微積分筆記1

來自專欄 英語學習及其他

作為高中以後沒有學過數學的文科生，我很無知無畏的選擇了這本書。如果哪天我棄坑了，這是再正常不過的結果了。如果我最終竟然懵懵懂懂的讀完了，我要嚴重的表揚自己。

1. Preparation for Calculus

P.1 Graphs and Models

Solution Point:

Consider the equation 3x + y = 7. The point (2,1) is a solution point of the equation.

1. The Graph of an Equation

Ex.1. Sketching a Graph by Point Plotting

One disadvantage of point plotting is that to get a good idea about the shape of a graph, you may need to plot many points. With only a few points, you could badly misrepresent the graph.

b. Intercepts of a Graph

Two types of solution points that are especially useful in graphing an equation are those having zero as their x- or y-coordinate. Such points are called intercepts because they are the points at which the graph intersects the x- or y-axis.

It is possible for a graph to have no intercepts, or it might have several.

c. Symmetry of a Graph

Ex.1 Testing for Symmetry

Origin Symmetry:

y = 2x^3 - x (original equation)

- y = 2(- x)^3 - (- x) (replace x by - x and y by - y)

y = 2x^3 - x (equivalent equation)

Ex.2 Using Intercepts and Symmetry to Sketch a Graph

d. Points of Intersection

A point of intersection of the graphs of two equations is a point that satisfied both equations.

Ex.1 Finding Points of Intersection

Find all points of intersection of the graphs of x^2 - y = 3 and x - y = 1.

Solution:

x^2 - 3 = x - 1 (equate y-values)

x^2 - x - 2 = 0

(x- 2)(x + 1) = 0

x = 2 or -1

(2, 1) and (-1, -2) (Points of intersection)

e. Mathematical Models

Real-life applications of mathematics often use equations as mathematical models.

Comparing Graphical and Analytic Approaches:

A purely graphical approach to this problem would involve a simple 「guess, check, and revise」 strategy. What types of things do you think an analytic approach might involve? For instance, does the graph has symmetry? Does the graph have turns? If so, where are they? As you proceed through Chapter 1, 2, and 3 of this text, you will study new analytic tools that will help you analyse graphs of equations such as these.

P.2 Linear Models and Rates of Change

a. The Slope of a Line

The slope m of the nonvertical line passing through (x1, y1) and (x2, y2) is

m = (y2-y1) / (x2-x1) x1不等於x2

Slope is not defined for vertical lines.

b. Equations of Lines

Point-slope equation of a line

An equation of the line with slope m passing through the point (x1, y1) is given by:

y - y1 = m(x - x1)

c. Ratios and Rates of Change

The slope of a line can be interpreted as either a ratio or a rate. If the x- and y-axes have the same unit of measure, the slope has no units and is a ratio. If the x- and y-axes have different units of measure, the slope is a rate or rate of change.

An average rate of change is always calculated over an interval. In Chapter 2 you will study another type of rate of change called an instantaneous rate of change.

d. Graphing Linear Models

The Slope-Intercept Equation of a Line

The graph of the linear equation:

y = mx + b

is a line having a slope of m and a y-intercept at (0, b).

Ex.1 Sketch the graph of the equation: y = 2x + 1

Solution:

Because b = 1, the y-intercept is (0, 1). Because the slope is m = 2, you know that the line rises two units for each unit it moves to the right, as shown in Figure P.18(a).

Slope的定義：

The slope of a nonvertical line is a measure of the number of units the line rises (or falls) vertically for each unit of horizontal change from left to right.

The equation of any line can be written in the general form

Ax + By + C = 0

where A and B are not both zero.

SUMMARY OF EQUATIONS OF LINES

1. General form: Ax + By + C = 0 (A, B ≠ 0)

1. Vertical line: x = a

1. Horizontal line: y = b

1. Point-slope form: y - y1 = m(x - x1)

1. Slope-intercept form: y = mx + b

d. Parallel and Perpendicular Lines

The slope of a line is a convenient tool for determine whether two lines are parallel or perpendicular.

1. Two distinct nonvertical lines are parallel if and only if their slopes are equal—that is, if and only if m1 = m2.

1. Two nonvertical lines are perpendicular if and only if their slopes are negative reciprocals of each other—that is, if and only if m1 = - 1/m2.

P.3 Functions and Their Graphs

a. Functions and Function Notation

Many real-life situations can be modelled by functions. For instance, the area A of a circle is a function of the circle』s radius r.

A = (Pi)*r^2 A is a function of r.

In this case r is the independent variable and A is the dependant variable.

DEFINITION OF A REAL-WORLD FUNCTION OF A REAL VARIABLE

Let X and Y be sets of real numbers. A real-valued function f of a real variable x from X to Y is a correspondence that assigns to each number x in X exactly one number y in Y.

