Slide 1: Derivatives - Derivative of Function in Parametric Forms
Introduction to derivatives in parametric forms
Definition of a parametric equation
Notation for a parametric equation
Relationship between a parametric equation and its corresponding set of parametric equations
Example: Parametric equation of a circle
Slide 2: Derivatives - Special Cases of Derivatives in Parametric Forms
Derivative of a constant function in parametric form
Derivative of a linear function in parametric form
Derivative of a quadratic function in parametric form
Derivative of a cubic function in parametric form
Example: Finding the derivative of a parametric equation
Slide 3: Derivatives - Finding Derivatives of Parametric Equations
Parametric differentiation rule
Derivative of a parametric equation with respect to the parameter
Finding dy/dx using the parametric differentiation rule
Example: Finding dy/dx for a given parametric equation
Slide 4: Derivatives - Implicit Differentiation of Parametric Equations
Introduction to implicit differentiation
Applying implicit differentiation to parametric equations
Finding dy/dx using implicit differentiation
Example: Finding dy/dx for a parametric equation using implicit differentiation
Slide 5: Derivatives - Second Derivative of Parametric Equations
Definition of the second derivative
Finding the second derivative of a parametric equation
Relation between the second derivatives of x and y with respect to the parameter
Example: Calculating the second derivative of a given parametric equation
Slide 6: Derivatives - Higher Order Derivatives of Parametric Equations
Definition of higher order derivatives
Finding higher order derivatives of parametric equations
Notation for higher order derivatives
Example: Calculating the third derivative of a parametric equation
Slide 7: Derivatives - Velocity and Acceleration in Parametric Equations
Introduction to velocity and acceleration in parametric equations
Finding velocity using derivatives of parametric equations
Expressing acceleration in terms of the first and second derivatives of parametric equations
Example: Finding velocity and acceleration for a given parametric equation
Slide 8: Derivatives - Tangents and Normals to a Parametric Curve
Definition of tangents and normals to a curve
Finding the equation of the tangent to a parametric curve
Finding the equation of the normal to a parametric curve
Example: Finding the equation of tangent and normal to a given parametric curve
Slide 9: Derivatives - Length of a Parametric Curve
Introduction to the length of a parametric curve
Deriving the formula for calculating the length of a parametric curve
Example: Finding the length of a given parametric curve
Slide 10: Derivatives - Shortest Distance between Two Parametric Curves
Introduction to finding the shortest distance between two parametric curves
Deriving the formula for calculating the shortest distance
Example: Finding the shortest distance between two given parametric curves
Slide 11: Derivatives - Derivative of Function in Parametric Forms
Introduction to derivatives in parametric forms
Definition of a parametric equation
Notation for a parametric equation
Relationship between a parametric equation and its corresponding set of parametric equations
Example: Parametric equation of a circle
Slide 12: Derivatives - Special Cases of Derivatives in Parametric Forms
Derivative of a constant function in parametric form
Derivative of a linear function in parametric form
Derivative of a quadratic function in parametric form
Derivative of a cubic function in parametric form
Example: Finding the derivative of a parametric equation
Slide 13: Derivatives - Finding Derivatives of Parametric Equations
Parametric differentiation rule
Derivative of a parametric equation with respect to the parameter
Finding dy/dx using the parametric differentiation rule
Example: Finding dy/dx for a given parametric equation
Slide 14: Derivatives - Implicit Differentiation of Parametric Equations
Introduction to implicit differentiation
Applying implicit differentiation to parametric equations
Finding dy/dx using implicit differentiation
Example: Finding dy/dx for a parametric equation using implicit differentiation
Slide 15: Derivatives - Second Derivative of Parametric Equations
Definition of the second derivative
Finding the second derivative of a parametric equation
Relation between the second derivatives of x and y with respect to the parameter
Example: Calculating the second derivative of a given parametric equation
Slide 16: Derivatives - Higher Order Derivatives of Parametric Equations
Definition of higher order derivatives
Finding higher order derivatives of parametric equations
Notation for higher order derivatives
Example: Calculating the third derivative of a parametric equation
Slide 17: Derivatives - Velocity and Acceleration in Parametric Equations
Introduction to velocity and acceleration in parametric equations
Finding velocity using derivatives of parametric equations
Expressing acceleration in terms of the first and second derivatives of parametric equations
Example: Finding velocity and acceleration for a given parametric equation
Slide 18: Derivatives - Tangents and Normals to a Parametric Curve
Definition of tangents and normals to a curve
Finding the equation of the tangent to a parametric curve
Finding the equation of the normal to a parametric curve
Example: Finding the equation of tangent and normal to a given parametric curve
Slide 19: Derivatives - Length of a Parametric Curve
Introduction to the length of a parametric curve
Deriving the formula for calculating the length of a parametric curve
Example: Finding the length of a given parametric curve
Slide 20: Derivatives - Shortest Distance between Two Parametric Curves
Introduction to finding the shortest distance between two parametric curves
Deriving the formula for calculating the shortest distance
Example: Finding the shortest distance between two given parametric curves
Slide 21: Derivatives - Derivative of Function in Parametric Forms
Introduction to derivatives in parametric forms
Definition of a parametric equation
Notation for a parametric equation
Relationship between a parametric equation and its corresponding set of parametric equations
Example: Parametric equation of a circle
Slide 22: Derivatives - Special Cases of Derivatives in Parametric Forms
Derivative of a constant function in parametric form
Derivative of a linear function in parametric form
Derivative of a quadratic function in parametric form
Derivative of a cubic function in parametric form
Example: Finding the derivative of a parametric equation
Slide 23: Derivatives - Finding Derivatives of Parametric Equations
Parametric differentiation rule
Derivative of a parametric equation with respect to the parameter
Finding dy/dx using the parametric differentiation rule
Example: Finding dy/dx for a given parametric equation
Slide 24: Derivatives - Implicit Differentiation of Parametric Equations
Introduction to implicit differentiation
Applying implicit differentiation to parametric equations
Finding dy/dx using implicit differentiation
Example: Finding dy/dx for a parametric equation using implicit differentiation
Slide 25: Derivatives - Second Derivative of Parametric Equations
Definition of the second derivative
Finding the second derivative of a parametric equation
Relation between the second derivatives of x and y with respect to the parameter
Example: Calculating the second derivative of a given parametric equation
Slide 26: Derivatives - Higher Order Derivatives of Parametric Equations
Definition of higher order derivatives
Finding higher order derivatives of parametric equations
Notation for higher order derivatives
Example: Calculating the third derivative of a parametric equation
Slide 27: Derivatives - Velocity and Acceleration in Parametric Equations
Introduction to velocity and acceleration in parametric equations
Finding velocity using derivatives of parametric equations
Expressing acceleration in terms of the first and second derivatives of parametric equations
Example: Finding velocity and acceleration for a given parametric equation
Slide 28: Derivatives - Tangents and Normals to a Parametric Curve
Definition of tangents and normals to a curve
Finding the equation of the tangent to a parametric curve
Finding the equation of the normal to a parametric curve
Example: Finding the equation of tangent and normal to a given parametric curve
Slide 29: Derivatives - Length of a Parametric Curve
Introduction to the length of a parametric curve
Deriving the formula for calculating the length of a parametric curve
Example: Finding the length of a given parametric curve
Slide 30: Derivatives - Shortest Distance between Two Parametric Curves
Introduction to finding the shortest distance between two parametric curves
Deriving the formula for calculating the shortest distance
Example: Finding the shortest distance between two given parametric curves