1.2 Evaluating Limits Analytically 1.3 Continuity and One-sided Limits 1.4 Infinite Limits 1.5 The Derivative and the Tangent Line Problem 2.1 Basic Differentiation Rules and Rates of Change 2.2 The Product and Quotient Rule and Higher-Order Derivatives

2.3

The Chain Rule 2.4 Implicit Differentiation 2.5 Related Rates 2.6 Extrema on an Interval 3.1 Rolle’s Theorem and the Mean Value Theorem 3.2 Increasing and Decreasing Functions and the First Derivative Test

3.3

Concavity and the Second Derivative Test 3.4 Limits at Infinity 3.5 A Summary of Curve Sketching 3.6 Optimization Problems 3.7 Newton’s Method 3.8 Differentials 3.9 Antiderivatives and Indefinite Integration 4.1 Area 4.2 Riemann Sums and Definite Integrals 4.3 The Fundamental Theorem of Calculus 4.4 Integration by Substitution 4.5 Numerical Integration 4.6

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PreReq

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Calculus 1 Name:____________________________________ Review- Precalculus Material Date:_____________________________________ These are all examples of the topics that you should know and understand. You should be able to do all the material in this review packet without a calculator. If you are not fluent in this material you will be hampered in learning the new stuff. Additional practice can be found in the opening chapter of the calculus text book as well. Do all work on separate paper.

1. Use the binomial theorem or Pascal’s triangle to expand each of the following polynomials:

a. (a + b)4 c. (1 – y)6

b. (x – 3)3 d. (2x + 1)5

2. a. Knowing the polynomial 2x4 – 5x3 – 6x2 + 19x – 10 has a double root at x = 1, use synthetic division to find the other roots. b. Use the rational zero theorem and synthetic division to find the roots of

x3 + 3x2 – 4x – 12.

3. Find the value of each expression without the use of a calculator a. d. b. e.

c. f.

4. Factor each expression completely

a. x3 + 8 c. 8x3 + 64

b. x3 – 27 d. 27x3 – 125y3

5. Answer each of the following questions about logs and exponentials: a. What is the domain of y = ln x? b. What is the range of y = ln x? c. What is the y-intercept of y = ln x? d. What is the x-intercept of y = ln x? e. What is the domain of y = ex? f. What is the range of y = ex? g. What is the y-intercept of y = ex? h. What is the x-intercept of y = ex? i. Solve ex = 0 j. ln 1 = ? k. ln e = ? l. elnx = ? m. ln ex = ? n. ln 0 = ? o. Rewrite 2lnx using properties p. Explain how ex+ln3 = 3ex

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6. What does it mean for a function to be odd? even? Answer these questions not only with what it means for the function but what it means with regards to its symmetry.

7. Which trig functions are even and which are odd? Which basic polynomials are even and which are

odd?

8. Knowing which trig functions are even and odd should aid you in rewriting the following expressions in terms of x:

a. cos(-x) = b. sin(-x) =

9. Find each without the use of a calculator (think UNIT CIRCLE):

a. sin 0 f. cos 0 k. tan 0

b. sin g. cos l. tan

c. sin h. cos m. tan

d. sin i. cos n. tan

e. sin j. cos o. tan

10. Find each without the use of a calculator (think UNIT CIRCLE):

a. sec d. cot g. sec2

b. csc e. csc2 h. cot

c. sec f. cot 0 i. csc

11. On top of being able to evaluate trig expressions you should know stuff about them too…

a. What is the range of sin-1x? b. What is the range of cos-1x? c. What is the range of tan-1x? d. When you evaluate sin-1x, what quadrants does it give you the answer in? e. When you evaluate cos-1x, what quadrants does it give you the answer in? f. When you evaluate tan-1x, what quadrants does it give you the answer in? g. Why is sine positive in quadrants I and II but negative in III and IV? h. Why is cosine positive in quadrants I and IV but negative in II and III? i. Why is tangent positive in quadrants I and III but negative in II and IV?

12. Solve the trig equation in the given interval:

a. tan x = 1 b. sec x = -2

c. sin x = -0.5

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13. Which of the following is equivalent to 20 sin5x cos5x:

a. 10sin10x b. 20sin5x c. 10sin5x d. 4cos5x e. 20cos5x

14. Simplify each of the following expressions.

a.

c.

b. d.

