Calculus Early Transcendental Functions PDF

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Calculus: Early Transcendental Functions Lecture Notes for Calculus 101

Feras Awad Mahmoud

Last Updated: August 2, 2012 1

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Feras Awad Mahmoud Department of Basic Sciences Philadelphia University JORDAN 19392

Textbook: This book is strongly recommended for Calculus 101 as well as a reference text for Calculus 102. The pdf soft−copy of the five chapters remain available for free download.

Typesetting: The entire document was written in LaTeX, implemented for Windows using the MiKTeX 2.9 distribution. As for the text editor of my choice, I fancy WinEdt 6.0.

Moral Support: My wife has had a very instrumental role in providing moral support. Thank God for her patience, understanding, encouragement, and prayers throughout the long process of writing and editing.

c 2012 Feras Awad Mahmoud. All Rights Reserved. www.philadelphia.edu.jo/academics/fawad

Contents Contents 1 Functions 1.1 Introduction . . . . . . . . 1.2 Essential Functions . . . . 1.3 Combinations of Functions 1.4 Inverse Functions . . . . . 1.5 Hyperbolic Functions . . .

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2 Limits and Continuity 2.1 An Introduction to Limits . . . . . . . . 2.2 Calculating Limits using the Limit Laws 2.3 Limits at Infinity and Infinite Limits . . 2.4 Limits Involving (sin θ) /θ . . . . . . . . 2.5 Continuous Functions . . . . . . . . . . .

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3 The Derivative 3.1 The Derivative as a Function . . . . . . . . . 3.2 Differentiation Rules and Higher Derivatives 3.3 The Chain Rule . . . . . . . . . . . . . . . . 3.4 Implicit Differentiation . . . . . . . . . . . . 3.5 Tangent Line . . . . . . . . . . . . . . . . . 4 Applications of Differentiation 3

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127

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CONTENTS 4.1 4.2 4.3 4.4 4.5

Indeterminate Forms and L’Hˆospital’s Rule . The Mean Value Theorem . . . . . . . . . . Extreme Values of Functions . . . . . . . . . Monotonic Functions . . . . . . . . . . . . . Concavity and Curve Sketching . . . . . . .

5 Integration 5.1 Antiderivatives . . . . . . . . . . . . . 5.2 Indefinite Integrals . . . . . . . . . . . 5.3 Integration by Substitution . . . . . . . 5.4 The Definite Integral . . . . . . . . . . 5.5 The Fundamental Theorem of Calculus 5.6 Area Between Two Curves . . . . . . .

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127 134 139 146 151

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159 159 160 170 175 181 188

A Solving Equations and Inequalities

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B Absolute Value

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C Equation of Line

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D Final Answers of Exercises

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Chapter

Functions 1.1

Introduction

Functions arise whenever one quantity depends on another. Definition 1.1.1. A function f is a rule that assigns to each element x in a set D exactly one element called f (x) in a set E . • We usually consider functions for which the sets D and E are sets of real numbers. • The set D is called the domain of the function. • The number f (x) is the value of f at x and is read f of x. • The range of f is the set of all possible values of f (x) as x varies throughout the domain • A symbol that represents an arbitrary number in the domain of a function f is called an independent variable. • A symbol that represents a number in the range of f is called a dependent variable. Since the y−coordinate of any point (x, y) on the graph is y = f (x), we can read the value of f (x) from the graph as being the height of the graph above the point x (see Figure 1.1).

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1

6

CHAPTER 1. FUNCTIONS

Figure 1.1:

The graph of f also allows us to picture the domain of f on the x−axis and its range on the y−axis as in Figure 1.2.

Figure 1.2:

Example 1.1. The graph of a function f is shown in Figure 1.3. (a) Find the values of f (1) and f (7). (b) What are the domain and range of f ? Solution 1.1. a) We see from Figure 1.3 that the point (1, 3) lies on the graph of f , so the value of f at 1 is f (1) = 3. (In other words, the point on the graph that lies above x = 1 is 3 units above the x−axis.) When x = 7, the graph lies on the x−axis, so we say that f (7) = 0. (In other words, x = 7 is a real root of f (x).)

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1.1. INTRODUCTION

Figure 1.3:

b) We see that f (x) is defined when 0 ≤ x ≤ 7, so the domain of f is the closed interval [0, 7]. Notice that f takes on all values from −2 to 4, so the range of f is the closed interval [−2, 4].  Representations of Functions There are four possible ways to represent a function: 1. verbally (by a description in words) 2. numerically (by a table of values) 3. visually (by a graph) 4. algebraically (by an explicit formula) The Vertical Line Test The graph of a function is a curve in the xy−plane. But the question arises: Which curves in the xy−plane are graphs of functions? This is answered by the Vertical Line Test: A curve in the xy−plane is the graph of a function of x if and only if no vertical line intersects the curve more than once. The reason for the truth of the Vertical Line Test can be seen in Figure 1.4. If each vertical line x = a intersects a curve only once, at (a, b), then exactly one functional value is defined by f (a) = b. But if a line x = a intersects the curve twice, at (a, b) and (a, c), then the curve cant represent a function because a function cant assign two different values to a.

