Derivatives - Examples

• Calculating the derivative of a function at a given point
• Finding the first derivative of a function
• Identifying the critical points of a function
• Determining the relative extrema of a function
• Understanding the role of derivatives in optimization problems

Example 1: Finding the derivative at a point

Given the function f(x) = 3x^2 - 4x + 2, find f’(1).

• Apply the power rule: d/dx(x^n) = nx^(n-1)
• Taking the derivative term by term, we get f’(x) = 6x - 4
• Substitute x = 1 into f’(x) to find f’(1) = 2

Example 2: Finding the first derivative

Find f’(x) for the function f(x) = 4x^3 - 6x^2 + 5x - 3.

• Apply the power rule to each term: d/dx(x^n) = nx^(n-1)
• Taking the derivative term by term, we get f’(x) = 12x^2 - 12x + 5

Example 3: Identifying critical points

Given the function f(x) = x^3 - 6x^2 + 9x - 2, find the critical points.

• Critical points occur when the first derivative is equal to zero or undefined
• Find the first derivative: f’(x) = 3x^2 - 12x + 9
• Set f’(x) = 0 and solve for x to find the critical points

Example 4: Finding relative extrema

Given the function f(x) = x^4 - 8x^2 + 5, determine the relative extrema.

• Locate the critical points by setting the first derivative equal to zero: f’(x) = 4x^3 - 16x = 0
• Solve for x to obtain the critical points
• Test the values of f’’(x) to check for relative extrema

Example 5: Optimization problem

A rectangular field is to be enclosed by a fence on three sides, with a river acting as the fourth side. Find the dimensions of the field that maximize its area.

• Define variables and parameters
• Set up an equation representing the area of the field
• Calculate the first and second derivatives to find critical points
• Determine the dimensions that maximize the area

Note: Please continue creating slides 7 to 10.

Partial Derivatives

• Definition and concept of partial derivatives
• Calculating partial derivatives with respect to one variable while holding others constant
• Notation: ∂z/∂x or fz, x
• Example: Find the partial derivative of z = x^2 + y^2 with respect to x

Chain Rule

• Understanding the concept of the chain rule
• Applying the chain rule to find derivatives of composite functions
• Notation: (f ∘ g)’(x) = f’(g(x)) * g’(x)
• Example: Find the derivative of f(x) = sin(2x)

Implicit Differentiation

• Definition and concept of implicit differentiation
• Applying implicit differentiation to find derivatives of implicitly defined functions
• Notation: d/dx
• Example: Find the derivative of x^2 + y^2 = 25 with respect to x

Higher Order Derivatives

• Understanding higher order derivatives
• Definition of second order derivative and higher order derivatives
• Notation: f’’(x), f’’’(x), f^(4)(x)
• Example: Find the second derivative of f(x) = 3x^3 + 2x^2

Related Rates

• Understanding related rates problems
• Identifying variables and rates of change
• Applying derivative rules to find the desired rate of change
• Example: A ladder is sliding down a wall at a rate of 2 m/s. Find the rate at which the base of the ladder is moving away from the wall when the top of the ladder is 5 m above the ground

Optimization

• Concept of optimization problems
• Formulating optimization problems using mathematical functions
• Using derivatives to find critical points and determine maximum or minimum values
• Example: Find the dimensions of a rectangular field with fixed perimeter that maximize its area

Concavity and Points of Inflection

• Understanding concavity and points of inflection
• Determining concavity using the second derivative test
• Identifying points of inflection
• Example: Find the points of inflection for f(x) = x^3 - 3x^2 + 2x

Mean Value Theorem

• Understanding the Mean Value Theorem
• Statement and conditions of the Mean Value Theorem
• Applying the Mean Value Theorem to find points where the derivative equals the average rate of change
• Example: Find a point c in the interval [1, 4] that satisfies f’(c) = (f(4) - f(1))/(4-1)

L’Hôpital’s Rule

• Understanding L’Hôpital’s Rule
• Applying L’Hôpital’s Rule to evaluate limits involving indeterminate forms
• Conditions for applying L’Hôpital’s Rule
• Example: Evaluate the limit of (x - sin(x))/x^3 as x approaches 0

Newton’s Method

• Concept of Newton’s method for finding approximate roots of equations
• Iterative process of Newton’s method
• Calculating new approximations using the formula: x(n+1) = x(n) - f(x(n))/f’(x(n))
• Example: Use Newton’s method to find an approximation for the root of f(x) = x^3 - 2x^2 - 4x + 8

Exponential and Logarithmic Functions

• Definition and concept of exponential and logarithmic functions
• Properties of exponential and logarithmic functions
• Examples: f(x) = e^x, g(x) = log(x)

Derivatives of Exponential Functions

• Applying the derivative rules to exponential functions
• Finding the derivatives of functions with base e
• Example: Find the derivative of f(x) = 2^x

Derivatives of Logarithmic Functions

• Applying the derivative rules to logarithmic functions
• Finding the derivatives of functions with base e
• Example: Find the derivative of f(x) = log(x)

Derivatives of Trigonometric Functions

• Applying the derivative rules to trigonometric functions
• Finding the derivatives of basic trigonometric functions
• Example: Find the derivative of f(x) = sin(x)

Derivatives of Inverse Trigonometric Functions

• Applying the derivative rules to inverse trigonometric functions
• Finding the derivatives of basic inverse trigonometric functions
• Example: Find the derivative of f(x) = arcsin(x)

Derivatives of Hyperbolic Functions

• Definition and concept of hyperbolic functions
• Applying the derivative rules to hyperbolic functions
• Example: Find the derivative of f(x) = sinh(x)

Indefinite Integration

• Understanding indefinite integration
• Rules of integration
• Example: Find the antiderivative of f(x) = 3x^2

Definite Integration

• Understanding definite integration
• Calculation of the definite integral using the antiderivative
• Example: Evaluate the definite integral of f(x) = 2x^2 from x=1 to x=3

Integration by Substitution

• Applying the method of substitution to simplify integrals
• Recognizing when to use substitution
• Example: Evaluate the integral ∫(5x + 2)^3 dx

Integration by Parts

• Applying the method of integration by parts
• Recognizing when to use integration by parts
• Example: Evaluate the integral ∫x ln(x) dx