15. What do you know about a function and its inverse? (2 related things)

16. You should be able to graph each of these basic functions without the aid of a calculator:

a. y = x2 b. y = x3 c. y = d. y = e. y = sin x f. y = cos x

17. Graph each of these equations using your knowledge of the graphs of basic functions and their transformations without the aid of a calculator (think a, h, k!)

a. y = 2x2 – 4 b. y = 2(x-4)2 c. y = sin(2x) + 1 d. y = (x+1)3 – 3 e. y = f. y =

g. y = 2cos -5

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18. You also need to be able to write a piecewise function. Be able to write a piecewise function for each of the following: a. b.

19. You should also be able to graph a piecewise function. Graph each of the following:

a.

b.

20. It is important too that you are able to understand and be able to write absolute value functions as a piecewise function.

a. b.

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Limits and Continuity

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Calculus 1 Name:__________________________________ Lesson- Graphing Piecewise/Absolute Value Functions Date:___________________________________ Objective: To learn how to graph a piecewise and absolute value function Do Now:

State the domain in interval notation and determine any asymptotes for the

______________________________________________________________________________________ Piecewise functions This is a graph that is exactly what it sounds like. It is a graph that is basically in pieces. Graph the following:

The procedure is to graph each part of the function separately. Absolute Value Functions Graph the Following

x

y

x

y

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(1)

(2)

(3)

(4)

x

y

x

y

x

y

x

y

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(5)

(6)

(7)

(8)

x

y

x

y

x

y

x

y

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Name________________________________ Calculus 1 Date_________________

Continuity

A function is continuous at a point c in its domain if and only if the limit of the function as x approaches c is equal to the value of the function at c, or mathematically

f(x) is continuous at c iff

A function is continuous at an endpoint if and only if the one sided limit at that endpoint equals the value of the function there, or mathematically

f(x) is continuous at left end point a iff

f(x) is continuous at right end point b iff

Figure (a) gives an example of a continuous function at x = 0

At which points does the limit not exist or does the limit not equal the value on the graph below?

* Remember * A limit only exists if the left hand limit is equal to the right hand limit, or mathematically

There are various types of discontinuities:

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a. Removable Discontinuities- These are discontinuities where the limit exists at a point, but it doesn’t equal the value of the function at that point. Figures (b) and (c) are examples of removable discontinuities at x = 0.

The mathematical expression for a removable discontinuity is:

Let’s see how to remove a discontinuity:

undefined, but it’s an indeterminate form

So the limit exists, but does not equal the value of the function, making it a removable discontinuity. So you can write an extended function, which is simply a function that fills in the hole (removable discount).

Method 1: Write a piecewise function Method 2: Rewrite as a new, simplified, function (if possible)

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b. Jump Discontinuities- These are discontinuities where the left and right hand limit at a point both exist but are not equal to each other. Figure (d) is an example of a jump discontinuity at x = 0.

The mathematical expression for a jump discontinuity is:

Questions: Is f(x) = int x continuous on [ 0 , 2 ]? How about the interval ( 0 , 1 )?

c. Infinite Discontinuities- These are discontinuities where the left and right hand limits at a point go to positive or negative infinity. Figures (e) and 2.22 are examples of infinite discontinuities.

The Mathematical expression for an infinite discontinuity is:

d. Oscillating Discontinuities- These are the most difficult to define as they are simply functions which oscillate around a point too much to have a limit at that point. Figure (f) is an example of an oscillating discontinuity.

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Calculus 1 Name:__________________________________ Lesson- Continuity & Increasing/decreasing/constant functions Date:___________________________________ Objectives: • determine whether a function is continuous or discontinuous

• determine whether a function is increasing, decreasing, or constant within an interval

Continuous graphs: Discontinuous graphs: Point Discontinuity:

Example:

Jump Discontinuity:

Example:

Infinite Discontinuity:

Example:

x

y

x

y

x

y

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ex) For each graph:

(a) Find the domain

(b) Find the range

(c) Find the intervals over which the function is increasing

(d) Find the intervals over which the function is decreasing

(e) Find the intervals over which the function is constant

(f) State any points of discontinuity

(1) (2)

x

y

x

y

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Calculus 1 Name:__________________________________ Lesson- Limits and Vertical Asymptotes Date:___________________________________ Objectives: • determine the limit as the graph approaches a vertical asymptote Limits:

(1) Graph and find the following limits:

(a)

(b)

(c)

(2) Graph and find the following limits:

(a)

(b)

