МАТЕМАТИКА В ТАБЛИЦАХ. ЧАСТЬ I. Министерство образования и науки Российской Федерации Новгородский государственный университет имени Ярослава Мудрого А. О. Кондюков, И. А. Судаков, Т. Г. Сукачева МАТЕМАТИКА В ТАБЛИЦАХ. ЧАСТЬ I. ВЕЛИКИЙ НОВГОРОД 2014

Математика в таблицах содержит материалы на английском языке к занятиям по основам математического анализа, одного из главных и трудных разделов высшей математики. Пособие может быть полезно как преподавателям, читающим соответствующий курс лекций, так и студентам, изучающим основы математического анализа (темы - предел, непрерывность, производная). Использование этих материалов позволит сделать преподавание основ указанного раздела более доступным и наглядным. Поскольку таблицы представлены на английском языке, то они могут быть использованы преподавателями, обучающими студентов-математиков английскому языку, а также их учениками. Пособие предназначено прежде всего для преподавателей и студентов высших учебных заведений, хотя отдельные его части можно применять в работе с учениками в классах с углубленным изучением математики.

1.1 Intro to Limits Gottfried Wilhelm Leibniz Sir Isaac Newton Joseph-Louis Lagrange Leonhard Euler

Example from sociology

Measurement of a Circle Archimedes The area of a circle is the limit of the areas of the inscribed polygons as n (the number of sides in the polygon) increases without bound

Def. Intuitive Meaning of Limit To say that means that when x is near but different from c then f (x) is near L.

Notations We write and say “ the limit of f (x), as x approaches c, equals L” this says that the values of f (x) tend to go closer and closer to the number L as x gets closer and closer to the number c but x ≠ c. An alternative notation for is as “f (x) approaches L as x approaches c”

Example: Step Function Functions whose graphs resemble sets of stairsteps are known as . the greatest integer less than or equal to x

One-sided limits Def. Right- and Left-Hand Limits To say that means that when x is near but to the right of c then f (x) is near L. Similarly, to say that means that when x is near but to the left of c then f (x) is near L.

Theorem A if and only if and

Absolute value as a Distance 1.2. Rigorous Study of Limits Absolute value as a Distance

Absolute value as a Distance

Def. Precise Meaning of Limit To say that means that for each given (no matter how small) there is a corresponding such that provided that that is,

Karl Wilhelm Weierstrass The “Fathers of modern mathematical  analysis“ Augustin-Louis Cauchy Karl Wilhelm Weierstrass

Theorem A: Main Limit Theorem 1.3. Limit Theorems Theorem A: Main Limit Theorem Let n a positive integer, k be a constant, and f (x) and g (x) be functions that have limits at c. Then The limit of a constant - the limit of a constant is just the constant - a constant factor may pass through the limit sign

Theorem A: Main Limit Theorem 1.3. Limit Theorems Theorem A: Main Limit Theorem Let n a positive integer, k be a constant, and f (x) and g (x) be functions that have limits at c. Then The limit of a sum or difference - the limit of a sum (or difference) is equal to the sum (or difference) of the limits

Theorem A: Main Limit Theorem 1.3. Limit Theorems Theorem A: Main Limit Theorem Let n a positive integer, k be a constant, and f (x) and g (x) be functions that have limits at c. Then The limit of a product or quotient provided - the limit of a product is equal to the product of the limits - the limit of a quotient is equal to the quotient of the limits,  provided the limit of the denominator is not 0.

Theorem A: Main Limit Theorem 1.3. Limit Theorems Theorem A: Main Limit Theorem Let n a positive integer, k be a constant, and f (x) and g (x) be functions that have limits at c. Then The limit of a power provided - the limit of a positive integer power of a function is the power of the limit of the function when n is even - the limit of the n-th root of a function is the n-th root of the limit of the function, if the n-th root of the limit is a real number

Theorem B: Substitution Theorem 1.3. Limit Theorems Theorem B: Substitution Theorem If f a polynomial function or a rational function, then provided f (c) is defined. In the case of a rational function, this means that the value the denominator at c is not 0. - a polynomial function - a rational function is the quotient of two polynomial functions Evaluating a Limit “by Substitution”!

