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This document discusses methods for finding relative extrema of functions: 1. The First Derivative Test (FDT) states that a critical point is a relative maximum if the derivative changes from positive to negative, and a relative minimum if the derivative changes from negative to positive. 2. The Second Derivative Test (SDT) states that a critical point is a relative maximum if the second derivative is negative, and a relative minimum if the second derivative is positive. 3. Examples are provided to demonstrate using the FDT and SDT to find relative extrema of functions.

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Increasing and decreasing functions ap calc sec 3.3

The document discusses increasing and decreasing functions and the first derivative test. It defines that a function is increasing if the derivative is positive, decreasing if the derivative is negative, and constant if the derivative is zero. It provides examples of finding the intervals where a function is increasing or decreasing by identifying critical numbers and testing points in each interval. The document also summarizes the first derivative test, stating that a critical point is an extremum if the derivative changes sign there, and whether it is a maximum or minimum depends on if the derivative changes from negative to positive or positive to negative.

Probability mass functions and probability density functions

This document discusses probability mass functions (pmf) and probability density functions (pdf) for discrete and continuous random variables. A pmf fX(x) gives the probability of a discrete random variable X taking on the value x. A pdf fX(x) defines the probability that a continuous random variable X falls within an interval via its cumulative distribution function FX(x). The pdf must be non-negative and have an area/sum of 1 under the curve/over all x values.

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.

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1) The document provides examples of solving linear equations involving fractions.
2) Several multi-step linear equations are solved through addition, subtraction, multiplication and division.
3) The solutions are provided after showing the step-by-step work.

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The document discusses discrete and continuous random variables. It defines discrete random variables as variables that can take on countable values, like the number of heads from coin flips. Continuous random variables can take any value within a range, like height. The document explains how to calculate and interpret the mean, standard deviation, and probabilities of events for both types of random variables using examples like Apgar scores for babies and heights of young women.

Inverse Function.pptx

This document provides information about inverse functions including:
- The inverse of a function is formed by reversing the coordinates of each ordered pair. A function has an inverse only if it is one-to-one.
- The domain of the inverse function is the range of the original function, and the range of the inverse is the domain of the original.
- To find the inverse of a one-to-one function, replace f(x) with y, interchange x and y, then solve for y in terms of x and replace y with f^-1(x).
- Several examples are provided to demonstrate finding the inverse of different one-to-one functions by following the given steps.

Differentiation jan 21, 2014

Stationary Points,
To determine Maximum / Minimum / Inflection Points
First Derivative Test
Second Derivative Test

Introduction to Function, Domain and Range - Mohd Noor

This slide introduce the concept of mathematical function as well as the concepts of domain and range.

Increasing and decreasing functions ap calc sec 3.3

The document discusses increasing and decreasing functions and the first derivative test. It defines that a function is increasing if the derivative is positive, decreasing if the derivative is negative, and constant if the derivative is zero. It provides examples of finding the intervals where a function is increasing or decreasing by identifying critical numbers and testing points in each interval. The document also summarizes the first derivative test, stating that a critical point is an extremum if the derivative changes sign there, and whether it is a maximum or minimum depends on if the derivative changes from negative to positive or positive to negative.

Probability mass functions and probability density functions

This document discusses probability mass functions (pmf) and probability density functions (pdf) for discrete and continuous random variables. A pmf fX(x) gives the probability of a discrete random variable X taking on the value x. A pdf fX(x) defines the probability that a continuous random variable X falls within an interval via its cumulative distribution function FX(x). The pdf must be non-negative and have an area/sum of 1 under the curve/over all x values.

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.

1.3 Solving Linear Equations

1) The document provides examples of solving linear equations involving fractions.
2) Several multi-step linear equations are solved through addition, subtraction, multiplication and division.
3) The solutions are provided after showing the step-by-step work.

Discrete and Continuous Random Variables

The document discusses discrete and continuous random variables. It defines discrete random variables as variables that can take on countable values, like the number of heads from coin flips. Continuous random variables can take any value within a range, like height. The document explains how to calculate and interpret the mean, standard deviation, and probabilities of events for both types of random variables using examples like Apgar scores for babies and heights of young women.

Inverse Function.pptx

This document provides information about inverse functions including:
- The inverse of a function is formed by reversing the coordinates of each ordered pair. A function has an inverse only if it is one-to-one.
- The domain of the inverse function is the range of the original function, and the range of the inverse is the domain of the original.
- To find the inverse of a one-to-one function, replace f(x) with y, interchange x and y, then solve for y in terms of x and replace y with f^-1(x).
- Several examples are provided to demonstrate finding the inverse of different one-to-one functions by following the given steps.

Differentiation jan 21, 2014

Stationary Points,
To determine Maximum / Minimum / Inflection Points
First Derivative Test
Second Derivative Test

Introduction to Function, Domain and Range - Mohd Noor

This slide introduce the concept of mathematical function as well as the concepts of domain and range.

Chapter 4 part3- Means and Variances of Random Variables

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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

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The closed interval method tells us how to find the extreme values of a continuous function defined on a closed, bounded interval: we check the end points and the critical points.

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The document discusses binomial, Poisson, and hypergeometric probability distributions. It provides examples of experiments that follow each distribution and how to calculate probabilities using the respective formulas. For binomial experiments, the probability of success must be constant on each trial and trials must be independent. Poisson experiments involve rare, independent events with a known average rate. Hypergeometric probabilities are used when the probability of success changes on each dependent trial, such as sampling without replacement.

3.1 derivative of a function

The document defines the derivative of a function and discusses:
- The definition of the derivative as the limit of the slope between two points as they approach each other.
- Notation used to represent derivatives, including f'(x), dy/dx, and df/dx.
- How the graph of a function's derivative f' relates to the graph of the original function f - where f' is positive/negative/zero corresponds to parts of f that are increasing/decreasing/at an extremum.
- How to graph f given a graph of its derivative f' by sketching the curve that matches the behavior of f' at each point.
- One-sided derivatives at endpoints of functions defined

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This document contains copyrighted content from Pearson Education discussing logarithmic functions. It includes examples of evaluating logarithmic expressions and solving logarithmic equations. The document covers properties of logarithmic functions including their domains and the process of changing between exponential and logarithmic form.

