(Parte 1 de 2)

Contents

1.1 The Concept of a Function2
1.2 Trigonometric Functions12
1.3 Inverse Trigonometric Functions19
1.4 Logarithmic, Exponential and Hyperbolic Functions26
2.1 Intuitive treatment and definitions35
2.1.1 Introductory Examples35
2.1.2 Limit: Formal Definitions41
2.1.3 Continuity: Formal Definitions43
2.1.4 Continuity Examples48
2.2 Linear Function Approximations61
2.3 Limits and Sequences72
2.4 Properties of Continuous Functions84
2.5 Limits and Infinity94

2 Limits and Continuity 35

3.1 The Derivative9
3.2 The Chain Rule1
3.3 Differentiation of Inverse Functions118
3.4 Implicit Differentiation130
3.5 Higher Order Derivatives137

3 Differentiation 9

4.1 Mathematical Applications146
4.2 Antidifferentiation157
4.3 Linear First Order Differential Equations164

4 Applications of Differentiation 146 i

4.4 Linear Second Order Homogeneous Differential Equations169

i CONTENTS 4.5 Linear Non-Homogeneous Second Order Differential Equations 179

5.1 Area Approximation183
5.2 The Definite Integral192
5.3 Integration by Substitution210
5.4 Integration by Parts216
5.5 Logarithmic, Exponential and Hyperbolic Functions230
5.6 The Riemann Integral242
5.7 Volumes of Revolution250
5.8 Arc Length and Surface Area260

5 The Definite Integral 183

6.1 Integration by formulae267
6.2 Integration by Substitution273
6.3 Integration by Parts276
6.4 Trigonometric Integrals280
6.5 Trigonometric Substitutions282
6.6 Integration by Partial Fractions288
6.7 Fractional Power Substitutions289
6.8 Tangent x/2 Substitution290
6.9 Numerical Integration291

6 Techniques of Integration 267

7.1 Integrals over Unbounded Intervals294
7.2 Discontinuities at End Points299
7.3304
7.4 Improper Integrals314

7 Improper Integrals and Indeterminate Forms 294

8.1 Sequences315
8.2 Monotone Sequences320
8.3 Infinite Series323
8.4 Series with Positive Terms327
8.5 Alternating Series341
8.6 Power Series347
8.8 Applications360

CONTENTS 1

9.1 Parabola361
9.2 Ellipse362
9.3 Hyperbola363
9.4 Second-Degree Equations363
9.5 Polar Coordinates364
9.6 Graphs in Polar Coordinates365
9.7 Areas in Polar Coordinates366

Chapter 1 Functions

In this chapter we review the basic concepts of functions, polynomial functions, rational functions, trigonometric functions, logarithmic functions, exponential functions, hyperbolic functions, algebra of functions, composition of functions and inverses of functions.

1.1 The Concept of a Function

Basically, a function f relates each element x of a set, say Df, with exactly one element y of another set, say Rf. We say that Df is the domain of f and

Rf is the range of f and express the relationship by the equation y = f(x). It is customary to say that the symbol x is an independent variable and the symbol y is the dependent variable.

graph

Let Df be the set of all real numbers and Rf be the set of all non-negative real numbers. For each x in Df, let y = |x| in Rf. In this case, f(x) = |x|,

1.1. THE CONCEPT OF A FUNCTION 3 the absolute value of x. Recall that

If the domain Df and the range Rf of a function f are both subsets of the set of all real numbers, then the graph of f is the set of all ordered pairs (x,f(x)) such that x is in Df. This graph may be sketched in the xycoordinate plane, using y = f(x). The graph of the absolute value function in Example 2 is sketched as follows:

graph

In order that the range of f contain real numbers only, we must impose the restriction that x ≥ 4. Thus, the domain Df contains the set of all real numbers x such that x ≥ 4. The range Rf will consist of all real numbers y such that y ≥ 0. The graph of f is sketched below.

graph

Example 1.1.4 A useful function in engineering is the unit step function, u, defined as follows:

The graph of u(x) has an upward jump at x = 0. Its graph is given below.

4 CHAPTER 1. FUNCTIONS graph

Example 1.1.5 Sketch the graph of

It is clear that Df consists of all real numbers x 6= ±2. The graph of f is given below.

graph

We observe several things about the graph of this function. First of all, the graph has three distinct pieces, separated by the dotted vertical lines x = −2 and x = 2. These vertical lines, x = ±2, are called the vertical asymptotes. Secondly, for large positive and negative values of x,f(x) tends to zero. For this reason, the x-axis, with equation y = 0, is called a horizontal asymptote.

