4.3 The Definite Integral
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The document discusses notation and properties for definite integrals. It defines the definite integral from a to b of f(x) dx as the area under the curve of f(x) between the x-axis and the limits of a and b. It lists five properties of definite integrals: 1) the order of integration does not matter, 2) the integral from a to a of any function f(x) is equal to zero, 3) a constant can be pulled out of the integral, 4) integrals can be added or subtracted, and 5) a definite integral over an interval can be broken into a sum of integrals over subintervals.
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4) Relating properties of the first and second derivatives to characteristics of the original function like concavity, inflection points, and local extrema.
Integration
This document provides an overview of integration and how it relates to calculating areas under curves and between functions. Some key points covered include:
- Integration uses the concept of reverse differentiation to calculate the area under a function.
- The area under a function between two x-values a and b is calculated as the definite integral from a to b.
- Integrals may be used to find the area between two curves by subtracting their integrals between the intersection points.
- Integrals can represent either positive or negative areas depending on whether the area is above or below the x-axis.
Calculus- Basics
It includes all the basics of calculus. It also includes all the formulas of derivatives and how to carry it out. It also includes function definition and different types of function along with relation.
Integration. area undera curve
The document discusses rules of integration, including:
1) The definite integral calculates the area under a curve between two bounds a and b.
2) If the function is negative in some intervals, the integral is the sum of the areas of regions where the function is positive minus the areas where it is negative.
3) The fundamental theorem of calculus relates the definite integral to antiderivatives.
Lesson 3 Oct 30
This document discusses definite integrals, including how to estimate them using left and right sums with subdivisions, how to find the exact value by drawing the graph and calculating the area, and how the sign of the integral relates to the sign of the function. It also provides examples of evaluating definite integrals using a graphing calculator or estimating the error bounds for monotone functions based on the number of intervals.
Calculus cheat sheet_integrals
This calculus cheat sheet provides definitions and formulas for:
1) Integrals including definite integrals, anti-derivatives, and the Fundamental Theorem of Calculus.
2) Common integration techniques like u-substitution and integration by parts.
3) Standard integrals of common functions like polynomials, trigonometric functions, logarithms, and exponentials.
Functions 1 - Math Academy - JC H2 maths A levels
Slides for Functions lecture 1.
JC H2 Maths
A levels Singapore
www.mathacademy.sg
Copyright 2015 Math Academy
03 convexfunctions
The document discusses convex functions and related concepts. It defines convex functions and provides examples of convex and concave functions on R and Rn, including norms, logarithms, and powers. It describes properties that preserve convexity, such as positive weighted sums and composition with affine functions. The conjugate function and quasiconvex functions are also introduced. Key concepts are illustrated with examples throughout.
Riemann sumsdefiniteintegrals
This document discusses Riemann sums and the definite integral. It explains that the definite integral is defined as the limit of Riemann sums as the size of the subintervals approaches zero. It provides examples of calculating Riemann sums and shows how the definite integral can be approximated by Riemann sums. The document also outlines some key properties of the definite integral, such as how to integrate sums and how the integral relates to calculating the area under a curve.
Fourier series 1
The document discusses periodic functions and their properties. The key points are:
- A periodic function f(x) satisfies f(x) = f(x + T) for some fixed period T and all real x.
- Periodic functions repeat their values at intervals of their period, including integer multiples of the period.
- Functions are defined as even if f(-x) = f(x) and odd if f(-x) = -f(x).
- Several important formulas are provided for integrating exponential and trigonometric functions.
Lesson 28: The Fundamental Theorem of Calculus
This document contains lecture notes on the Fundamental Theorem of Calculus. It begins with announcements about the final exam date and current grade distribution. The outline then reviews the Evaluation Theorem and introduces the First and Second Fundamental Theorems of Calculus. It provides examples of how the integral can represent total change in concepts like distance, cost, and mass. Biographies are also included of mathematicians like Gregory, Barrow, Newton, and Leibniz who contributed to the development of calculus.
Lesson 28: The Fundamental Theorem of Calculus
This document contains notes from a calculus class. It provides the outline and key points about the Fundamental Theorem of Calculus. It discusses the first and second Fundamental Theorems of Calculus, including proofs and examples. It also provides brief biographies of several important mathematicians that contributed to the development of calculus, including the Fundamental Theorem of Calculus, such as Isaac Newton, Gottfried Leibniz, James Gregory, and Isaac Barrow.
boyd 3.1
This document discusses convex functions and some of their key properties. It defines a convex function as a function where the epigraph is a convex set. Some key properties are: convex functions have sublevel sets that are convex, their epigraph representation links them to convex geometry, and Jensen's inequality extends to them. Examples are given of convex functions on R and Rn like norms, quadratic over linear, log-sum-exp, and log-determinant functions. Convex functions satisfy useful first-order and second-order conditions and their properties help derive important inequalities like Holder's inequality.