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This chapter discusses continuous random variables and probability distributions, including the normal distribution. It introduces continuous random variables and their probability density functions. It describes the key characteristics and properties of the uniform and normal distributions. It also discusses how to calculate probabilities using the normal distribution, including how to standardize a normal distribution and use normal distribution tables.

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The document discusses hypergeometric probability distribution. It provides examples of hypergeometric experiments involving selecting items from a population without replacement, where the probability of success changes with each trial. The key points are:
- A hypergeometric experiment has a fixed population with a specified number of successes, samples items without replacement, and the probability of success changes on each trial.
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- Examples demonstrate calculating hypergeometric probabilities and approximating it as a binomial when the population is large compared to the sample size.

Functions limits and continuity

This document discusses functions, limits, and continuity. It begins by defining functions, domains, ranges, and some standard real functions like constant, identity, modulus, and greatest integer functions. It then covers limits of functions including one-sided limits and properties of limits. Examples are provided to illustrate evaluating limits using substitution and factorization methods. The overall objectives are to understand functions, domains, ranges, limits of functions and methods to evaluate limits.

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1) The document discusses limits, properties of limits, one-sided limits, and continuity in functions. It provides examples of calculating limits as variables approach certain values.
2) One-sided limits are defined as left and right hand limits, depending on whether the variable approaches the point from the left or right.
3) For a function to be continuous at a point, its limit must exist at that point and be equal to the function value. Examples are given to demonstrate continuity.

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1. The document defines geometric probability as probability based on ratios of geometric measures like length and area, where outcomes are represented by points or regions.
2. Examples are provided to demonstrate calculating geometric probabilities for situations like choosing a random point on a line segment or in a plane figure, the probability of light cycles, and spinners.
3. Additional examples find the probabilities of points chosen in a rectangle landing in specific shapes like a circle, trapezoid, or one of two squares.

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Deriving the composition of functions

1) A composite function is formed by combining two functions, where one function is substituted into the other.
2) Notations like fg(x) indicate that function g(x) is substituted into function f(x).
3) Composite functions are non-commutative, meaning the order of the functions matters - fg(x) may not equal gf(x).

FM calculus

The document discusses differentiation and tangents. It explains that differentiation finds the gradient of a curve at a point and is needed because curves have changing gradients. It provides examples of differentiating simple functions like y=3x^5. Tangents are lines that touch a curve at a single point, with the same gradient as the curve at that point. To find the equation of a tangent, take the derivative and plug in the x-value to get the slope, then use the point to find the y-intercept. Normals are perpendicular to tangents, with a gradient equal to the negative reciprocal of the tangent's gradient.

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The document discusses various types of limits of functions including:
- One-sided limits, which describe the limiting behavior of a function as the independent variable approaches a given value from one side.
- Two-sided limits, which require the function values to get closer to a number as the variable approaches the value from both sides.
- Limits at infinity, which describe the behavior of a function as the variable increases or decreases without bound.
The document provides definitions, examples, and theorems related to evaluating different types of limits algebraically and graphically.

Calc 1.4a

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Chapter 4 part3- Means and Variances of Random Variables

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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 19: Maximum and Minimum Values

The closed interval method tells us how to find the extreme values of a continuous function defined on a closed, bounded interval: we check the end points and the critical points.

Discrete probability distributions

The document discusses binomial, Poisson, and hypergeometric probability distributions. It provides examples of experiments that follow each distribution and how to calculate probabilities using the respective formulas. For binomial experiments, the probability of success must be constant on each trial and trials must be independent. Poisson experiments involve rare, independent events with a known average rate. Hypergeometric probabilities are used when the probability of success changes on each dependent trial, such as sampling without replacement.

3.1 derivative of a function

The document defines the derivative of a function and discusses:
- The definition of the derivative as the limit of the slope between two points as they approach each other.
- Notation used to represent derivatives, including f'(x), dy/dx, and df/dx.
- How the graph of a function's derivative f' relates to the graph of the original function f - where f' is positive/negative/zero corresponds to parts of f that are increasing/decreasing/at an extremum.
- How to graph f given a graph of its derivative f' by sketching the curve that matches the behavior of f' at each point.
- One-sided derivatives at endpoints of functions defined

Section 5.4 logarithmic functions

This document contains copyrighted content from Pearson Education discussing logarithmic functions. It includes examples of evaluating logarithmic expressions and solving logarithmic equations. The document covers properties of logarithmic functions including their domains and the process of changing between exponential and logarithmic form.

Exponential and logarithmic functions

This document provides an overview of exponential and logarithmic functions. It defines one-to-one functions and inverse functions. It explains how to find the inverse of a one-to-one function and shows that the inverse of f(x) is f-1(x). Properties of exponential functions like f(x)=ax and logarithmic functions like f(x)=logax are described. The product, quotient, and power rules for logarithms are outlined along with examples. Finally, it discusses how to solve exponential and logarithmic equations using properties of these functions.

Chap05 continuous random variables and probability distributions

This chapter discusses continuous random variables and probability distributions, including the normal distribution. It introduces continuous random variables and their probability density functions. It describes the key characteristics and properties of the uniform and normal distributions. It also discusses how to calculate probabilities using the normal distribution, including how to standardize a normal distribution and use normal distribution tables.

Hypergeometric probability distribution

The document discusses hypergeometric probability distribution. It provides examples of hypergeometric experiments involving selecting items from a population without replacement, where the probability of success changes with each trial. The key points are:
- A hypergeometric experiment has a fixed population with a specified number of successes, samples items without replacement, and the probability of success changes on each trial.
- The hypergeometric distribution gives the probability of getting x successes in n draws from a population of N items with K successes.
- Examples demonstrate calculating hypergeometric probabilities and approximating it as a binomial when the population is large compared to the sample size.