1.3. Limit Theorems Theorem C If for all x in an open interval containing the number c, except possibly at the number c itself, and if exists, then exist and

Theorem D: Squeeze Theorem 1.3. Limit Theorems Theorem D: Squeeze Theorem Let f, g, and h be functions satisfying for all x near c, except possibly at c. If then

1.3. Limit Theorems Show that does not exist!!! because f(x) h(x) g(x) does not exist!!! because ??? at x→0 f(x) h(x)

1.4. Limits Involving Trigonometric Functions

1.4. Limits Involving Trigonometric Functions Theorem A. Limits of Trigonometric Functions 1. 2. 3. 4. 5. 6.

1.4. Limits Involving Trigonometric Functions Proof: Special case: c = 0 → → The Squeeze Theorem General case: Addition Identity Note:

1.4. Limits Involving Trigonometric Functions Theorem B. Special Trigonometric Limits 1. 2.

1.4. Limits Involving Trigonometric Functions Proof: Note: multiplying by 2 and dividing by the positive number as well as taking into account that The Squeeze Theorem - the are of a sector of a circle r – the radius, t – central angel - the are of a triangle a – the base, h – the height

1.5. Limits Involving Infinity Infinity is a very special idea. We know we can't reach it, but we can still try to work out the value of functions that have infinity in them. Division by infinity gives infinitesimal (formally we should write 0) ..... We are now faced with an interesting situation: We can't say what happens when x gets to infinity But we can see that 1/x is going towards 0 “the limit as x becomes infinite”, not as "x approaches infinity" As x becomes infinite, then 1/x approaches 0 (it is not equal to 0 !!!)  When you see "limit", think "approaching"

1.5. Limits Involving Infinity Rigorous Def. Limits as Let f be defined on for some number c. We say that if for each there is a corresponding number M such that Rigorous Def. Limits as M Let f be defined on for some number c. We say that if for each there is a corresponding number M such that

1.5. Limits Involving Infinity Limit of a Sequence: So again we have an odd situation: We don't know what the value is when n=infinity But we can see that it settles towards 2.71828... Euler's number 2.7182818284590452353602874713527… 

1.5. Limits Involving Infinity Division by infinitesimal gives infinity If x approaches 0 from the left, then the values of 1/x become large negative numbers But don't be fooled by the "=". You can’t actually get to infinity, but in "limit“ language the limit is infinity (which is really saying the function is limitless) Rigorous Def. Infinite Limit We say that if for every positive number M, there exists a corresponding such that

1.5. Limits Involving Infinity Def. Vertical Asymptote The line is called a vertical asymptote of the curve if at least one of the following statements is true: c c c c c c c c Example Def. Horizontal Asymptote The line is called a horizontal asymptote of the curve if either or

1.6. Continuity of Functions Def. Continuity at a point Let f be defined on an open interval containing c. We say that f is continuous at c if We mean by this definition to require three things: exists, f (c) exists (i.e., c is in the domain of f ), and the expected value agrees with the value f (c) which value f actually takes at c This definition basically means that there is no missing point, gap, or split for f (x) at c. In other words, you can move your pencil along the image of the function and you would not have to lift up the pencil. These functions are called smooth functions. If the limit does not exist, or if it exists but is not equal to f (c), we say that f has a discontinuity (or is discontinuous) at x = c.

1.6. Continuity of Functions Atmospheric temperature and world population are represented by continuous graphs The creation of 3D animation using discontinuous function The Hertzsprung–Russell diagram is a discontinuous graph of stars showing the relationship between the stars' absolute magnitudes or luminosities versus their spectral types or classifications and effective temperatures.

The kinds of discontinuity 1.6. Continuity of Functions The kinds of discontinuity A point of discontinuity c is called removable if the function can be defined or redefined at c so as to make function continuous. Otherwise, a point of discontinuity is called nonremovable. Infinite discontinuities break the 2nd condition: They have an asymptote instead of a specific f(c) value. Jump discontinuities break the 1st condition: The limit approaching from a specific c from the left is not the same as the limit approaching c from the right. Point discontinuities break the 3rd condition: The limit of c is not the same as (c).

Theorems A,B,D. Continuity of familiar functions 1.6. Continuity of Functions Theorems A,B,D. Continuity of familiar functions Any polynomial is continuous everywhere; that is, it is continuous at every real number. Any rational function is continuous wherever it is defined; that is, it is continuous on its domain (except where its denominator is zero). The following types of functions are continuous at every number in their domains: absolute value functions; root functions; trigonometric functions; inverse trigonometric functions.

Theorem C. Continuity under function operations 1.6. Continuity of Functions Theorem C. Continuity under function operations Let f and g continuous at c then so are (provided that f (c) >0 if n is even) Theorem E. Composite limit theorem If and if f is a continuous at L, then In particular, if g is continuous at c and f is continuous at g(c), then the composite is continuous at c.