The domain of f is the set X. The number y is the image of x under f and is denoted by f(x), which is called the value f at x. The range of f is a subset of Y and consists of all images of numbers in X.

x^2 + 2y = 1 Equation in implicit form

y = 1/2(1-x^2) Equation in explicit form

f(x) = 1/2(1 - x^2) Function notation

Ex1. Evaluation a Function

For the function f defined by f(x) = x^2 + 7, evaluate the expression:

f(3a) = (3a)^2 +7

= 9a^2 + 7

b. The Domain and Range of a Function

The function given by

f(x) = 1/(x^2 - 4), 4 <= x <= 5

has a explicitly defined domain. On the other hand, the function given by

f(x) = 1/(x^2 - 4)

has an implied domain that is the set {x: x ≠＋－2｝

c. The Graph of a Fuction

A vertical line intersect the graph of a function of x at most once. This observation provides a convenient visual test, called the Vertical Line Test, for functions of x. That is, a graph in the coordinate plane is the graph of a function of x if and only if no vertical line intersects the graph at more than one point.

d. Transformations of Functions

BASIC TYPES OF TRANSFORMATIONS (c > 0)

Original graph: y = f(x)

Horizontal shift c units to the right: y = f(x - c)

Horizontal shift c units to the left: y = f(x + c)

Vertical shift c units downward: y = f(x) - c

Vertical shift c units upward: y = f(x) + c

Reflection (about the x-axis): y = - f(x)

Reflection (about the y-axis): y = f(-x)

Reflection (about the origin): y = - f(-x)

e. Classifications and Combinations of Functions

The modern notion of a function is derived from the efforts of many seventeenth- and eighteenth-century mathematicians. Of particular note was Leonhard Euler, to whom we are indebted for the function notation y = f(x). By the end of the eighteenth century, mathematicians and scientists had concluded that many real-world phenomena could be represented by mathematical models taken from a collection of functions called elementary functions. Elementary functions fall into three categories.

1. Algebraic functions (polynomial, radical, rational)

1. Trigonometric functions(sin, cosine, tangent, and so on)

1. Exponential and logarithmic functions

The most common type of algebraic function is a polynomial function

f(x) = anxn + an-1xn-1 + … + a2x2 + a1x + a0

where n is a nonnegative integer. The numbers ai are coefficients, with an the leading coefficient and a0 the constant term of the polynomial function. If an ≠ 0, then n is the degree of the polynomial function.

Zeroth degree: f(x) = a Constant function

First degree: f(x) = ax + b Linear function

Second degree: f(x) = ax^2 + bx + c Quadratic function

Third degree: f(x) = ax^3 + bx^2 + cx + d Cubic function

Although the graph of a nonconstant polynomial function can have several turns, eventually the graph will rise or fall without bound as x moves to the right or left. Whether the graph of f(x) eventually rises or falls can be determined by the function』s degree (odd or even) and by the leading coefficient an, as indicated in Figure P.29.

Two functions can be combined in various ways to creat new functions. For example, given f(x) = 2x - 3 and g(x) = x^2 + 1, you can form the functions shown.

(f + g)(x) = f(x) + g(x) = (2x - 3) + (x^2 + 1)

(f - g)(x) = f(x) - g(x) = (2x - 3) - (x^2 + 1)

(fg)(x) = f(x)g(x) = (2x - 3) (x^2 + 1)

(f/q)(x) = f(x)/g(x) = (2x - 3)/(x^2 + 1)

You can combine two functions in yet another way, called composition. The resulting function is called a composite function.

DEFINITION OF COMPOSITE FUNCTION

Let f and g be functions. The function given by (f○g)(x) = f(g(x)) is called the composite of f with g. The domain of f○g is the set of all x in the domain of g such that g(x) is in the domain of f.

ex1. Finding Composite Functions

Given f(x) = 2x - 3 and g(x) = cos x, find each composite function.

1. (f○g)(x) = f(g(x)) Definition of f○g

= f(cos x) Substitute cos x for g(x)

= 2(cos x) - 3 Definition of f(x)

= 2cos x -3 Simplify

b. (g○f)(x) = g(f(x))

= g(2x - 3)

= cos(2x - 3)

In the terminology of functions, a function is even if its graph is symmetric with respect to the y-axis, and is odd if its graph is symmetric with respect to the y-axis, and is odd if its graph is symmetric with respect the the origin.

TEST FOR EVEN AND ODD FUNCTIONS

The function y = f(x) is even if f(-x) = f(x).

The function y = f(x) is odd if f(-x) = -f(x).

Note: Except for the constant function f(x) = 0, the graph of a function of x cannot have symmetry with respect to the x-axis because it then would fail the Vertical Line Test for the graph of the function.

P.4 Fitting Models to Data

A basic premise of science is that much of the physical world can be described mathematically and that many physical phenomena are predictable. This scientific outlook was part of the scientific revolution that took place in Europe during the late 1500s. Two early publications connected with this revolution were On the Revolutions of the Heavenly Spheres by the Polish astronomer Nicolaus Copernicus and On the Structure of the Human Body by the Belgian anatomist Andreas Vesalius. Each of these books was published in 1543, and each broke with prior tradition by suggesting the use of a scientific method rather than unquestioned reliance on authority.

One basic technique of modern science is gathering data and then describing the data with a mathematical model.

越學習，越覺得16-19世紀的迷人之處。