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CHAPTER 1. FUNCTIONS

Figure 1.4:

Symmetry If a function f satisfies f (−x) = f (x) for every number x in its domain, then f is called an even function. For instance, the function f (x) = x2 is even because f (−x) = (−x)2 = x2 = f (x). If f satisfies f (−x) = −f (x) for every number x in its domain, then f is called an odd function. For example, the function f (x) = x3 is odd because f (−x) = (−x)3 = −x3 = f (x).

Figure 1.5: Even Function

Figure 1.6: Odd Function

1.2. ESSENTIAL FUNCTIONS

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The geometric significance of an even function is that its graph is symmetric with respect to the y−axis as in Figure 1.5, while the graph of an odd function is symmetric about the origin, see Figure 1.6. Example 1.2. Determine whether each of the following functions is even, odd, or neither even nor odd. (a) f (x) = x5 + x. (b) g(x) = 1 − x4 . (c) h(x) = 2x − x2 . Solution 1.2. (a) f (−x) = (−x)5 + (−x) = (−1)5 x5 + (−x) = −x5 −x = −(x5 + x) = −f (x). Therefore f is an odd function. (b) g(−x) = 1 − (−x)4 = 1 − x4 = g(x). So g is even. (c) h(−x) = 2(−x) − (−x)2 = −2x − x2 . Since h(−x) 6= h(x) and h(−x) 6= −h(x), we conclude that h is neither even nor odd. 

1.2

Essential Functions

There are many different types of functions that can be used to model relationships observed in the real world. In what follows, we discuss the behavior and graphs of these functions and give examples of situations appropriately modeled by such functions.

Linear Function When we say that y is a linear function of x, we mean that the graph of the function is a line, so we can use the slope−intercept form of the equation of a line (see Appendix C) to write a formula for the function as y = f (x) = mx+ b where m is the slope of the line and b is the y−intercept. Example 1.3. As dry air moves upward, it expands and cools. If the ground temperature is 20 ◦ C and the temperature at a height of 1 km is 10 ◦ C.

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CHAPTER 1. FUNCTIONS

(a) Express the temperature T (in ◦ C) as a function of the height h (in kilometers), assuming that a linear model is appropriate. (b) Draw the graph of the function in part (a). What does the slope represent? (c) What is the temperature at a height of 2.5 km? Solution 1.3. (a) Because we are assuming that T is a linear function of h, we can write T = mh + b. We are given that T = 20 when h = 0, so 20 = m × 0 + b = b. In other words, the y−intercept is b = 20. We are also given that T = 10 when h = 1, so 10 = m × 1 + 20. The slope of the line is therefore m = 10 − 20 = −10 and the required linear function is T = −10h + 20. (b) The graph is sketched in Figure 1.7. The slope is m = −10 ◦ C/km, and this represents the rate of change of temperature with respect to height.

Figure 1.7:

(c) At a height of h = 2.5 km, the temperature is T = −10(2.5) + 20 = −5 ◦ C. 

Polynomials A function P is called a polynomial if P (x) = an xn + an−1 xn−1 + ... + a2 x2 + a1 x + a0 where n is a nonnegative integer and the numbers a0 , a1 , a2 , ..., an

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1.2. ESSENTIAL FUNCTIONS are constants called the coefficients of the polynomial.

The domain of any polynomial is R = (−∞, ∞). If the leading coefficient an 6= 0, then the degree of the polynomial is n. For example, - A polynomial of degree 1 is of the form P (x) = mx + b and so it is a linear function. - A polynomial of degree 2 is of the form P (x) = ax2 + bx + c and is called a quadratic function. - A polynomial of degree 3 is of the form P (x) = ax3 + bx2 + cx + d and is called a cubic function. Remark 1.2.1. A polynomial of degree n has at most n zeros (roots).

Piecewise Defined Functions Example 1.4. A function f is defined by f (x) =



1 − x : x ≤ −1 x2 : x > −1

Evaluate f (−2), f (−1), and f (0) and sketch the graph. Solution 1.4. Remember that a function is a rule. For this particular function the rule is the following: First look at the value of the input x. If it happens that x ≤ −1, then the value of f (x) is 1 − x. On the other hand, if x > −1, then the value of f (x) is x2 . Since −2 ≤ −1, we have f (−2) = 1 − (−2) = 3. Since −1 ≤ −1, we have f (−1) = 1 − (−1) = 2. Since 0 > −1, we have f (0) = 02 = 0. The graph of this functions appears in Figure 1.8.  The absolute value (see Appendix B) is an example of a piecewise defined function.