Functions limits and continuity

This document discusses functions, limits, and continuity. It begins by defining functions, domains, ranges, and some standard real functions like constant, identity, modulus, and greatest integer functions. It then covers limits of functions including one-sided limits and properties of limits. Examples are provided to illustrate evaluating limits using substitution and factorization methods. The overall objectives are to understand functions, domains, ranges, limits of functions and methods to evaluate limits.

Differentiation

The document discusses differentiation and its history. It was independently developed in the 17th century by Isaac Newton and Gottfried Leibniz. Differentiation allows the calculation of instantaneous rates of change and is used in many areas including mathematics, physics, engineering and more. Key concepts covered include calculating speed, estimating instantaneous rates of change, the rules for differentiation, and differentiation of expressions with multiple terms.

limits and continuity

1) The document discusses limits, properties of limits, one-sided limits, and continuity in functions. It provides examples of calculating limits as variables approach certain values.
2) One-sided limits are defined as left and right hand limits, depending on whether the variable approaches the point from the left or right.
3) For a function to be continuous at a point, its limit must exist at that point and be equal to the function value. Examples are given to demonstrate continuity.

Lesson 11: Limits and Continuity

The concept of limit is a lot harder for functions of several variables than for just one. We show the more dramatric ways that a limit can fail.

Obj. 41 Geometric Probability

1. The document defines geometric probability as probability based on ratios of geometric measures like length and area, where outcomes are represented by points or regions.
2. Examples are provided to demonstrate calculating geometric probabilities for situations like choosing a random point on a line segment or in a plane figure, the probability of light cycles, and spinners.
3. Additional examples find the probabilities of points chosen in a rectangle landing in specific shapes like a circle, trapezoid, or one of two squares.

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

Composite functions

The document discusses operations and composition of functions. It explains that to find the sum of two functions f and g, you add them together and combine like terms. To find the difference, you subtract the second function from the first and distribute negatives. To find the product, you multiply corresponding terms of f and g. For the quotient f/g, you divide the first function by the second. The domain of sums, differences and products is where x is in the domains of both f and g, while the domain of the quotient excludes values where the denominator would be 0. Composition means substituting one function into another, written as f(g) or g(f).

Factor theorem

The document discusses the factor theorem and how to determine if a polynomial is a factor of another polynomial. It provides examples of using the factor theorem to show that (x + 1) is a factor of 2x^3 + 5x^2 - 3 and that (x - 2) is a factor of x^4 + x^3 - x^2 - x - 18. It also gives an example of finding a polynomial function given its zeros as -2, 1, -1. The document provides exercises for using the factor theorem to determine unknown values in polynomials.

Deriving the composition of functions

1) A composite function is formed by combining two functions, where one function is substituted into the other.
2) Notations like fg(x) indicate that function g(x) is substituted into function f(x).
3) Composite functions are non-commutative, meaning the order of the functions matters - fg(x) may not equal gf(x).

FM calculus

The document discusses differentiation and tangents. It explains that differentiation finds the gradient of a curve at a point and is needed because curves have changing gradients. It provides examples of differentiating simple functions like y=3x^5. Tangents are lines that touch a curve at a single point, with the same gradient as the curve at that point. To find the equation of a tangent, take the derivative and plug in the x-value to get the slope, then use the point to find the y-intercept. Normals are perpendicular to tangents, with a gradient equal to the negative reciprocal of the tangent's gradient.

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This file is about Polynomial Function and Synthetic Division. A project passed to Mrs. Marissa De Ocampo. Submitted by Group 6 of Grade 10-Galilei of Caloocan National Science and Technology High School '15-'16

Chapter 4 part3- Means and Variances of Random Variables

Chapter 4 part3- Means and Variances of Random Variables

Lesson 3.3 First Derivative Information

Lesson 3.3 First Derivative Information

Lesson 19: Maximum and Minimum Values

Lesson 19: Maximum and Minimum Values

Discrete probability distributions

Discrete probability distributions

3.1 derivative of a function

3.1 derivative of a function

Section 5.4 logarithmic functions

Section 5.4 logarithmic functions

Exponential and logarithmic functions

Exponential and logarithmic functions

Chap05 continuous random variables and probability distributions

Chap05 continuous random variables and probability distributions

Hypergeometric probability distribution

Hypergeometric probability distribution

Functions limits and continuity

Functions limits and continuity

Differentiation

Differentiation

limits and continuity

limits and continuity

Lesson 11: Limits and Continuity

Lesson 11: Limits and Continuity

Obj. 41 Geometric Probability

Obj. 41 Geometric Probability

3.1 Extreme Values of Functions

3.1 Extreme Values of Functions

Composite functions

Composite functions

Factor theorem

Factor theorem

Deriving the composition of functions

Deriving the composition of functions

FM calculus

FM calculus

Polynomial Function and Synthetic Division

Polynomial Function and Synthetic Division

L4 one sided limits limits at infinity

The document discusses various types of limits of functions including:
- One-sided limits, which describe the limiting behavior of a function as the independent variable approaches a given value from one side.
- Two-sided limits, which require the function values to get closer to a number as the variable approaches the value from both sides.
- Limits at infinity, which describe the behavior of a function as the variable increases or decreases without bound.
The document provides definitions, examples, and theorems related to evaluating different types of limits algebraically and graphically.

Calc 1.4a

This document discusses continuity of functions and one-sided limits. It defines continuity at a point as having no interruptions, holes, jumps or gaps. Discontinuities can be removable, where defining the function at a point can make it continuous, or nonremovable due to asymptotes or gaps. A function is continuous on an open interval if it is continuous at every point except possibly at the endpoints. For a closed interval, a function is continuous if it is continuous in the interior and exhibits one-sided continuity at the endpoints. Examples are provided to illustrate these concepts and to test functions for continuity over different intervals.