(c)

x

y

x

y

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(3) Graph and find the following limits:

(a)

(b)

(c)

(4) Graph and find the following limits:

(a)

(b)

(c)

(5) Graph and find the following limits:

(a)

(b)

(c)

x

y

x

y

x

y

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Calculus 1 Name:__________________________________ Lesson- Determining limits graphically Date:___________________________________ Objectives: • determine the limit of a function graphically

(1) Graph and find the following limits:

(a)

(b)

(c)

(2) Graph and find the following limits:

(a)

(b)

(c)

(d)

(e)

(f)

x

y

x

y

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Given the value of c, use the graph of f to find each of the following values:

(a) (b) (c) (d)

(1) (2)

(4) (3)

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Calculus 1 Name:__________________________________ Lesson - Limits and horizontal asymptotes Date:___________________________________ Objectives: • determine the limit as the graph approaches a horizontal asymptote Limits where x approaches infinity:

Limits where x approaches a constant

Find each of the following limits:

(1)

(2)

(3)

(4)

(5)

(6)

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Find the limit for each of the following algebraically:

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

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Graphs, Functions, and Limits & Continuity Definitions, Properties & Formulas

Linear Equation equation of a straight line

Slope the slope, m, of the line through (x1, y1) and (x2, y2) is given by the following

equation, if x1 ≠ x2:

Types of Slope

y-intercept where the graph crosses the y-axis

x-intercept where the graph crosses the x-axis

Slope-Intercept Form

y = mx + b

where m represents the slope and b represents the y-intercept of the linear equation

Standard Form Ax + By = C where A, B, and C are constants and A ≥ 0 (positive, whole number)

Point-Slope Form

y – y1 = m(x – x1)

where m represents the slope and (x1, y1) are the coordinates of a point on the line of the linear equation

Parallel Lines

Two nonvertical lines in a plane are parallel if and only if their slopes are equal and they have no points in common. (Two vertical lines are always parallel.)

ex) y = 2x + 3 and y = 2x – 4 equal slopes m = 2 m = 2 // lines

Perpendicular Lines

Two nonvertical lines in a plane are perpendicular if and only if their slopes are negative reciprocals. (A horizontal and a vertical line are always perpendicular.)

ex) and neg. recip. slopes

m = m = ⊥ lines

Relation a set of ordered pairs (x, y)

Domain the set of all x-values of the ordered pairs

Range the set of all y-values of the ordered pairs

y

Negative

y

Positive

y

Zero horizontal line:

y = b

y

Undefined vertical line: x

= a

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Function a relation in which each element of the domain is paired with exactly one element in the range.

Vertical Line Test (VLT)

If any vertical line passes through two or more points on the graph of a relation, then it does not define a function.

Horizontal Line Test (HLT)

If any horizontal line passes through two or more points on the graph of a relation, then its inverse does not define a function.

One-to-One Functions

a function where each range element has a unique domain element

(use HLT to determine)

Inverse Relations & Functions

f -1(x) is the inverse of f(x), but f -1(x) may not be a function

(use HLT to determine)

Writing Inverse Functions

To find f -1(x): (1) let f(x) = y (2) switch the x and y variables (3) solve for y (4) let y = f -1(x)

Odd Functions symmetric with respect to the origin TEST: f(-x) = -f(x)

Even Functions symmetric with respect to the y-axis TEST: f(x) = f(-x)

Symmetry Tests

symmetric with respect to the: the given equation is equivalent when: y-axis x is replaced with -x x-axis y is replaced with –y origin x and y are replaced with –x and -y

Operations with Functions

sum: (f + g)(x) = f(x) + g(x) difference: (f – g)(x) = f(x) – g(x)

product: (f • g)(x) = f(x) • g(x)

quotient:

Composition of Functions

given functions f and g, the composite function is , where g(x) is substituted for x in the f(x) function

Correlation and

Regression

• Press Stat, Right Arrow, [4 through C] • , , Enter (Correlation and Regression data is now presented) • Zoom 9 (Line should appear on the graph verifying that equation is correct)

**Make sure Diagnostics are ON to get r and r-squared**

Piecewise Function

A function in which different equations are used for different intervals of the domain.