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CHAPTER 1. FUNCTIONS

Figure 1.8:

Rational Functions Definition 1.2.1. A function in the form f (x) =

P (x) , Q(x)

where P and Q are polynomials, is called a rational function. The domain of the rational function f (x) is the set D = R − {x ∈ R : Q(x) = 0} Example 1.5. Find the domain of f (x) =

2x4 −x2 +1 . x2 −4

Solution 1.5.   D = R − x ∈ R : x2 − 4 = 0 = R − {−2, 2} The graph of the function is shown in Figure 1.9.  2

x −9 Example 1.6. Find the domain of f (x) = 1−|x| .

Solution 1.6. D = R − {x ∈ R : 1 − |x| = 0} = R − {−1, 1} 

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1.2. ESSENTIAL FUNCTIONS

Figure 1.9:

Exercise 1.1. What is the domain of each of the following functions. (a) f (x) =

1 |1−x|

−4x (b) g(x) = 10+x 2

Root Function Definition 1.2.2. For any integer n ≥ 2, p f (x) = n g(x)

is the nth root function of g(x). The domain of the root function depends on the value of n if it is even or odd. n is odd: The domain of f (x) in this case is the same as the domain of g(x). The range of f (x) will be R. n is even: In this case, the domain of f (x) is the set D = {x ∈ R : g (x) ≥ 0} ∩ {g (x) domain} The range of f is [0, ∞). Example 1.7. Find the domain of f (x) =

√3

x2 − 4.

Solution 1.7. Since f is odd root function, then D = domain (x2 − 4) = R

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CHAPTER 1. FUNCTIONS

Example 1.8. Find the domain of f (x) =

√

x2 − 4.

Solution 1.8. Since f is even root function, then   D = x ∈ R : x2 − 4 ≥ 0 ∩ { domain of x2 − 4} = (−∞, −2] ∪ [2, ∞) ∩ R = (−∞, −2] ∪ [2, ∞)

 Example 1.9. Find the domain of f (x) =

p 6

|x|.

Solution 1.9. Since f is even root function, then D = {x ∈ R : |x| ≥ 0} ∩ { domain of |x|} = R∩R = R  Example 1.10. Find the domain of f (x) =

√ 1 . 9−x2

Solution 1.10. This function is rational and its denominator is even root. The root’s domain is the dominant here. The domain of f is n o n o √ √ D = domain of 9 − x2 − x ∈ R : 9 − x2 = 0   = x ∈ R : 9 − x2 > 0 ∩ { domain of 9 − x2 }   = x ∈ R : x2 < 9 ∩ R = {x ∈ R : |x| < 3} ∩ R = (−3, 3) ∩ R = (−3, 3)

 Example 1.11. Find the domain of f (x) =

1√ . 1− x

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1.2. ESSENTIAL FUNCTIONS

Solution 1.11. This example is quite different from the previous example. The function f here is rational but its denominator contains an even root. The root’s domain is also the dominant here. So, the domain of f is   √   √ D = domain of x − x ∈ R : 1 − x = 0 = {x ∈ R : x ≥ 0} ∩ { domain of x} − {1} = [0, ∞) ∩ R − {1} = [0, ∞) − {1}

= [0, 1) ∪ (1, ∞)

 Example 1.12. Find the domain of f (x) =

p

2−

√

x.

Solution 1.12. Since f is even root function that contains an even root function inside it, then both roots are dominant here. Hence, the domain of f is   √ √ D = x ∈ R : 2 − x ≥ 0 ∩ { domain of 2 − x}   √ = x ∈ R : x ≤ 2 ∩ {x ∈ R : x ≥ 0} ∩ { domain of x} = {x ∈ R : 0 ≤ x ≤ 4} ∩ [0, ∞) ∩ R = [0, 4] ∩ [0, ∞) ∩ R = [0, 4]

 Exercise 1.2. Find the domain of the following. (a) f (x) =

1√ 1+ x

(b) g(x) =

√

(c) h(x) =

q

−x 1 x

−1

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CHAPTER 1. FUNCTIONS

Figure 1.10:

Trigonometric Functions In calculus the convention is that radian measure is always used (except when otherwise indicated). Figure 1.10 shows a sector of a circle with central angle θ and radius r subtending an arc with length s. Then the radian measure of the central angle A′ CB ′ is the number θ = rs . Example 1.13. Find the radian measure of 60◦ . Solution 1.13. To convert degrees to radians, multiply degrees by (π rad )/180◦ .  π  π ◦ = . 60 = 60 3 180 

Example 1.14. Express 5π/4 in degrees. Solution 1.14. To convert radians to degrees, multiply radians by 180◦ /(π rad ).    5π 5π 180◦ rad = = 225◦ . 4 4 π  Nonzero radians measures can be positive or negative and can go beyond 2π = 360◦ . The standard position of an angle occurs when we place its vertex at the origin of a coordinate system and its initial side on the positive x−axis as in Figure 1.11. A positive angle is obtained by rotating the initial side counterclockwise until it coincides with the terminal side. Likewise, negative angles are obtained by clockwise rotation.