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The document discusses how the first derivative can be used to analyze curves geometrically. It indicates that the first derivative measures the slope of the tangent line to a curve. If the derivative is positive, the curve is increasing; if negative, decreasing; and if zero, the curve is stationary. As an example, it finds the stationary points of the curve y = 3x^2 - x^3 and determines that they are a minimum at (0,0) and an inflection point at (2,4).

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The document discusses using the first derivative to analyze the geometric properties of curves. The first derivative at a point measures the slope of the tangent line to the curve at that point. If the first derivative is greater than 0 at a point, the curve is increasing there, and if it is less than 0, the curve is decreasing. If the first derivative is 0, the point is stationary. This information can be used to sketch the nature of curves and find all stationary points. For the example curve y=x^2, the stationary points are found to be 0 and 2, with 0 being a minimum turning point.

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This document outlines strategies for differentiating instruction to meet the needs of high-ability learners. It discusses assessing students' readiness, interests, and learning profiles through pre-assessments. Differentiation strategies presented include tiered activities, learning centers, compacting, independent projects, acceleration, and mentorships. The document emphasizes starting small with differentiation and giving students choices that appeal to their varying skills, interests, and preferences. The goal of differentiation is to customize instruction so all students continuously learn.

Limits And Derivative

The document discusses key concepts in calculus including functions, limits, derivatives, and derivatives of trigonometric functions. It provides examples of calculating derivatives from first principles using the definition of the derivative and common derivative rules like the product rule and quotient rule. Formulas are also derived for the derivatives of the sine, cosine, and tangent functions.

Math major 14 differential calculus pw

This document provides an overview of topics covered in a differential calculus course, including:
1. Limits and differential calculus concepts such as derivatives
2. Special functions and numbers used in calculus
3. A brief history of calculus and its founders Newton and Leibniz
4. Explanations and examples of key calculus concepts such as variables, constants, functions, and limits

Application of differentiation

This presentation explains how the differentiation is applied to identify increasing and decreasing functions,identifying the nature of stationary points and also finding maximum or minimum values.

The Application of Derivatives

This document discusses several topics related to calculus including:
1) Derivatives of position, velocity, and acceleration and how they relate to each other.
2) An example problem calculating velocity from a position function.
3) The Mean Value Theorem and how to apply it to find critical points of a function.
4) How the first and second derivatives of a function relate to critical points, maxima, minima, and points of inflection or concavity.
5) Related rates problems and how to set them up using derivatives and relationships between variables.

L4 one sided limits limits at infinity

L4 one sided limits limits at infinity

Calc 1.4a

Calc 1.4a

Chapter 02 differentiation

Chapter 02 differentiation

11 x1 t10 01 first derivative (2012)

11 x1 t10 01 first derivative (2012)

Sulpcegu5e ppt 14_3

Sulpcegu5e ppt 14_3

11 x1 t12 01 first derivative (2013)

11 x1 t12 01 first derivative (2013)

Kalkulus II (5 - 6)

Kalkulus II (5 - 6)

Lar calc10 ch01_sec4

Lar calc10 ch01_sec4

neoplastic disruptions alterations in cell function & differentiation pp

neoplastic disruptions alterations in cell function & differentiation pp

Vocabulary

Vocabulary

IB Maths. Graphs of a funtion and its derivative

IB Maths. Graphs of a funtion and its derivative

Lesson 3: The Limit of a Function (slides)

Lesson 3: The Limit of a Function (slides)

Energy minimization

Energy minimization

Indeterminate Forms and L' Hospital Rule

Indeterminate Forms and L' Hospital Rule

Lesson 3: The Limit of a Function

Lesson 3: The Limit of a Function

Differentiation For High Ability Learners

Differentiation For High Ability Learners

Limits And Derivative

Limits And Derivative

Math major 14 differential calculus pw

Math major 14 differential calculus pw

Application of differentiation

Application of differentiation

The Application of Derivatives

The Application of Derivatives

Calc 3.4b

This document discusses using the second derivative test to find relative extrema of a function. It explains that you first find the critical numbers where the first derivative is equal to zero or undefined. You then take the second derivative at those critical numbers. If the second derivative is positive, it is a relative minimum, and if negative, it is a relative maximum. The document provides an example of using this process to find the relative extrema of the function f(x) = -3x^5 + 5x^3, determining it has a relative minimum at x = -1 and a relative maximum at x = 1.

Day 1a examples

The document discusses how to determine if a function is increasing or decreasing based on the sign of its derivative, and provides an example of finding the intervals where a function is increasing or decreasing. It also discusses how to find and classify critical points of a function by taking the derivative, setting it equal to 0, and using the first derivative test to determine if the critical points are local maxima or minima.

Calc 3.4

The document discusses using the second derivative test to determine the concavity and points of inflection of functions. It explains that the second derivative test involves finding intervals where the second derivative is positive or negative to determine if a graph is concave up or down. Points of inflection occur when the second derivative is equal to zero. Examples are provided to demonstrate how to apply this test to locate intervals of concavity and points of inflection of various functions.

Derivatives in graphing-dfs

This document provides an overview of key concepts in calculus related to derivatives, including: analyzing functions to determine if they are increasing or decreasing; finding relative extrema, critical points, and inflection points; using the first and second derivative tests to determine concavity; and graphing polynomials. Examples are provided to illustrate how to apply these concepts to specific functions in order to analyze intervals of increase/decrease, locate critical points, identify relative maxima and minima, and determine intervals of concavity. Videos and Khan Academy links are also included for supplemental instruction on related topics.

Curve sketching

This document outlines the 7 steps for sketching the curve of a function: 1) Determine the domain, 2) Find critical points, 3) Determine graph direction and max/min, 4) Use the second derivative to find concavity and points of inflection, 5) Find asymptotes, 6) Find intercepts and important points, 7) Combine evidence to graph the function. Key tests are outlined for max/min, concavity, and points of inflection using the first and second derivatives.

