MS2 Max and Min Points
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The document discusses how to determine if a function is increasing or decreasing on an interval using the derivative. It states that if the derivative is positive on an interval, the function is increasing on that interval, and if the derivative is negative, the function is decreasing. It provides steps to determine where a function is increasing or decreasing: 1) take the derivative, 2) find critical points where the derivative is 0 or undefined, 3) plot critical points to get intervals, 4) check sign of derivative in intervals. An example problem demonstrates finding the intervals where a cubic function is increasing or decreasing.
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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.
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Linked in: https://www.linkedin.com/in/tajimiitju
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Similar to MS2 Max and Min Points
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.
Applying the derivative
The document discusses using the derivative to determine whether a function is increasing or decreasing over an interval. It provides examples of using the sign of the derivative to determine if a function is increasing or decreasing. It also discusses using the second derivative test to determine if a stationary point is a relative maximum or minimum. Specifically:
- The sign of the derivative indicates whether the function is increasing or decreasing over an interval. Positive derivative means increasing, negative means decreasing.
- Stationary points where the derivative is zero require the second derivative test to determine if it is a relative maximum or minimum. Positive second derivative means a relative minimum, negative means a maximum.
- Examples demonstrate finding stationary points, using the first and second derivative
Ch04
The document discusses analyzing functions using calculus concepts like derivatives. It introduces analyzing functions to determine if they are increasing, decreasing, or constant on intervals based on the sign of the derivative. The sign of the derivative indicates whether the graph of the function has positive, negative, or zero slope at points, relating to whether the function is increasing, decreasing, or constant. It also introduces the concept of concavity, where the derivative indicates whether the curvature of the graph is upward (concave up) or downward (concave down) based on whether tangent lines have increasing or decreasing slopes. Examples are provided to demonstrate these concepts.
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 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.
Extreme Value
1) A continuous function on a closed interval will have both a maximum and minimum value on that interval.
2) Critical points of a function occur where the derivative is equal to 0 or undefined, and can indicate local maxima or minima.
3) The second derivative determines concavity and points where it changes sign indicate inflection points.
maxima & Minima thoeyr&solved.Module-4pdf
In this chapter we shall study those points of the domain of a function where its graph changes its direction from upwards to downwards or from downwards to upwards. At such points the derivative of the function, if it exists, is necessarily zero.
The value of a function f (x) is said to be maximum at x = a, if there exists a very small positive number h, such that
f(x) < f(a) x (a – h,a + h) , x a
In this case the point x = a is called a point of maxima for the function f(x).
Simlarly, the value of f(x) is said to the minimum at x = b, If there exists a very small positive number, h, such that
f(x) > f(b), x (b – h,b + h), x b
In this case x = b is called the point of minima for the function f(x).
Hene we find that,
(i) x = a is a maximum point of f(x)
f(a) f(a h) 0
f(a) f(a h) 0
(ii) x = b is a minimum point of f(x)
Rf(b) f(b h) 0
f(b) f(b h) 0
(iii) x = c is neither a maximum point nor a minimum point
Note :
(i) The maximum and minimum points are also known as extreme points.
(ii) A function may have more than one maximum and minimum points.
(iii) A maximum value of a function f(x) in an interval [a,b] is not necessarily its greatest value in that interval. Similarly, a minimum value may not be the least value of the function. A minimum value may be greater than some maximum value for a function.
(iv) If a continuous function has only one maximum (minimum) point, then at this point function has its greatest (least) value.
(v) Monotonic functions do not have extreme points.
Ex. Function y = sin x, x (0, ) has a maximum point at x = /2 because the value of sin /2 is greatest in the given interval for sin x.
Clearly function y = sin x is increasing in the interval (0, /2) and decreasing in the interval ( /2, ) for that reason also it has maxima at x = /2. Similarly we can see from the graph of cos x which has a minimum point at x = .
Ex. f(x) = x2 , x (–1,1) has a minimum point at x = 0 because at x = 0, the value of x2 is 0, which is
less than the all the values of function at different points of the interval.
Clearly function y = x2 is decreasing in the interval
Rf(c) f(c h)
Sand
Tf(c) f(c h) 0
|V| have opposite signs.
(–1, 0) and increasing in the interval (0,1) So it has minima at x = 0.
Ex. f(x) = |x| has a minimum point at x = 0. It can be easily observed from its graph.
A. Necessary Condition : A point x = a is an extreme point of a function f(x) if f’(a) = 0, provided f’(a) exists. Thus if f’ (a) exists, then
x = a is an extreme point f’(a) = 0
or
f’ (a) 0 x = a is not an extreme point.
But its converse is not true i.e.
f’ (a) = 0 x = a is an extreme point.
For example if f(x) = x3 , then f’ (0) = 0 but x = 0 is not an extreme point.
B. Sufficient Condition :
(i) The value of the function f(x) at x = a is maximum, if f’ (a) = 0 and f” (a) < 0.
