Derivatives - Increasing & Decreasing Functions

Slide 1: Derivatives - Increasing & Decreasing Functions

Introduction to derivatives
Definition of increasing and decreasing functions
Importance of determining increasing and decreasing intervals Slide 2: Increasing Functions
Definition of increasing function: when the derivative is positive
Example: f(x) = x^2, f’(x) = 2x, is an increasing function as f’(x) > 0 for all x
Graphical representation of an increasing function Slide 3: Decreasing Functions
Definition of decreasing function: when the derivative is negative
Example: g(x) = -2x, g’(x) = -2, is a decreasing function as g’(x) < 0 for all x
Graphical representation of a decreasing function Slide 4: Critical Points
Definition of critical points: where the derivative is equal to zero or undefined
Examples of critical points and their significance in determining increasing and decreasing intervals
Critical points as local maximums or minimums Slide 5: First Derivative Test
Definition of the first derivative test
Steps for applying the first derivative test
Example: f(x) = x^3 - 3x^2 + 2x, find the increasing and decreasing intervals Slide 6: Second Derivative Test
Definition of the second derivative test
Steps for applying the second derivative test
Example: f(x) = x^4 - 5x^2, determine the increasing and decreasing intervals using the second derivative test Slide 7: Determining the Sign of the Derivative
Sign chart: a helpful tool for determining the sign of the derivative
Example: f(x) = x^3 + 2x^2 - x - 2, determine the sign of f’(x) using a sign chart Slide 8: Application Problems 1
Real-life examples of increasing and decreasing functions
Example problem: A company’s profit function is given by P(x) = 150x - x^2 - 200. Determine the increasing and decreasing intervals of P(x) and interpret the results. Slide 9: Application Problems 2
Example problem: The rate of change of the volume V(t) of a spherical balloon is given by dV/dt = 4t - 6t^2. Determine the increasing and decreasing intervals of V(t) and interpret the results. Slide 10: Summary
Recap of the main concepts covered in the lecture
Importance of understanding increasing and decreasing functions in various applications
Encouragement for further practice and exploration in the topic Slide 11: Concavity and Points of Inflection
Definition of concave up and concave down functions
Concave up functions: second derivative is positive
Concave down functions: second derivative is negative
Points of inflection: where the concavity changes
Example: f(x) = x^3, f’’(x) = 6x, determine the concavity and points of inflection Slide 12: Tests for Concavity
Definition of the second derivative test for concavity
Steps for applying the second derivative test for concavity
Example: f(x) = x^4 - 4x^3 + 3x^2, determine the intervals of concavity and points of inflection Slide 13: Analysis of Equations
Analysis of equations involving derivatives
Example: f(x) = x^3 - 3x^2 + 2x, find the critical points, intervals of increase and decrease, and intervals of concavity Slide 14: Maxima and Minima
Definition of maximum and minimum points
Local maximum: where the function reaches a high point within a specific interval
Local minimum: where the function reaches a low point within a specific interval
Global maximum and minimum: where the function reaches the highest and lowest points over its entire domain
Example: f(x) = x^3 - 3x^2 + 2x, find the local and global maximum and minimum points Slide 15: First Derivative Test for Maxima and Minima
Application of the first derivative test for identifying maxima and minima
Steps for applying the first derivative test for maxima and minima
Example: f(x) = x^3 - 3x^2 + 2x, determine the local maxima and minima using the first derivative test Slide 16: Second Derivative Test for Maxima and Minima
Application of the second derivative test for confirming maxima and minima
Steps for applying the second derivative test for maxima and minima
Example: f(x) = x^4 - 4x^3 + 3x^2, determine the local maxima and minima using the second derivative test Slide 17: Points of Inflection
Definition of points of inflection
Examples of points of inflection and their graphical representation
Nature of points of inflection when the second derivative changes sign
Example: f(x) = x^3, determine the points of inflection Slide 18: Optimization Problems
Introduction to optimization problems
Definition of optimization: finding the maximum or minimum value of a function within a given domain
Steps for solving optimization problems
Example problem: A farmer wants to construct a rectangular pen with 200 ft of fencing, what should be the dimensions of the pen to maximize the enclosed area? Slide 19: Optimization Problems - Continued
Example problem continued: A farmer wants to construct a rectangular pen with 200 ft of fencing, what should be the dimensions of the pen to maximize the enclosed area?
Application of derivatives to find the critical points and determine the maximum area Slide 20: Summary
Recap of the main concepts covered in the lecture
Importance of understanding concavity, maxima, minima, and optimization problems
Encouragement for further practice and exploration in the topic Slide 21: Applications of Derivatives
Introduction to the applications of derivatives
Importance of derivatives in various real-life situations
Examples of applications of derivatives:
- Finding maximum and minimum values
- Modeling and optimization problems
- Calculating rates of change
- Determining concavity and inflection points Slide 22: Maximum and Minimum Problems
Definition of maximum and minimum problems
Steps for solving maximum and minimum problems using derivatives
Example problem: A rectangular box with an open top has a volume of 64 cubic units. Find the dimensions of the box that minimize the amount of material used for the sides. Slide 23: Related Rates Problems
Introduction to related rates problems
Definition of related rates: when two or more variables are changing with respect to time and their rates of change are related
Steps for solving related rates problems
Example problem: A balloon is being inflated at a rate of 3 cubic feet per minute. How fast is the radius of the balloon increasing when the radius is 4 feet? Slide 24: Tangent Lines and Linear Approximations
Definition of tangent lines and linear approximations
Importance of tangent lines in understanding the behavior of functions
Steps for finding tangent lines and using linear approximations
Example problem: Find the equation of the tangent line to the curve y = x^2 at the point (1, 1). Slide 25: The Chain Rule
Definition of the chain rule
Importance and application of the chain rule in finding derivatives of composite functions
Examples of using the chain rule:
- Derivative of sin(x^2)
- Derivative of e^(3x) Slide 26: Implicit Differentiation
Definition of implicit differentiation
Importance and application of implicit differentiation in finding derivatives of implicitly defined functions
Steps for implicit differentiation
Example: Find dy/dx for the equation x^2 + y^2 = 25 Slide 27: Derivatives of Exponential and Logarithmic Functions
Derivatives of exponential functions: f(x) = a^x, where a is a constant
Derivative of the natural exponential function: f(x) = e^x
Derivatives of logarithmic functions: f(x) = log_a(x), where a is a constant
Example: Find the derivative of f(x) = 3^x + ln(x) Slide 28: Derivatives of Trigonometric Functions
Derivatives of sine and cosine functions: f(x) = sin(x), f(x) = cos(x)
Derivatives of tangent, cotangent, secant, and cosecant functions
Example: Find the derivative of f(x) = 2sin(3x) - cos(2x) Slide 29: Higher Order Derivatives
Introduction to higher order derivatives
Definition of second order derivatives and beyond
Application of higher order derivatives in analyzing the behavior of functions
Examples of finding higher order derivatives Slide 30: Summary
Recap of the main concepts covered in the lecture
Importance of derivatives in various applications and problem-solving
Encouragement for further practice and exploration in the topic
End of lecture