Ap calculus extrema v2

The document discusses extreme values (maximums and minimums) of functions. It defines local and absolute extrema and notes that not all functions have extrema. Critical points, where the derivative is zero or undefined, are discussed. The relationship between critical points and extrema is explained by Fermat's Theorem. Methods for finding extrema on closed intervals using critical points and endpoints are presented, along with an example. Rolle's Theorem relating critical points to functions with equal values at endpoints is also introduced.

Mac2311 study guide-tcm6-49721

This document provides an overview of key calculus concepts and formulas taught in a Calculus I course at Miami Dade College - Hialeah Campus. The topics covered include limits and derivatives, integration, optimization techniques, and applications of calculus to economics, business, physics, and other fields. The document is intended as a study guide for students in the Calculus I class taught by Professor Mohammad Shakil.

2301MaxMin.ppt

The document defines absolute and local maximum and minimum values of functions. An absolute (or global) maximum is the highest value a function can take over its entire domain, while an absolute minimum is the lowest value. A local maximum/minimum is the highest/lowest value a function takes in the neighborhood of a particular point. The Extreme Value Theorem states that continuous functions on closed intervals attain both absolute maximum and minimum values. Fermat's Theorem provides that local extrema occur where the derivative is 0 or undefined. To find absolute extrema, one evaluates the function at critical points and endpoints.

Probability-1.pptx

This document provides an overview of key concepts in probability and probability distributions. It introduces random variables and their probability distributions, and covers discrete and continuous random variables. Specific probability distributions discussed include the binomial, Poisson, and normal distributions. Expected value and variance are defined as measures of the central tendency and variability of random variables. Examples are provided to illustrate calculating probabilities and parameters for different probability distributions.

Applications of Differentiation

This document discusses various applications of differentiation including finding extrema, Rolle's theorem, the mean value theorem, and determining whether a function is increasing or decreasing. It provides examples of using the derivative to find relative extrema, applying Rolle's theorem to show a horizontal tangent exists between two roots, using the mean value theorem to find a point where the tangent line is parallel to the secant line, and determining the intervals where a function is increasing or decreasing using the first derivative test.

Lesson 20: Derivatives and the Shapes of Curves (slides)

f is decreasing on (-∞, 0] and (-4/5, 0] and increasing on [0, ∞) and [-4/5, 0).

Lesson 20: Derivatives and the Shapes of Curves (slides)

This document contains lecture notes on derivatives and the shapes of curves from a Calculus I class taught by Professor Matthew Leingang at New York University. The notes cover using derivatives to determine the intervals where a function is increasing or decreasing, classifying critical points as maxima or minima, using the second derivative to determine concavity, and applying the first and second derivative tests. Examples are provided to illustrate finding intervals of monotonicity for various functions.

Limit and continuity

The document discusses limits, continuity, and related concepts. Some key points:
1) It defines the concept of a limit and explains how to evaluate one-sided and two-sided limits. A limit exists only if the left and right-sided limits are equal.
2) Continuity is defined as a function being defined at a point, and the limit existing and being equal to the function value. Functions like tan(x) are only continuous where the denominator is not 0.
3) Theorems are presented for evaluating limits of polynomials, sums, products, quotients of continuous functions, and the squeeze theorem. Piecewise functions may or may not be continuous depending on behavior at points of discontin

Maximums and minimum

Applications of maximums and minimums in Calculus. This includes infection points, critical points, increasing and decreasing, etc.

Lecture 12(point of inflection and concavity)

This document discusses techniques for analyzing functions based on their derivatives. It explains how to determine if a function is increasing or decreasing based on the sign of the first derivative. It also describes how to identify relative maxima and minima by examining changes in the sign of the first derivative. Additionally, it covers how to determine if a function is concave up or down using the second derivative and how to identify points of inflection where the concavity changes. The second derivative test is presented for classifying critical points as maxima or minima.

Lecture 12(point of inflection and concavity)

This document discusses techniques for analyzing functions based on their derivatives. It explains how to determine if a function is increasing or decreasing based on the sign of the first derivative. It also describes how to identify relative maxima and minima by examining changes in the sign of the first derivative. Additionally, it covers how to determine if a function is concave up or down using the second derivative and how to identify points of inflection where the concavity changes. The second derivative test is presented to classify critical points as maxima or minima.

Calculus

The document discusses methods for finding the maximum and minimum values of functions with two independent variables, including:
1) Using the second derivative test to determine if a critical point is a local maximum or minimum.
2) Finding absolute maxima and minima, which are the overall largest and smallest function values in the domain.
3) Identifying stationary points where the first partial derivatives are equal to zero.
4) Using the Hessian matrix and eigenvalues to generalize the second derivative test to functions of more than one variable.

Application of derivative

Additional Applications of the Derivative related to engineering maths
this presentation is only for the knowledge...............

Application of derivatives

The document discusses several key concepts regarding derivatives:
(1) It explains how to use the derivative to determine if a function is increasing, decreasing, or neither on an interval using the signs of the derivative.
(2) It provides theorems and rules for finding local extrema (maxima and minima) of functions using the first and second derivative tests.
(3) It also discusses absolute extrema, monotonic functions, and the Rolle's Theorem and Mean Value Theorem which relate the derivative of a function to values of the function.

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.