Function evaluation, termination, vertical line test etc
The document discusses functions and their inverses. It defines key terms like domain, range, and inverse functions. It provides examples of determining if a relationship is a function using the vertical line test or one-to-one mapping. The document also covers finding inverses of straight lines, parabolas, exponential graphs and logarithmic graphs. It shows how to sketch these graphs and determine their domains and ranges. Finally, it discusses solving cubic expressions through finding factors and remainders.
3.2 Derivative as a Function
This document covers key concepts in calculus including:
- Computing derivatives using notation such as f'(a) and dy/dx
- Relationships between differentiability and continuity
- Using derivatives to find horizontal tangents, max/min points, and inflection points
- Applying derivative rules such as the sum and constant multiple rules
- Examples of computing derivatives of various functions and determining differentiability
4.4 Fundamental Theorem of Calculus
This document discusses the mean value theorem and how it can be used to find the average value of a function over an interval. It states that for a continuous function, there exists some value c within the interval where the area under the curve between the bounds is equal to the value of the function at c multiplied by the width of the interval. It then works through an example of finding the average value of the function f(x)=3x^2-2x over the interval [1,4].
Common derivatives integrals_reduced
This document provides summaries of common derivatives and integrals, including:
- Basic properties and formulas for derivatives and integrals of functions like polynomials, trig functions, inverse trig functions, exponentials/logarithms, and more.
- Standard integration techniques like u-substitution, integration by parts, and trig substitutions.
- How to evaluate integrals of products and quotients of trig functions using properties like angle addition formulas and half-angle identities.
- How to use partial fractions to decompose rational functions for the purpose of integration.
So in summary, this document outlines essential derivatives and integrals for many common functions, along with standard integration strategies and techniques.
2.1 Calculus 2.formulas.pdf.pdf
The document provides information on various integration techniques including the midpoint rule, trapezoidal rule, Simpson's rule, integration by parts, trigonometric substitutions, and applications of integrals such as finding the area between curves, arc length, surface area of revolution, and volume of revolution. It also covers integrals of common functions, properties of integrals, and techniques for parametric and polar coordinates.
Manyformulas
1. The document outlines various formulas and concepts related to trigonometric, exponential, and calculus functions including differentiation, integration, asymptotes, derivatives, inverse functions, and volumes of revolution.
2. Formulas are provided for trigonometric functions, exponential growth and decay, and the definitions of continuity, derivatives, and inverse functions.
3. Theorems and properties are described for mean value theorem, L'Hopital's rule, fundamental theorem of calculus, and volumes generated by revolving regions about axes.
Similar to 4.3 The Definite Integral (20)
Function evaluation, termination, vertical line test etc
Function evaluation, termination, vertical line test etc
More from Sharon Henry
6.1 Law of Sines Ambiguous Case
This document discusses the ambiguous case of the Law of Sines when given an acute angle (A) and sides a and c of a triangle, where a < c. It explains that:
1) When a < h, where h is the altitude of the triangle, no triangle can be formed.
2) When a = h, one right triangle is formed.
3) When a > h, two triangles are formed - one acute triangle and one obtuse triangle.
So in summary, given the measurements A, a and c where a < c, there will be 0 triangles formed if a < h, 1 triangle if a = h, and 2 triangles if a > h.
6.2 law of cosines
This document provides instruction on using the Law of Cosines to solve triangles. It begins by explaining that the Law of Sines cannot be used to solve triangles when the side-side-side (SSS) or side-angle-side (SAS) configurations are given. The Law of Cosines formulas are then introduced. Examples are worked through demonstrating how to use the Law of Cosines to solve triangles in SSS and SAS formations. Key steps like verifying a triangle can be formed from given SSS measurements and properly applying order of operations are emphasized. Students are assigned homework problems applying these concepts.
Second Derivative Information
The second derivative of a function f(x) provides information about the concavity and points of inflection of f(x). It indicates the intervals where f(x) is concave up or down. A curve is concave up where the second derivative is positive and concave down where the second derivative is negative. Points of inflection are where the concavity changes from up to down or vice versa, and occur where the second derivative is zero or undefined.