Let f be a function whose domain Df and range Rf are sets of real numbers. Then f is said to be even if f(x) = f(−x) for all x in Df. And f is said to be odd if f(−x) = −f(x) for all x in Df. Also, f is said to be one-to-one if f(x1) = f(x2) implies that x1 = x2.

Example 1.1.6 Sketch the graph of f(x) = x4 − x2. This function f is even because for all x we have

The graph of f is symmetric to the y-axis because (x,f(x)) and (−x,f(x)) are on the graph for every x. The graph of an even function is always symmetric to the y-axis. The graph of f is given below.

graph

1.1. THE CONCEPT OF A FUNCTION 5

Example 1.1.7 Sketch the graph of g(x) = x3 − 3x. The function g is an odd function because for each x,

The graph of this function g is symmetric to the origin because (x,g(x)) and (−x,−g(x)) are on the graph for all x. The graph of an odd function is always symmetric to the origin. The graph of g is given below.

graph

It can be shown that every function f can be written as the sum of an even function and an odd function. Let

Then,

We define

6 CHAPTER 1. FUNCTIONS and

Then clearly g(x) is even and h(x) is odd.

We note that

It is not always easy to tell whether a function is one-to-one. The graphical test is that if no horizontal line crosses the graph of f more than once, then f is one-to-one. To show that f is one-to-one mathematically, we need

1.1. THE CONCEPT OF A FUNCTION 7

This is only possible if x1 is not a real number. This contradiction proves

graph

If a function f with domain Df and range Rf is one-to-one, then f has a unique inverse function g with domain Rf and range Df such that for each

This function g is also written as f−1. It is not always easy to express g explicitly but the following algorithm helps in computing g.

Step 1 Solve the equation y = f(x) for x in terms of y and make sure that there exists exactly one solution for x.

Step 3 If it is desirable to have x represent the independent variable and y represent the dependent variable, then exchange x and y in Step 2 and write y = g(x).

We already know from Example 9 that f is one-to-one and, hence, it has a unique inverse. We use the above algorithm to compute g = f−1.

Step 1 We solve y = x3 for x and get x = y1/3, which is the unique solution.

8 CHAPTER 1. FUNCTIONS

Step 3 We plot y = x3 and y = x1/3 on the same coordinate axis and compare their graphs.

graph

A polynomial function p of degree n has the general form

The polynomial functions are some of the simplest functions to compute. For this reason, in calculus we approximate other functions with polynomial functions. A rational function r has the form

where p(x) and q(x) are polynomial functions. We will assume that p(x) and q(x) have no common non-constant factors. Then the domain of r(x) is the set of all real numbers x such that q(x) 6= 0.

Exercises 1.1 1. Define each of the following in your own words.

(a) f is a function with domain Df and range Rf (b) f is an even function

(c) f is an odd function (d) The graph of f is symmetric to the y-axis (e) The graph of f is symmetric to the origin. (f) The function f is one-to-one and has inverse g.

1.1. THE CONCEPT OF A FUNCTION 9 2. Determine the domains of the following functions

3. Sketch the graphs of the following functions and determine whether they are even, odd or one-to-one. If they are one-to-one, compute their inverses and plot their inverses on the same set of axes as the functions.

4. If {(x1, y1), (x2, y2),, (xn+1, yn+1)} is a list of discrete data points in

the plane, then there exists a unique nth degree polynomial that goes through all of them. Joseph Lagrange found a simple way to express this polynomial, called the Lagrange polynomial.

10 CHAPTER 1. FUNCTIONS

5. A linear function has the form y = mx + b. The number m is called the slope and the number b is called the y-intercept. The graph of this function goes through the point (0,b) on the y-axis. In each of the following determine the slope, y-intercept and sketch the graph of the given linear function:

6. A quadratic function has the form y = ax2 + bx + c, where a 6= 0. On completing the square, this function can be expressed in the form

The graph of this function is a parabola with vertex ( − b

and line of symmetry axis being the vertical line with equation x = −b

The graph opens upward if a > 0 and downwards if a < 0. In each of the following quadratic functions, determine the vertex, symmetry axis and sketch the graph.

7. Sketch the graph of the linear function defined by each linear equation and determine the x-intercept and y-intercept if any.

1.1. THE CONCEPT OF A FUNCTION 1

8. Sketch the graph of each of the following functions:

9. Sketch the graph of each of the following piecewise functions.

10. The reflection of the graph of y = f(x) is the graph of y = −f(x). In each of the following, sketch the graph of f and the graph of its reflection on the same axis.