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Increasing, Decreasing, and

Constant Functions

Let I be an interval in the domain of a function f. Then: f is increasing on I if

f(b) > f(a) whenever b > a in I

f is decreasing on I if f(b) < f(a) whenever b

> a in I

f is constant on I if f(b) = f(a) for all a and

b in I

increasing on (-∞,∞)

decreasing on (-∞,∞)

constant on (-∞,∞)

Properties of Limits

If f is the constant function f(x) = k (the function whose outputs have the constant value k), then for any value of c, .

If f is the identity function f(x) = x, then for any value of c, .

If is any polynomial function, then limits can be found by a substitution method: .

If f(x) and g(x) are polynomials in a rational function (substitution method may

work, but not always), then: .

As : if f is a constant function f(x) = k, then: ; and

if f is the reciprocal function , then .

Right-Hand and Left-Hand Limits

A function f(x) has a limit as x approaches c if and only if the right-hand and left-hand limits at c exist and are equal:

Vertical and Horizontal

Asymptotes

A line y = b is a horizontal asymptote of the graph of a function y = f(x) if either: or

A line x = a is a vertical asymptote of the graph of a function y = f(x) if either: or

Oblique Asymptotes Use Long division to determine the quotient (drop any remainder)

x

y

x

y

x

y

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Calculus 1 Name:______________________________ Review- Units P & 1 Test Date:_______________________________

SHOW ALL WORK!

(1) Which of the following equations define functions? Explain your reasoning.

(a) y = x + 6

(b) y2 = x + 1

(c) y3 = x + 4

(d) y = x – 2

(e) y3 = x – 3

(f) y2 = x – 5

(2) Which of the following functions are one-to-one? Explain your reasoning.

(a) f(x) = x5

(b) g(x) = 2x + 7

(c) h(x) = x2 – 4

(3) Write the linear equation in standard form Ax + By = C that passes through the points (3, 5) and (4, -3).

(4) Write the linear equation in slope-intercept form y = mx + b that passes through the point (1, 3) and is parallel to the line 2x + 2y = 5.

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(5) Write the linear equation in slope-intercept form y = mx + b that passes through the point (1, 3) and is perpendicular to the line 3x – 3y = 4.

(6) Write the linear equation in standard form Ax + By = C that is a horizontal line and passes through the point (-9, 2).

__________________________________________________________________________________________

(7) Find the point(s) of intersection between: &

(8) Given f(x) = 2x2 – x + 3 find f(k – 2)

(9) Given f(x) = 4x – 7 and g(x) = 2x – x2, evaluate f(2) + g(-1)

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(10) For the function f(x) = -6x + 5, find and simplify:

(a) (b)

(11) Find the domain of each of the following:

(a)

(b)

(12) Algebraically determine the symmetry with respect to the y-axis, x-axis, and origin, if any exists, for each of the given equations:

(a) 2x – 4y = 7 (b) 9x2 – 4y2 = 36

(13) Algebraically determine if each of the given functions are odd, even, or neither: (a) f(x) = x2 – 6x (b) f(x) = x6 + 7

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(14) Perform the four basic operations with the functions f(x) = 4x and g(x) = x2 + 2

(f + g)(x) = (f – g)(x) = (f • g)(x) =

(15) Given f(x) = x2 – 3 and , find and and give each domain.

(16) For each of the following functions, f(x), find the inverse, f -1(x): (a) f(x) = 5x + 2

(b)

(17) Given the graph on the right, answer the following questions: (a) write the linear equation in slope-intercept form: (b) write in slope-intercept form the equation of the parallel line

through (2, -1) and graph: (c) write in slope-intercept form the equation of the perpendicular

line through the x-intercept and graph: _______________________________________________________________________________________

x

y

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Graph each function and use interval notation to answer the following questions: (a) Find the domain (b) Find the range (c) Find the intervals over which the function is increasing

(d) Find the intervals over which the function is decreasing (e) Find the intervals over which the function is constant (f) State any points of discontinuity

(18) (19)

(20) (21)

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x

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(22) (23)

(24) Graph the following piecewise function:

(25) Based on your graph of f(x) from question (7), find the value of each of the following limits: (a) =

(b) =

(c) =

(d) =

(e) =

(f) =

(g) =

(h) =

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y

x

y

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y

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(26) Based on the given graph of f(x), find the value of each of the following: (a) =

(b) =

(c) =

(d) =

(e) =

(f) f(3) =

(g) =

(h) =

Algebraically determine the value each of the following limits or explain why it does not exist:

(27) (28)

(29) (30)

y

x

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(31) (32)

(33) (34)

(35) (36)

(37) (38)

(39) (40)