1.6. Continuity of Functions Open interval Closed interval Def. Continuity at an interval The function f is right continuous at a if and left continuous at b if We say f is continuous on an open interval if it is continuous at each point of that interval. It is continuous on the closed interval [a, b] if it is continuous on (a, b), right continuous at a, and left continuous at b. it is neither left- nor right-continuous left continuous at c right continuous at c

Theorem F. Intermediate Value Theorem 1.6. Continuity of Functions Theorem F. Intermediate Value Theorem Let f be a function defined on [a, b] and let W be a number between f (a) and f (b). If f is continuous on [a, b], then there is at least one number c between a and b such that f (c) = W. The Intermediate Value Theorem says that a continuous function does not skip values. By contrast, a discontinuous function can skip values. W In fact, the Intermediate Value Theorem plays a role in the very way these graphing devices work. A computer calculates a finite number of points on the graph and turns on the pixels that contain these calculated points. It assumes that the function is continuous and takes on all the intermediate values between two consecutive points. The computer therefore connects the pixels by turning on the intermediate pixels.

Limit Main Theorems Special Limits Continuity of Functions derivative Rigorous Definition Trigonometric Functions Continuity of Functions For each given ε > 0 (no matter how small) there is a corresponding δ >0 such that provided that that is, Continuity A function f is continuous at the point x = c if exists. 2. f (c) exists. 3. We say f is continuous on an open interval (a, b) if it is continuous at each point of that interval. It is continuous on the closed interval [a, b] if it is continuous on (a, b), right continuous at a, and left continuous at b. One-sided limits Limits at Infinity if and only if If p is a positive number, then Division by infinity gives infinitesimal (formally 0). Limits and Operations The special limit which is giving Euler's number: Kinds of discontinuity Limits at infinity for rational functions A point of discontinuity c is called removable if the function can be defined or redefined at c so as to make function continuous. Otherwise, a point of discontinuity is called nonremovable. a) Jump discontinuities break the 1st condition: The limit approaching from a specific c from the left is not the same as the limit approaching c from the right. b) Infinite discontinuities break the 2nd condition: They have an asymptote instead of a specific f (c) value. c) Point discontinuities break the 3rd condition: The limit of c is not the same as (c). Infinite Limits If p is a positive number, then Division by infinitesimal gives infinity when n is even The function f (x) will have a vertical asymptote at  x = a if we have any of the following limits: The line y = b is called a horizontal asymptote of the curve y = f (x) if either Substitution Method If P(x) and Q(x) are algebraic expressions and, Q(c)≠0 Intermediate Value Theorem then: Let f be a function defined on [a, b] and let s be a number between f (a) and f (b). If f is continuous on [a, b], then there is at least one number c between a and b such that f (c) = s. Squeeze (sandwich) theorem Let f, g, and h be functions satisfying for all x near c, except possibly at c. The Intermediate Value Theorem says that a continuous function does not skip values derivative

2.1. Rates of Change and Tangent Lines Euclid: a tangent as a line touching a curve at just point is all right for circles but completely unsatisfactory for most others curves Def. Tangent Line The tangent line to the curve y = f (x) at the point P (c, f(c)) is that line through P with slope provided that this limit exists is not infinity. The line through P and Q is called a secant line Movable point on a curve Q The tangent line at P is the limiting position (if it exists) of the secant line as Q moves toward P along the curve

Police average velocity instantaneous velocity

2.1. Rates of Change and Tangent Lines Def. Instantaneous velocity In an object moves along a coordinate line with position function f (t), then its instantaneous velocity at time c is provided that this limit exists is not infinity. Instantaneous rate of change =

2.1. Rates of Change and Tangent Lines Suppose we have a circuit with a source of supply. To find a current intensity at t (time)? (q - quantity of electricity) Plan What there are ? What is an average current intensity? What is an instantaneous current intensity? Suppose we have a chemical reaction. N (t) - amount of matter (which is undergo reaction). To find rate of chemical reaction at t (time)? Plan What there are ? What is an average rate of chemical reaction ? What is an instantaneous rate of chemical reaction? Suppose we produce merchandise. C (t) – costs of producing. To find rate of change the marginal cost. (Q – amount of merchandise? Plan What there are ? What is an average rate of change the marginal cost? What is an instantaneous rate of change the marginal cost?