Calc 3.4b

Calc 3.4b

Day 1a examples

Day 1a examples

Calc 3.4

Calc 3.4

Derivatives in graphing-dfs

Derivatives in graphing-dfs

Curve sketching

Curve sketching

Ap calculus extrema v2

Ap calculus extrema v2

Mac2311 study guide-tcm6-49721

Mac2311 study guide-tcm6-49721

2301MaxMin.ppt

2301MaxMin.ppt

Probability-1.pptx

Probability-1.pptx

Applications of Differentiation

Applications of Differentiation

Lesson 20: Derivatives and the Shapes of Curves (slides)

Lesson 20: Derivatives and the Shapes of Curves (slides)

Lesson 20: Derivatives and the Shapes of Curves (slides)

Lesson 20: Derivatives and the Shapes of Curves (slides)

Limit and continuity

Limit and continuity

Maximums and minimum

Maximums and minimum

Lecture 12(point of inflection and concavity)

Lecture 12(point of inflection and concavity)

Lecture 12(point of inflection and concavity)

Lecture 12(point of inflection and concavity)

Calculus

Calculus

Application of derivative

Application of derivative

Application of derivatives

Application of derivatives

Lesson 3.3 3.4 - 1 st and 2nd Derivative Information

Lesson 3.3 3.4 - 1 st and 2nd Derivative Information

8.7 numerical integration

The trapezoidal rule is used to approximate the area under a curve by dividing it into trapezoids. It takes the average of the function values at the beginning and end of each sub-interval multiplied by the sub-interval width. The general formula sums these values over all sub-intervals divided by the number of intervals. An example calculates the area under y=1+x^3 from 0 to 1 using n=4 sub-intervals and gets an approximate value of 1.26953125.

8.2 integration by parts

Integration by parts is a technique for evaluating integrals of the form ∫udv, where u and v are differentiable functions. It works by expressing the integral as uv - ∫vdu. Some examples of integrals solved using integration by parts include ∫xe^xdx, ∫lnxdx, and ∫xe^-xdx. The technique can also be used repeatedly and for definite integrals between limits a and b using the formula ∫abudv = uv|_a^b - ∫avdu.

7.3 volumes by cylindrical shells

This document discusses calculating the volume of solids of revolution formed by rotating an area bounded by graphs around an axis. It provides the formula for finding the volume of a cylindrical shell as well as the formula for finding the total volume of a solid of revolution by summing the volumes of infinitely thin cylindrical shells. It includes two example problems demonstrating how to set up and solve the integrals to find the volume of solids of revolution.

7.2 volumes by slicing disks and washers

This document discusses different methods for calculating the volumes of solids of revolution: the disk method and washer method. It provides step-by-step explanations of how to set up and evaluate the definite integrals needed to calculate these volumes, whether the region is revolved about an axis that forms a border or not. Examples are given to illustrate each method. The key steps are to divide the solid into slices, approximate the volume of each slice, add the slice volumes using a limit of a Riemann sum, and evaluate the resulting definite integral.

7.1 area between curves

The document discusses calculating the area between two curves. It explains that this area is defined as the limit of sums of the areas of rectangles between the curves as the number of rectangles approaches infinity, which is represented by a definite integral. It provides examples of finding the area between curves defined by various functions through setting up and evaluating the appropriate definite integrals.

6.3 integration by substitution

This document discusses integration by substitution. It provides an example of recognizing a composite function and rewriting the integral in terms of the inside and outside functions. Specifically, it shows rewriting the integral of (x2 +1)2x dx as the integral of the outside function (x2 + 1) with the inside function (x) plugged in, plus a constant. It then provides additional practice problems applying the technique of substitution to rewrite integrals in terms of u-substitutions.

6.2 the indefinite integral

This document discusses the indefinite integral and antiderivatives. It defines an antiderivative as a function whose derivative is the original function, and notes that there are infinitely many antiderivatives that differ by a constant. The process of finding antiderivatives is called indefinite integration or antidifferentiation. Initial conditions can be used to determine a unique particular solution by solving for the constant of integration.

6.1 & 6.4 an overview of the area problem area

The document discusses different methods for approximating the area under a curve:
- Lower estimate (LAM) uses the left endpoints of intervals
- Upper estimate (RAM) uses the right endpoints
- Average estimate (MAM) uses the midpoints
Formulas are provided for calculating the area using each method by summing the areas of rectangles. Examples are shown for finding the area under y=x^2 from 0 to 2 using each method. Finally, the document introduces using the antiderivative method to find the exact area under a curve by calculating the antiderivative and evaluating it over the bounds.

5.8 rectilinear motion

This document discusses rectilinear motion and concepts related to position, velocity, speed, and acceleration for objects moving along a straight line. It defines velocity as the rate of change of position with respect to time and speed as the magnitude of velocity. Acceleration is defined as the rate of change of velocity with respect to time. Examples are given to show how to calculate position, velocity, speed, and acceleration functions from a given position function. The document also analyzes position versus time graphs to determine characteristics of the particle's motion at different points in time.

5.7 rolle's thrm & mv theorem

This document discusses Rolle's theorem and the mean value theorem. It provides the definitions and formulas for each theorem. It then gives examples of applying each theorem to find values of c where a derivative is equal to zero or a tangent line is parallel to a secant line. Rolle's theorem examples find values of c where the derivative of a function over an interval is zero. The mean value theorem examples find values of c where the slope of a tangent line equals the slope of a secant line over an interval.

5.5 optimization

1. Optimization problems involve finding the maximum or minimum value of a function subject to certain constraints.
2. To solve maximum/minimum problems: draw a figure, write the primary equation relating quantities, reduce to one variable if needed, take the derivative(s) to find critical points, and check solutions in the domain.
3. Examples show applying this process to find the dimensions that maximize volume of an open box, minimize cost of laying pipe between points, maximize area of two corrals with a fixed fence length, and find the largest volume cylinder that can fit in a cone.

5.4 absolute maxima and minima

The document provides information on finding absolute maximum and minimum values (absolute extrema) of functions on different interval types. It discusses determining absolute extrema on closed, infinite, and open intervals. Examples are provided finding the absolute extrema of specific functions on given intervals, including finding any critical points and limits to determine if absolute extrema exist. Practice problems are also provided at the end to find the absolute extrema of additional functions on specified intervals.