(ii) The value of
Application of derivative
Additional Applications of the Derivative related to engineering maths
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Lesson 5: Continuity
Continuous function have an important property that small changes in input do not produce large changes in output. The Intermediate Value Theorem shows that a continuous function takes all values between any two values. From this we know that your height and weight were once the same, and right now there are two points on opposite sides of the world with the same temperature!
differentiate free
This document discusses the concept of the derivative and differentiation. It begins by explaining how the slope of a curve changes at different points, unlike the constant slope of a line. It then defines the derivative of a function f at a point x0 as the limit of the difference quotient as h approaches 0. If this limit exists, then f is said to be differentiable at x0. The derivative f'(x) then represents the slope of the curve y=f(x) at each point x and is a measure of how steeply the curve is rising or falling at that point. Several examples are provided to illustrate how to compute derivatives using this limit definition.
.
The document discusses functions and their characteristics including domain, range, and inverse functions. It provides examples of evaluating, adding, multiplying, and dividing functions. It also covers compound functions, using graphs to determine domain and range, and recognizing functions using the vertical line test. Logarithms are also briefly introduced.
mc-ty-polynomial-2009-1.pdf
This document discusses polynomial functions. It defines a polynomial as a function involving only non-negative integer powers of x, and states that the degree of a polynomial is the highest power of x in its expression. It examines the graphs of polynomials up to degree 4 and how changing coefficients affects the graph. It also discusses turning points, roots, and multiplicity of roots of polynomials.
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.
Lemh105
This document provides an introduction to the concepts of continuity and differentiability in calculus. It begins by giving two informal examples of functions that are and aren't continuous at a point to build intuition. It then provides a formal definition of continuity as the limit of a function at a point equaling the function value at that point. Several examples are worked through to demonstrate checking continuity at points and for entire functions. The document introduces the concept of limits approaching infinity to discuss the continuity of functions like 1/x. Overall, it lays the groundwork for understanding continuity and differentiability through examples and definitions.
Introduction to Functions
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- For a rule to be a valid function, it must map each input to only one output. The domain is the set of valid inputs, and the range is the set of corresponding outputs.
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Calculus Sections 4.1 and 4.3
This document discusses finding extreme values and critical points of functions by examining their first and second derivatives. It provides examples of using the first derivative test to find local maximum and minimum values when the derivative is equal to zero or undefined. The second derivative test is introduced to determine concavity and relative maximum or minimum values based on whether the second derivative is positive, negative, or zero. Graphs, sign lines, and examples are provided to demonstrate these concepts.
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This document discusses exponents and exponential functions. It defines what exponents are, how they are read, and some of their key properties like the product rule for exponents. It also examines exponential functions where the base is raised to the power of x. The graphs of exponential functions where the base is greater than 1, between 0 and 1, and equal to 1 are explored. The graphs are always increasing, have a horizontal asymptote at y=0, and a y-intercept of (0,1). Exponential equations are also briefly covered.
Calc 3.4
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MS2 Max and Min Points
- 1. Section 5.1 Increasing and Decreasing Functions
- 2. Objectives Upon completion of this lesson, you should be able to: • find where functions are increasing or decreasing
- 3. Increasing/Decreasing Functions So far, we have only been able to determine if a function is increasing or decreasing by plotting points to graph the function. Now that we know how to find the derivative of a function, we will learn how the derivative can be used to determine the intervals where a function is increasing or decreasing.
- 4. Increasing/Decreasing Functions Remember, the derivative of a function represents the slope of the tangent line at a particular point on the graph. So, if the derivative is positive on an open interval (a, b), then the slope of the tangent line is positive, which means the function is increasing on the interval (a, b). So, if the derivative is negative on an open interval (a, b), then the slope of the tangent line is negative, which means the function is decreasing on the interval (a, b).
- 5. Increasing/Decreasing Functions A function f is increasing on (a, b) if f (x1) < f (x2) whenever x1 < x2. A function f is decreasing on (a, b) if f (x1) > f (x2) whenever x1 < x2. Increasing Decreasing Increasing
- 6. Increasing/Decreasing/Constant Functions If f ′( x ) > 0 for each value of x in an interval ( a, b ) , then f is increasing on ( a, b ). If f ′( x ) < 0 for each value of x in an interval ( a, b ) , then f is decreasing on ( a, b ). If f ′( x ) = 0 for each value of x in an interval ( a, b ) , then f is constant on ( a, b ).
- 7. Example In the given graph of the function f(x), determine the interval(s) where the function is increasing, decreasing, or constant.
- 8. Example Solution: Looking at the graph from left to right, we would have the following three intervals. The function is decreasing on the interval (-4, -2) The function is increasing on the interval (-2, 0) The function is decreasing on the interval (0, 2)
- 9. Critical Numbers In order to find the intervals where a function is increasing, decreasing, or constant without first graph the function, we must find what are called critical numbers. The critical numbers are those contained in the domain of f(x) and which make the first derivative equal to zero or undefined.
- 10. Critical Points of f A critical point of a function f is a point in the domain of f where f ′( x) = 0 or f ′( x) does not exist. (horizontal tangent lines, vertical tangent lines and sharp corners)
- 11. Increasing/Decreasing Functions Steps in determining where a function is increasing or decreasing: 1. Find the derivative of the given function. 2. Locate any critical numbers by seeing where the derivative is either zero or undefined. 3. Plot the critical numbers on a number line to determine the open intervals. 4. Select a test point in each interval and evaluate the derivative at this point. 5. Use the sign of the derivative in each interval to determine whether it is increasing or decreasing.
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