<slide 1>

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>

Derivatives - Related Rates

• 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

<slide 11>

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>

Derivatives - Related Rates

• 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)

<slide 20>

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]