12 CHAPTER 1. FUNCTIONS

1. The graph of y = f(x) is said to be

(i) Symmetric with respect to the y-axis if (x,y) and (−x,y) are both on the graph of f;

(i) Symmetric with respect to the origin if (x,y) and (−x,−y) are both on the graph of f.

For the functions in problems 10 a) – 10 i), determine the functions whose graphs are (i) Symmetric with respect to y-axis or (i) Symmetric with respect to the origin.

12. Discuss the symmetry of the graph of each function and determine whether the function is even, odd, or neither.

1.2 Trigonometric Functions

The trigonometric functions are defined by the points (x,y) on the unit circle with the equation x2 + y2 = 1.

graph

Consider the points A(0,0),B(x,0),C(x,y) where C(x,y) is a point on the unit circle. Let θ, read theta, represent the length of the arc joining the points D(1,0) and C(x,y). This length is the radian measure of the angle CAB. Then we define the following six trigonometric functions of θ as

1.2. TRIGONOMETRIC FUNCTIONS 13 follows:

sinθ = y x = sin θ

cosθ ,cotθ = x

Since each revolution of the circle has arc length 2pi,sinθ and cosθ have period 2pi. That is,

The function values of some of the common arguments are given below:

A function f is said to have period p if p is the smallest positive number such that, for all x,

f(x + np) = f(x), n = 0,±1,±2,

Since cscθ is the reciprocal of sinθ and secθ is the reciprocal of cos(θ), their periods are also 2pi. That is,

csc(θ + 2npi) = csc(θ) and sec(θ + 2npi) = secθ, n = 0,±1,±2,
tan(θ + npi) = tanθ and cot(θ + npi) = cotθ, n = 0,±1,±2,

It turns out that tanθ and cotθ have period pi. That is,

Geometrically, it is easy to see that cosθ and secθ are the only even trigonometric functions. The functions sinθ,cosθ,tanθ and cotθ are all odd functions. The functions sinθ and cosθ are defined for all real numbers. The

14 CHAPTER 1. FUNCTIONS functions cscθ and cotθ are not defined for integer multiples of pi, and secθ and tanθ are not defined for odd integer multiples of pi/2. The graphs of the six trigonometric functions are sketched as follows:

graph

The dotted vertical lines represent the vertical asymptotes.

There are many useful trigonometric identities and reduction formulas. For future reference, these are listed here.

sin(x + y) = sinxcosy + cosxsiny, cos(x + y) = cosxcosy − sinxsiny sin(x − y) = sinxcosy − cosxsiny, cos(x − y) = cosxcosy + sinxsiny

1.2. TRIGONOMETRIC FUNCTIONS 15

In applications of calculus to engineering problems, the graphs of y =

Asin(bx+c) and y = Acos(bx+c) play a significant role. The first problem has to do with converting expressions of the form Asinbx + B cosbx to one of the above forms. Let us begin first with an example.

First of all, we make a right triangle with sides of length 3 and 4 and compute the length of the hypotenuse, which is 5. We label one of the acute angles as θ and compute sinθ,cosθ and tanθ. In our case,

graph

16 CHAPTER 1. FUNCTIONS

Thus, the problem is reduced to sketching a cosine function, ??? y = −5cos(2x + θ). We can compute the radian measure of θ from any of the equations

In order to sketch the graph, we first compute all of the zeros, relative maxima, and relative minima. We can see that the maximum values will be 5 and minimum values are −5. For this reason the number 5 is called the amplitude of the graph. We know that the cosine function has zeros at odd integer multiples of pi/2. Let

, n = 0,±1,±2

The max and min values of a cosine function occur halfway between the consecutive zeros. With this information, we are able to sketch the graph of the given function. The period is pi, phase shift is 1

2 and frequency is 1 pi .

graph

For the functions of the form y = Asin(ωt ± d) or y = Acos(ωt ± d) we make the following definitions:

1.2. TRIGONOMETRIC FUNCTIONS 17 period = ω

The motion of a particle that follows the curves Asin(ωt±d) or Acos(ωt±d) is called simple harmonic motion.