Lesson 3.3 First Derivative Information
The first derivative of a function f(x) indicates where the function is increasing, decreasing, or constant. If f'(x) is positive, f(x) is increasing, and if f'(x) is negative, f(x) is decreasing. If f'(x) is 0, f(x) is constant. To determine where a function is increasing or decreasing, identify the critical numbers and determine the sign of f'(x) in intervals between them. The first derivative test also allows identification of relative extrema - if f'(x) changes from negative to positive at a critical number c, then f(c) is a relative minimum, and if it changes from positive to negative, f(c
Lesson 3.2 Rolle and Mean Value Theorems
Rolle's Theorem states that if a function f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), with f(a) = f(b), then there exists at least one value c in the open interval where the derivative f'(c) is equal to 0. The Mean Value Theorem similarly guarantees the existence of at least one value c where the derivative is equal to the average rate of change of the function over the interval, (f(b)-f(a))/(b-a).
Lesson 3.3 3.4 - 1 st and 2nd Derivative Information
1) The document discusses tests for determining whether a function is increasing, decreasing, or constant based on the sign of the first derivative on an interval.
2) It also describes how to use the first and second derivative tests to determine if a critical point is a relative maximum or minimum. The tests analyze how the sign of the first or second derivative changes at the critical point.
3) Finally, the document outlines the test for concavity and points of inflection based on the sign of the second derivative on an interval. A change in the sign of the second derivative indicates an inflection point.
3.5 3.6 exp-log models 13-14
1. This document discusses several common mathematical models involving exponential and logarithmic functions, including exponential growth/decay models, logistic growth models, Gaussian models, logarithmic models, and Newton's Law of Cooling.
2. Examples are provided to illustrate how to set up and solve problems using each type of model. For instance, an example uses an exponential growth model to predict when the world's population will reach 7.1 billion people.
3. Guidance is also given on using graphing utilities and calculators to analyze exponential and logarithmic relationships. Instructions for performing exponential regression on a TI-84 calculator are provided.
3.1 Extreme Values of Functions
This document defines and discusses absolute and local extreme values of functions. It states that absolute extrema (global maximum and minimum values) can occur at endpoints or interior points of an interval, but a function is not guaranteed to have an absolute max or min on every interval. The Extreme Value Theorem says that if a function is continuous on a closed interval, it will have both an absolute maximum and minimum value. Local extrema are defined as maximum or minimum values within an open neighborhood of a point, and the theorem is presented that if a function has a local extremum at an interior point where the derivative exists, the derivative must be zero at that point. Critical points are defined as points where the derivative is zero or undefined. Methods
2.6 Related Rates
This document defines and provides examples of related rate problems. Related rate problems involve calculating the rates of change of two or more related variables with respect to time. The document outlines the steps to solve related rate problems: 1) identify changing quantities, 2) write an equation relating the quantities, 3) differentiate both sides with respect to time, 4) solve for the required quantity and substitute known values. Examples include calculating the rate of change of a dog's circular puddle area, a melting snowball's radius, the top of a slipping ladder, and the height of a conical sand pile.
3.5 EXP-LOG MODELS
The document discusses various mathematical models involving exponential and logarithmic functions, including exponential growth and decay models, logistic growth models, Gaussian models, and logarithmic models. It provides examples of how each type of model can be used, such as modeling world population growth with an exponential function, modeling the spread of a virus on a college campus with a logistic function, and modeling cooling of a heated object using Newton's Law of Cooling. The document also discusses how to graph and solve problems involving these different mathematical models.
Larson 4.4
This document discusses evaluating trigonometric functions of any angle by using reference angles. It defines trigonometric functions for any real value of the angle θ. For angles greater than 90° or less than 0°, the values can be determined from the acute reference angle. Examples show how to find the reference angle and then use it to evaluate trigonometric functions in different quadrants, including for nonacute angles.
Larson 4.3
This document discusses trigonometric functions and their applications. It defines the six trigonometric functions - sine, cosine, tangent, cotangent, secant, and cosecant - using ratios of sides of a right triangle. Examples are provided to evaluate the trig functions of given angles and to use identities to relate functions. The document also discusses applications of solving right triangles by using trig functions when given angle and side length information.
Larson 4.2
The unit circle relates real numbers to points on a circle of radius 1 centered at the origin. Each real number t corresponds to a point (x, y) on the circle, allowing the definition of the six trigonometric functions in terms of x and y. The trigonometric functions sine and cosine are periodic with period 2π, meaning their values repeat every 2π units. Their domains are all real numbers and their ranges are between -1 and 1.