1.2. ESSENTIAL FUNCTIONS

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Figure 1.11:

Figure 1.12:

Figure 1.12 shows several examples of angles in standard position. For an acute angle θ the six trigonometric functions are defined as ratios of lengths of sides of a right triangle as in Figure 1.13.

Figure 1.13:

This definition does not apply to obtuse or negative angles, so for a general angle θ in standard position we let P (x, y) be any point on the terminal side of θ and we let r be the distance |OP | as in Figure 1.14. The signs of the trigonometric functions for angles in each of the four quadrants can be remembered by means of the rule All Students Take Calculus shown in Figure 1.15. Example 1.15. Find the exact trigonometric ratios for θ = 2π/3.

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CHAPTER 1. FUNCTIONS

Figure 1.14:

Figure 1.15:

Solution 1.15. From 1.16 we see that a point on the terminal line  Figure √  √ −1 1 3 3 for θ = 2π/3 is P 2 , 2 . Therefore, taking x = − 2 and y = 2 in the definitions of the trigonometric ratios, we have √ 2π 3 sin = 2 3 2 2π =√ csc 3 3

2π 1 =− 3 2 2π = −2 sec 3

cos

√ 2π =− 3 3 2π −1 cot = √ 3 3

tan

 The following table gives some values of sin θ, cos θ and tan θ .

1.2. ESSENTIAL FUNCTIONS

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Figure 1.16:

Reference Angles The values of trigonometric functions of angles greater than 90◦ = π/2 or less than 0◦ can be determined from their values at corresponding acute angles called reference angles. Definition 1.2.3. Let θ be an angle in standard position. Its reference angle is the acute angle θ ′ formed by the terminal side of θ and the x−axis.

Example 1.16. Find the reference angle of θ = 5π/3.

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CHAPTER 1. FUNCTIONS Solution 1.16. Because 5π/3 = 300◦ lies in quadrant 4, the angle it makes with the x−axis is θ ′ = 2π − 5π/3 = π/3 = 60◦ .  Example 1.17. Find the reference angle of θ = −3π/4. Solution 1.17. First, note that −3π/4 = 135◦ is coterminal with 5π/4 = 225◦ which lies in quadrant 3. So, the reference angle is θ ′ = 5π/4 − π = π/4 = 45◦ .  Example 1.18. Evaluate each of the following. a) cos (4π/3) b) tan (−210◦ )

c) csc (11π/4)

Solution 1.18. a) Because θ = 4π/3 = 240◦ lies in quadrant 3, the reference angle is θ ′ = 4π/3 − π = π/3. Moreover, the cosine is negative in quadrant 3, so cos (4π/3) = (−) cos (π/3) = −1/2 b) Because −210◦ +360◦ = 150◦ , it follows that −210◦ is coterminal with the second-quadrant angle 150◦ . Therefore, the reference angle is θ ′ = 180◦ − 150◦ = 30◦ . Finally, because the tangent is negative in quadrant 2, you have √ √ tan (−210◦ ) = (−) tan (30◦ ) = − 3/3 = −1/ 3 c) Because 11π/4 − 2π = 3π/4, it follows that 11π/4 is coterminal with the second-quadrant angle 3π/4. Therefore, the reference angle is θ ′ = π − 3π/4 = π/4. Because the cosecant is positive in quadrant 2, you have √ csc (11π/4) = (+) csc (π/4) = 1/ sin (π/4) = 2

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1.2. ESSENTIAL FUNCTIONS

Trigonometric Identities A trigonometric identity is a relationship among the trigonometric functions. The most identities are the following. Part 1 The following identities are immediate consequences of the definitions of the trigonometric functions. 1 1 , sec θ = , sin θ cos θ sin θ cos θ tan θ = , cot θ = sin θ cos θ

csc θ =

cot θ =

1 tan θ

Part 2 The following are the most useful of all trigonometric identities: sin2 θ + cos2 θ = 1 tan2 θ + 1 = sec2 θ 1 + cot2 θ = csc2 θ Part 3 The following identities show that sin θ is an odd function and cos θ is an even function. sin(−θ) = − sin θ,

cos(−θ) = cos θ

Part 4 The next identities show that the sine and cosine functions are periodic with period 2π Since the angles θ and θ + 2π have the same terminal side. sin(θ + 2π) = sin θ,

cos(θ + 2π) = cos θ

Part 5 The addition and subtracting formulas are the following identities. sin(x + y)...