Exercises 1.2

1. Determine the amplitude, frequency, period and phase shift for each of the following functions. Sketch their graphs.

(e) y = sinx

2. Sketch the graphs of each of the following:

(a) y = tan(3x) (b) y = cot(5x) (c) y = xsinx (d) y = sin(1/x) (e) y = xsin(1/x)

3. Express the following products as the sum or difference of functions.

(a) sin(3x)cos(5x) (b) cos(2x)cos(4x) (c) cos(2x)sin(4x) (d) sin(3x)sin(5x) (e) sin(4x)cos(4x)

4. Express each of the following as a product of functions:

5. Consider the graph of y = sinx, −pi2 ≤ x ≤ pi 2 . Take the sample points

( pi

18 CHAPTER 1. FUNCTIONS

Compute the fourth degree Lagrange Polynomial that approximates and agrees with y = sinx at these data points. This polynomial has the form

6. Sketch the graphs of the following functions and compute the amplitude, period, frequency and phase shift, as applicable.

7. Sketch the graphs of the following functions over two periods. a) y = 2secx b) y = −3tanx c) y = 2cotx

tanx − tany

1.3. INVERSE TRIGONOMETRIC FUNCTIONS 19

h = cosx

h = sinx

13. Prove that

1.3 Inverse Trigonometric Functions

None of the trigonometric functions are one-to-one since they are periodic. In order to define inverses, it is customary to restrict the domains in which the functions are one-to-one as follows.

20 CHAPTER 1. FUNCTIONS

Its inverse function is denoted arcsinx, and we define y = arcsinx, −1 ≤

graph

graph

2 , is one-to-one and covers the range −∞ < y < ∞ Its inverse function is denoted arctanx, and we define y =

graph

graph

1.3. INVERSE TRIGONOMETRIC FUNCTIONS 21

2 < x ≤ pi is one-to-one and covers the range

2 or pi graph

Example 1.3.1 Show that each of the following equations is valid.

(a) arcsinx + arccosx = pi

(b) arctanx + arccotx = pi

(c) arcsecx + arccscx = pi

2 To verify equation (a), we let arcsinx = θ.

graph

Then x = sinθ and cos (pi2

−θ) = x, as shown in the triangle. It follows that pi2 − θ = arccosx, pi

The equations in parts (b) and (c) are verified in a similar way.

2 CHAPTER 1. FUNCTIONS

Example 1.3.2 If θ = arcsinx, then compute cosθ,tanθ,cotθ,secθ and csc θ.

If θ is −pi 2 , 0, or pi

2 , then computations are easy.

graph

2 . Then, from the triangle, we get

Example 1.3.3 Make the given substitutions to simplify the given radical expression and compute all trigonometric functions of θ.

(a) For part (a), sinθ = x

2 and we use the given triangle:

graph

Then

1.3. INVERSE TRIGONOMETRIC FUNCTIONS 23

(b) For part (b), secθ = x

3 and we use the given triangle:

graph

Then,

(c) For part (c), tanθ = x

2 and we use the given triangle:

graph

Then,

24 CHAPTER 1. FUNCTIONS

Remark 2 The three substitutions given in Example 15 are very useful in calculus. In general, we use the following substitutions for the given radicals:

Exercises 1.3 1. Evaluate each of the following:

(a) 3arcsin

(b) 4arctan

(d) cos(2arccos(x)) (e) sin(2arccos(x))

2. Simplify each of the following expressions by eliminating the radical by using an appropriate trigonometric substitution.

(Hint: In parts (d) and (e), complete squares first.)

3. Some famous polynomials are the so-called Chebyshev polynomials, defined by

1.3. INVERSE TRIGONOMETRIC FUNCTIONS 25

cos(nθ + θ),Tn−1(x) = cos(nθ − θ). Use the expansion formulas and then make substitutions in part (a)).

4. Show that for all integers m and n,

(Hint: use the expansion formulas as in problem 3.) 5. Find the exact value of y in each of the following

6. Solve the following equations for x in radians (all possible answers).

26 CHAPTER 1. FUNCTIONS

7. If arctant = x, compute sinx, cosx, tanx, cotx, secx and cscx in terms of t.

8. If arcsint = x, compute sinx, cosx, tanx, cotx, secx and cscx in terms of t.

9. If arcsect = x, compute sinx, cosx, tanx, cotx, secx and cscx in terms of t.

10. If arccost = x, compute sinx, cosx, tanx, cotx, secx and cscx in terms of t.

Remark 3 Chebyshev polynomials are used extensively in approximating functions due to their properties that minimize errors. These polynomials are called equal ripple polynomials, since their maxima and minima alternate between 1 and −1.

1.4 Logarithmic, Exponential and Hyperbolic Functions

(Parte 1 de 2)

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