2.2. The Derivative Def. Derivative Instantaneous velocity Slope of the tangent line Def. Derivative The derivative of a function f is another function f ′ (read “ f prime”) whose value at any number x is If this limit does exist, we say that f is differentiable at x. Finding a derivative is called differentiation. Equivalent forms for the derivative. Let us assume x = c + h and h = x −c

2.2. The Derivative Theorem A. Differentiability Implies Continuity !!!Continuous But Not Differentiable!!! If f ′ (c) exists, then f is continuous at c. Three ways for ƒ not to be differentiable at c c c c A corner c A discontinuity c A vertical tangent c The curve has a vertical tangent line when x = c; that is, f is continuous at c and This means that the tangent lines become steeper and steeper as x → c. c c f is differentiable at c f is not differentiable at c

2.2. The Derivative The derivative f ′ (x) gives us important information about the graph of f (x) . For example, the sign of f ′ (x) tells us whether the tangent line has positive or negative slope and the size of f ′ (x) reveals how steep the slope is. If f ′ (x) = 0, we can see the horizontal tangent line.

2.2. The Derivative - the change in x is an increment of x This is read “f prime of x” This is read “dee y dee x” or “dee y by dee x” and is also called the derivative of y with respect to x. Leibniz's notation Lagrange's notation The “quotient” is not really a quotient, although it is intuitively suggestive of the idea of calculating a slope as “rise over run”. So is an indivisible piece of notation. In other words, we can consider like . Here Δy means "change in y“ and Δ x means "change in x". So, the quotient is the ratio of the change in y to the change in x. The function f ′ can be evaluated at any point; To evaluate at a particular point, we write something like: The bar followed by the x = x0 means "evaluated at x0 ".

2.3. Rules for finding derivatives Euler’s notation Newton’s notation We use the symbol Dx to indicate the operation of differentiating. The Dx symbol says that we are to take the derivative (with respect to the variable x) of that follows. Dx is a differential operator. f f ′ Input Output An operator Newton’s notation uses dots above the name of the function. It is more commonly seen in physics books and often refers to derivatives with respect to time. Operation Dx

Theorem A. Constant function rule 2.3. Rules for finding derivatives Theorem A. Constant function rule If f (x) = k, where k is a constant, then for any x, The derivative of any constant is always zero. Theorem B. Identity function rule If f (x) = x, then The derivative of the function x is always one. Theorem C. Power rule If f (x) = x n, where n is any real number, then If you raise x to any constant power, you find the derivative by multiplying x raised to one less than that power by the power itself.

Theorem D. Constant multiple rule 2.3. Rules for finding derivatives Theorem D. Constant multiple rule If k is a constant and f is differentiable function, then A constant multiplier can be passed across the operator as well as the derivative of a constant times a function is just the constant times the derivative. Theorem E. Sum rule If f and g are differentiable functions, then The derivative of a sum is the sum of the derivatives. Theorem F. Difference rule If f and g are differentiable functions, then The derivative of a difference is the difference of the derivatives.

2.3. Rules for finding derivatives Theorem G. Product rule If f and g are differentiable functions, then The derivative of a product is equal to the first function times the derivative of the second function plus the second function times the derivative of the first function: The derivative of a product of two functions is the first function times the derivative of the second function plus the second function times the derivative of the first function. Theorem H. Quotient rule The derivative of a quotient is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator. Let f and g be differentiable functions, with Then The numerator in the Quotient Rule is equal to the bottom times the derivative of the top minus the top times the derivative of the bottom:

2.4. Derivatives of trigonometric functions

2.4. Derivatives of trigonometric functions Theorem A. If f (x) = sin x and g (x) = cos x are both differentiable and, The formula (sin x)′ = cos x makes sense when we compare the graphs of sine and cosine. The tangent lines to the graph of y = sin x have positive slope on the interval (- π/2 , π/2), and on this interval, the derivative y′ = cos x is positive. Similarly, the tangent lines have negative slope on the interval ( π/2 , 3π/2 ), where y′ = cos x is negative. The tangent lines are horizontal at x = −π/2 , π/2 , 3π/2 , where cos x = 0.

2.4. Derivatives of trigonometric functions Theorem B. For all points x in the function’s domain,

2.5. The chain rule Theorem A. Chain Rule Let y = f (u) and u = g (x). If g is differentiable at x and f is differentiable at u = g (x), then the composite function f ◦ g, defined by (f ◦ g)(x) = f (g (x)), is differentiable at x and That is, or The derivative of a composite function is the derivative of the outer function evaluated at the inner function, times the derivative of the inner function.

2.5. The chain rule The last step in calculation corresponds to the first step in differentiation the first step in differentiation the last step in calculation we differentiate the outer function [at the inner function ] and then we multiply by the derivative of the inner function.

TABLE OF DIFFERENTIATION FORMULAS