Derivatives - Intro

Definition of a derivative
Notation for derivatives
Importance of derivatives in calculus
Applications of derivatives in real life
Concept of instantaneous rate of change <slide 2>

Derivatives - Notation

Derivative as a limit
Notation: f’(x), y’, dy/dx, d/dx(f(x))
Differentiation rules: power rule, constant rule, sum rule, product rule, quotient rule <slide 3>

Derivatives - Rules

Power rule: finding derivatives of functions raised to a power
Constant rule: finding derivatives of constant values
Sum rule: finding derivatives of sums or differences of functions
Product rule: finding derivatives of products of functions
Quotient rule: finding derivatives of quotients of functions <slide 4>

Derivatives - Chain Rule

Applying the chain rule to find derivatives of composite functions
Basic concept of chain rule
Chain rule formula: (f(g(x)))’ = f’(g(x)) * g’(x)
Chain rule examples <slide 5>

Derivatives - Higher Order

Second order derivative: finding the derivative of the derivative
Notation: f’’(x), y’’, d²y/dx², d²/dx²(f(x))
Higher order derivatives
Leibniz’s notation: d²y/dx², d³y/dx³, … <slide 6>

Derivatives - Implicit Differentiation

Applying implicit differentiation to find derivatives of implicitly defined functions
Steps for implicit differentiation
Implicit differentiation examples
Solving for dy/dx in terms of x and y <slide 7>

Solving problems using related rates
Steps for solving related rates problems
Differentiating both sides of an equation
Substituting values and solving for the desired rate <slide 8>

Derivatives - Applications

Finding tangent lines and normal lines
Finding local extrema and points of inflection
Optimization problems
Finding the maximum or minimum value of a function <slide 9>

Derivatives - Application Examples

Example: finding the tangent line to a given curve at a specific point
Example: finding the maximum area of a rectangle given a fixed perimeter
Example: finding the rate of change of a volume of a sphere as its radius changes <slide 10>

Derivatives - Conclusion

Review of key concepts covered in this lecture
Importance of derivatives in calculus and real life applications
Practice problems and exercises to reinforce learning
Next topic: Integration

Derivatives - Differentiation Rules

Power rule: derivating functions like f(x) = x^n, where n is a constant
Constant rule: derivating a constant value, such as f(x) = 5
Sum rule: derivating the sum or difference of two functions, like f(x) = g(x) + h(x)
Product rule: derivating the product of two functions, such as f(x) = g(x) * h(x)
Quotient rule: derivating the quotient of two functions, like f(x) = g(x) / h(x) <slide 12>

Derivatives - Chain Rule

Applying the chain rule to find derivatives of composite functions
Basic concept of chain rule
Chain rule formula: (f(g(x)))’ = f’(g(x)) * g’(x)
Example: find the derivative of f(x) = sin(2x)
Example: find the derivative of f(x) = (2x^2 + 3x - 5)^3 <slide 13>

Derivatives - Higher Order

Second order derivative: finding the derivative of the derivative
Notation: f’’(x), y’’, d²y/dx², d²/dx²(f(x))
Higher order derivatives
Leibniz’s notation: d²y/dx², d³y/dx³, …
Example: find the second derivative of f(x) = 3x^4 - 2x^2 <slide 14>

Derivatives - Implicit Differentiation

Applying implicit differentiation to find derivatives of implicitly defined functions
Steps for implicit differentiation
Example: find dy/dx for the equation x^2 + y^2 = 1
Example: find dy/dx for the equation x^3 + y^3 = 6xy <slide 15>

Solving problems using related rates
Steps for solving related rates problems
Differentiating both sides of an equation
Example: a balloon is being inflated, and its radius is increasing at a rate of 2 cm/s. Find the rate at which the volume is increasing when the radius is 5 cm.
Example: a ladder is sliding down a wall, and its top is moving at a rate of 3 m/s. Find the rate at which the bottom of the ladder is moving away from the wall when the bottom is 4 m from the wall and the top is 5 m above the ground. <slide 16>

Derivatives - Applications

Finding tangent lines and normal lines
Finding local extrema and points of inflection
Optimization problems
Example: find the tangent line to the curve y = x^2 - 3x + 2 at the point (2, 1)
Example: find the critical points and determine if they are local maxima, local minima, or points of inflection for the function f(x) = x^3 - 3x^2 - 9x + 5 <slide 17>

Derivatives - Application Examples

Example: find the maximum area of a rectangle with a fixed perimeter of 20 cm
Example: find the rate of change of the volume of a sphere as its radius changes at a rate of 2 cm/s <slide 18>