Derivative rules ch2
This document provides the basic derivative rules from Chapter 2. It lists various expressions and their derivatives on subsequent slides, including the derivatives of sums, products, quotients, composite functions, powers, trigonometric functions like sine, cosine, tangent, cotangent, secant and cosecant, and constants. The derivatives covered are: du/dv, df(g(x))/dx, dn/dx, d/dx cot(u), d/dx cu, d/dx tan(u), d/dx (u*v), d/dx c, d/dx cos(u), d/dx sec(u), dx/dx, and d/dx csc(
1.4 Continuity
The document discusses continuity of functions at a point c. It defines continuity as a function being defined at c, the limit existing at c, and the function value at c equaling the limit value. Five examples are given and categorized as continuous, discontinuous with removable discontinuities, or discontinuous with nonremovable discontinuities. Removable discontinuities can be resolved by redefining the function value at c, while nonremovable discontinuities cannot be resolved.
Chapter 5 Definite Integral Topics Review
This document discusses several topics in calculus including RAM theory for approximating functions, computing definite integrals using formulas and a TI-89 calculator, properties of definite integrals like additivity and the mean value theorem, the fundamental theorem of calculus in two parts, and the trapezoidal rule for approximating definite integrals which overestimates for increasing functions and underestimates for decreasing functions.
Lesson 4.3 First and Second Derivative Theory
1) To determine if a function is increasing or decreasing, examine the sign of the first derivative f'(x) at points around any critical points. A change from positive to negative indicates a relative maximum, and a change from negative to positive indicates a relative minimum.
2) A curve is concave up if the second derivative f''(x) is positive, and concave down if f''(x) is negative. To determine concavity, examine the sign of f''(x) around any points where it is zero or undefined. A change in sign indicates an inflection point where the concavity changes.
Lesson 4.1 Extreme Values
1) A function has an absolute maximum value on its domain if its value is greater than or equal to its value at all other points in its domain, and an absolute minimum value if its value is less than or equal to its value at all other points.
2) By the Extreme Value Theorem, if a function is continuous on a closed interval, it will have both an absolute maximum and minimum value within that interval. These extreme values can occur at interior points or endpoints.
3) A local extreme value of a function is a maximum or minimum value within a neighborhood of some interior point, where the function's derivative is equal to 0 or undefined at that point, according to the Local Extreme Value Theorem.
YOSEMITE
Yosemite National Park is home to stunning natural wonders like El Capitan, Half Dome, and Yosemite Falls. John Muir, a famous conservationist, believed that spending time in nature was essential for human well-being and that mountains and wilderness areas should be protected. The document highlights several of Yosemite's scenic areas and includes inspiring quotes from Muir about the spiritual renewal found in nature.
1.3 SYMMETRY
The document discusses three types of symmetry for graphs: x-axis symmetry, where (x,y) corresponds to (x,-y); y-axis symmetry, where (x,y) corresponds to (-x,y); and origin symmetry, where (x,y) corresponds to (-x,-y). It provides tests for each type of symmetry by replacing variables in the graphing equation. An example at the end finds no x-axis or origin symmetry but y-axis symmetry.
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Lesson 3.3 3.4 - 1 st and 2nd Derivative Information
Lesson 3.3 3.4 - 1 st and 2nd Derivative Information
4.3 The Definite Integral
- 2. Notation for the definite integral… b f(x) dx a ◦ ◦ b is upper limit of integration a is lower limit of integration This equals the area under a curve, above the x-axis if… 1. f(x) is continuous 2. f(x) is nonnegative
- 3. b A rea f ( x ) d x w h e n f(x) 0 a b A rea f ( x ) d x w h e n f(x) 0 a Net Area: b f ( x ) d x = (a re a a b ove x-a xis) - (a re a b e low x-a xis) a
- 4. If f ( x ) c on [a ,b ], th e n b c d x = c(b -a ) a
- 5. b Syntax: To evaluate f(x) dx … a MATH 9 fnInt(
- 6. 1. Order of Integration a b f(x) dx b f(x) dx a
- 7. 2. Zero a f(x) dx = 0 a
- 8. 3. Constant Multiple b b k f(x) dx a k f (x ) dx a
- 9. 4. Sum and Difference b b [ f(x) a g ( x )] d x b f(x) dx a g( x ) d x a
- 10. 5. Additivity b c f(x) dx + a c f(x) dx b f (x) dx a
- 11. a b 1. Order of Integration f(x) dx f(x) dx b a a 2. Zero f(x) dx = 0 a b b 3. Constant Multiple k f(x) dx k a f (x ) dx a b 4. Sum and Difference b [ f(x) g ( x )] d x a b f(x) dx a b 5. Additivity c f(x) dx + a a c f(x) dx b g( x ) d x f (x) dx a