5.3 curve sketching

This document provides guidance on sketching graphs of functions by considering key features such as symmetries, intercepts, extrema, asymptotes, concavity, and inflection points. It then works through an example of sketching the graph of the function f(x) = (2x^2 - 8)/(x^2 - 16). Key steps include finding vertical and horizontal asymptotes, critical points and inflection points, intervals of increase/decrease, and finally sketching the graph.

5.1 analysis of function i

This document discusses increasing and decreasing functions, concavity of functions, and finding intervals where functions are increasing, decreasing, concave up, or concave down. It provides examples of finding the intervals for the functions f(x)=x-4x^2+3 and f(x)=x-5x^4+9x showing the steps to determine where the functions are increasing or decreasing and where they are concave up or concave down. It also discusses inflection points and provides an example of finding intervals of increase, decrease, concavity and the inflection point for the function f(x)=x^3-3x^2+1.

4.3 derivatives of inv erse trig. functions

This document discusses derivatives of inverse trigonometric functions and differentiability of inverse functions. It provides examples of finding the derivative of inverse trig functions like sin^-1(x^3) and sec^-1(e^x). It also explains that if a function f(x) is differentiable on an interval I, its inverse f^-1(x) will also be differentiable if f'(x) is not equal to 0. It gives the formula for the derivative of the inverse function and an example confirming this formula. It also discusses monotonic functions and how if f'(x) is always greater than 0 or less than 0, f(x) is one-to-one and its inverse will be different

8.2 integration by parts

Integration by parts is a technique for evaluating integrals of the form ∫udv, where u and v are differentiable functions. It works by expressing the integral as uv - ∫vdu. Some examples of integrals solved using integration by parts include ∫xe^xdx, ∫lnxdx, and ∫xe^-xdx. Repeated integration by parts may be necessary when the integral ∫vdu generated cannot be directly evaluated. Integration by parts also applies to definite integrals between limits a and b using the formula ∫_a^budv = uv|_a^b - ∫_a^bvdu.

8.7 numerical integration

The trapezoidal rule is used to approximate the area under a curve by dividing it into trapezoids. It takes the average of the function values at the beginning and end of each sub-interval. The area is calculated as the sum of the areas of each trapezoid multiplied by the width of the sub-interval. An example calculates the area under y=1+x^3 from 0 to 1 using n=4 sub-intervals, giving an approximate result of 1.26953125. The document also provides an example of using the trapezoidal rule with n=8 sub-intervals to estimate the area under the curve of the function y=x from 0 to 3.

7.3 volumes by cylindrical shells

This document discusses different methods for calculating the volumes of solids of revolution: the disk method and washer method. It provides step-by-step explanations of how to set up and evaluate the definite integrals needed to calculate these volumes, whether the region is revolved about an axis that forms a border or not. Examples are given to illustrate each method. The key steps are to divide the solid into slices, approximate the volume of each slice, add the slice volumes using a limit of a Riemann sum, and evaluate the resulting definite integral.

8.7 numerical integration

8.7 numerical integration

8.2 integration by parts

8.2 integration by parts

7.3 volumes by cylindrical shells

7.3 volumes by cylindrical shells

7.2 volumes by slicing disks and washers

7.2 volumes by slicing disks and washers

7.1 area between curves

7.1 area between curves

6.5 & 6.6 & 6.9 the definite integral and the fundemental theorem of calculus...

6.5 & 6.6 & 6.9 the definite integral and the fundemental theorem of calculus...

6.3 integration by substitution

6.3 integration by substitution

6.2 the indefinite integral

6.2 the indefinite integral

6.1 & 6.4 an overview of the area problem area

6.1 & 6.4 an overview of the area problem area

5.8 rectilinear motion

5.8 rectilinear motion

5.7 rolle's thrm & mv theorem

5.7 rolle's thrm & mv theorem

5.5 optimization

5.5 optimization

5.4 absolute maxima and minima

5.4 absolute maxima and minima

5.3 curve sketching

5.3 curve sketching

5.1 analysis of function i

5.1 analysis of function i

4.3 derivatives of inv erse trig. functions

4.3 derivatives of inv erse trig. functions

7.2 volumes by slicing disks and washers

7.2 volumes by slicing disks and washers

8.2 integration by parts

8.2 integration by parts

8.7 numerical integration

8.7 numerical integration

7.3 volumes by cylindrical shells

7.3 volumes by cylindrical shells

Authentically Social Presented by Corey Perlman

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ikea_woodgreen_petscharity_cat-alogue_digital.pdf

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In this keynote, Luan Wise will provide invaluable insights to elevate your employer brand on social media platforms including LinkedIn, Facebook, Instagram, X (formerly Twitter) and TikTok. You'll learn how compelling content can authentically showcase your company culture, values, and employee experiences to support your talent acquisition and retention objectives. Additionally, you'll understand the power of employee advocacy to amplify reach and engagement – helping to position your organization as an employer of choice in today's competitive talent landscape.

Tata Group Dials Taiwan for Its Chipmaking Ambition in Gujarat’s Dholera

The Tata Group, a titan of Indian industry, is making waves with its advanced talks with Taiwanese chipmakers Powerchip Semiconductor Manufacturing Corporation (PSMC) and UMC Group. The goal? Establishing a cutting-edge semiconductor fabrication unit (fab) in Dholera, Gujarat. This isn’t just any project; it’s a potential game changer for India’s chipmaking aspirations and a boon for investors seeking promising residential projects in dholera sir.
Visit : https://www.avirahi.com/blog/tata-group-dials-taiwan-for-its-chipmaking-ambition-in-gujarats-dholera/

Unveiling the Dynamic Personalities, Key Dates, and Horoscope Insights: Gemin...

Explore the fascinating world of the Gemini Zodiac Sign. Discover the unique personality traits, key dates, and horoscope insights of Gemini individuals. Learn how their sociable, communicative nature and boundless curiosity make them the dynamic explorers of the zodiac. Dive into the duality of the Gemini sign and understand their intellectual and adventurous spirit.