Derivatives - Conclusion

Review of key concepts covered in this lecture
Importance of derivatives in calculus and real life applications
Practice problems and exercises to reinforce learning
Next topic: Integration <slide 19>

Derivatives - Practice Problems

Solve the following derivatives:
- f(x) = 3x^2 - 7x + 4
- g(x) = 2sin(3x)
- h(x) = (8x^3 + 5x^2 - 2x + 1) / (3x^2 - 2x + 5)
Find the second derivative of the following functions:
- f(x) = 4x^3 - 2x^2 + 7x - 1
- g(x) = sin^2(x) + cos^2(x)

Derivatives - Practice Problems

Use implicit differentiation to find dy/dx for the following equations:
- x^2 + y^2 = 25
- 2x^3 + 3xy + y^3 = 0
Solve the following related rates problems:
- A ladder is being pulled away from the wall at a rate of 2 m/s. The bottom of the ladder is initially 4 m from the wall and slides along the ground. Find the rate at which the top of the ladder is sliding down the wall when the bottom of the ladder is 3 m from the wall.
- A spherical balloon is being inflated with air at a rate of 10 cm³/min. Find the rate at which the radius is increasing when the radius is 5 cm. <slide 21>

Integrals - Intro

Definition of an integral
Notation for integrals
Importance of integrals in calculus
Applications of integrals in real life
Concept of area under a curve <slide 22>

Integrals - Notation

Indefinite integrals: finding the antiderivative of a function
Notation: ∫f(x) dx, F(x) + C
Definite integrals: finding the area under a curve
Notation: ∫a^b f(x) dx <slide 23>

Integrals - Antiderivatives

Finding the antiderivative of a function
Basic rules of integration
Integration formulas: power rule, exponential rule, trigonometric rule
Examples:
- ∫x^n dx = (x^(n+1)) / (n+1) + C
- ∫e^x dx = e^x + C
- ∫sin(x) dx = -cos(x) + C <slide 24>

Integrals - Definite Integrals

Finding the area under a curve using definite integrals
Properties of definite integrals
Definite integral formulas: constant multiple rule, sum rule, change of limits
Examples:
- ∫[a,b] kf(x) dx = k * ∫[a,b] f(x) dx
- ∫[a,b] (f(x) + g(x)) dx = ∫[a,b] f(x) dx + ∫[a,b] g(x) dx
- ∫[a,b] f(x) dx = -∫[b,a] f(x) dx <slide 25>

Integrals - Integration Techniques

Substitution method: using a change of variable to simplify the integrand
Integration by parts: breaking down the integrand into a product of two functions
Partial fractions: decomposing a rational function into simpler fractions
Examples:
- ∫(2x + 1) dx using substitution method
- ∫x^2 e^x dx using integration by parts
- ∫(x + 3) / (x^2 + 4x + 3) dx using partial fractions <slide 26>

Integrals - Applications

Finding the area between two curves
Finding the volume of a solid of revolution
Finding the average value of a function
Examples:
- Find the area between y = x^2 and y = x
- Find the volume of the solid generated by rotating the region bounded by y = x^2 and the x-axis around the x-axis
- Find the average value of f(x) = 2x^3 - x^2 + 5 on the interval [0, 3] <slide 27>

Integrals - Application Examples

Example: find the area under the curve y = sin(x) between x = 0 and x = π
Example: find the volume of the solid generated by rotating the region bounded by y = x^2 and the y-axis around the y-axis
Example: find the average value of f(x) = 3x^2 - 2x + 1 on the interval [-1, 1] <slide 28>

Integrals - Techniques and Formulas

Integration techniques: trigonometric substitutions, partial fractions, integration by parts
Integration formulas: power rule, exponential rule, trigonometric rule
Tabular integration: a shortcut for integration by parts
Examples:
- ∫sec^2(x) dx using trigonometric substitution
- ∫(x^2 - 3x + 2) dx using tabular integration <slide 29>

Integrals - Conclusion

Review of key concepts covered in this lecture
Importance of integrals in calculus and real life applications
Practice problems and exercises to reinforce learning
Next topic: Differential Equations <slide 30>

Integrals - Practice Problems

Solve the following integrals:
- ∫(3x^2 - 2x + 1) dx
- ∫e^x sin(x) dx
- ∫(x + 2) / (x^2 + 3x + 2) dx
Find the area between the curves y = x^2 and y = x^3
Find the volume of the solid generated by rotating the region bounded by y = x^2 and the x-axis around the y-axis
Find the average value of f(x) = cos(x) on the interval [0, π]