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At Techbox Square, in Singapore, we're not just creative web designers and developers, we're the driving force behind your brand identity. Contact us today.

amptalk_RecruitingDeck_english_2024.06.05

amptalk_RecruitingDeck_english

Recruiting in the Digital Age: A Social Media Masterclass

In this masterclass, presented at the Global HR Summit on 5th June 2024, Luan Wise explored the essential features of social media platforms that support talent acquisition, including LinkedIn, Facebook, Instagram, X (formerly Twitter) and TikTok.

The Evolution and Impact of OTT Platforms: A Deep Dive into the Future of Ent...

This presentation provides a thorough examination of Over-the-Top (OTT) platforms, focusing on their development and substantial influence on the entertainment industry, with a particular emphasis on the Indian market.We begin with an introduction to OTT platforms, defining them as streaming services that deliver content directly over the internet, bypassing traditional broadcast channels. These platforms offer a variety of content, including movies, TV shows, and original productions, allowing users to access content on-demand across multiple devices.The historical context covers the early days of streaming, starting with Netflix's inception in 1997 as a DVD rental service and its transition to streaming in 2007. The presentation also highlights India's television journey, from the launch of Doordarshan in 1959 to the introduction of Direct-to-Home (DTH) satellite television in 2000, which expanded viewing choices and set the stage for the rise of OTT platforms like Big Flix, Ditto TV, Sony LIV, Hotstar, and Netflix. The business models of OTT platforms are explored in detail. Subscription Video on Demand (SVOD) models, exemplified by Netflix and Amazon Prime Video, offer unlimited content access for a monthly fee. Transactional Video on Demand (TVOD) models, like iTunes and Sky Box Office, allow users to pay for individual pieces of content. Advertising-Based Video on Demand (AVOD) models, such as YouTube and Facebook Watch, provide free content supported by advertisements. Hybrid models combine elements of SVOD and AVOD, offering flexibility to cater to diverse audience preferences.
Content acquisition strategies are also discussed, highlighting the dual approach of purchasing broadcasting rights for existing films and TV shows and investing in original content production. This section underscores the importance of a robust content library in attracting and retaining subscribers.The presentation addresses the challenges faced by OTT platforms, including the unpredictability of content acquisition and audience preferences. It emphasizes the difficulty of balancing content investment with returns in a competitive market, the high costs associated with marketing, and the need for continuous innovation and adaptation to stay relevant.
The impact of OTT platforms on the Bollywood film industry is significant. The competition for viewers has led to a decrease in cinema ticket sales, affecting the revenue of Bollywood films that traditionally rely on theatrical releases. Additionally, OTT platforms now pay less for film rights due to the uncertain success of films in cinemas.
Looking ahead, the future of OTT in India appears promising. The market is expected to grow by 20% annually, reaching a value of ₹1200 billion by the end of the decade. The increasing availability of affordable smartphones and internet access will drive this growth, making OTT platforms a primary source of entertainment for many viewers.

Company Valuation webinar series - Tuesday, 4 June 2024

This session provided an update as to the latest valuation data in the UK and then delved into a discussion on the upcoming election and the impacts on valuation. We finished, as always with a Q&A

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Tata Group Dials Taiwan for Its Chipmaking Ambition in Gujarat’s Dholera

Tata Group Dials Taiwan for Its Chipmaking Ambition in Gujarat’s Dholera

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Creative Web Design Company in Singapore

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amptalk_RecruitingDeck_english_2024.06.05

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Recruiting in the Digital Age: A Social Media Masterclass

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The Influence of Marketing Strategy and Market Competition on Business Perfor...

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The Evolution and Impact of OTT Platforms: A Deep Dive into the Future of Ent...

Company Valuation webinar series - Tuesday, 4 June 2024

Company Valuation webinar series - Tuesday, 4 June 2024

- 1. Relative Extrema First Derivative Test - FDT Second Derivative Test - SDT
- 3. Critical Points Critical points are points at which: •Derivative equals zero (also called stationary point). •Derivative doesn’t exist.
- 4. First Derivative Test Let f be a differentiable function with f '(c) = 0, then: •If f '(x) changes from positive to negative, then f has a relative maximum at c. •If f '(x) changes from negative to positive, then f has a relative minimum at c. •If f '(x) has the same sign from left to right, then f does not have a relative extremum at c.
- 5. Practice Time!!! Use First Derivative Test to find critical points and state whether they are 5 2 minimums or maximums. f ( x ) = 3x 3 −15x 3 2 −1 −1 2 1 − 5 3 2 3 3 3 f ' ( x ) = 3× x −15× x f ' ( x ) = 5x −10x = 5x 3 ( x − 2 ) 3 3 1 − 5 ( x − 2) 5x 3 ( x − 2 ) = 0 Critical points =0 1 x3 + 0 __ • 0 relative maximum • 2 relative minimum + 2(stationary)
- 6. Second Derivative Test Suppose that c is a critical point at which f’(c) = 0, that f(x) exists in a neighborhood of c, and that f(c) exists. Then: • f has a relative maximum value at c if f”(c) < 0. •f has a relative minimum value at c if f”(c) > 0. •If f(c) = 0, the test is not conclusive. Note: Second derivative test is still used to calculate max and min
- 7. Practice Time again !!! Use second derivative test to find extrema of f (x) = 3x 5 − 5x 3 f '(x) = 15x − 15x 4 2 15x − 15x = 0 4 2 critical points = 0, -1, 1 f "(x) = 60x 3 − 30x inconclusive f "(0) = 0 f "(− 1) = − 30 < 0 f has a maximum at x = -1 f has a minimum at x = 1 f "(1) = 30 > 0 15x 2 ( x 2 − 1) = 0
- 8. x2 Find extrema of f (x) = x 4 +16 A.Using first derivative test B.Using second derivative test