Math 123: Calculus on Parametric

Ryan Blair

CSU Long Beach

Tuesday April 26, 2016

Ryan Blair (CSULB) Math 123: Calculus on Parametric Curves Tuesday April 26, 2016 1 / 7 Outline

1 Parametric Curves

2 of parametric curves

Ryan Blair (CSULB) Math 123: Calculus on Parametric Curves Tuesday April 26, 2016 2 / 7 Example: Find the parametric for the unit in the . Example: Find the parametric equation for the portion of the circle of radius R in the 3rd quadrant. Give the terminal point and the initial point. Example: All graphs of functions in can be represented as a parametric .

Parametric Curves Parametric Curves

Curves in the plane that are not graphs of functions can often be represented by parametric curves. Definition A parametric curve in the xy-plane is given by x = f (t) and y = g(t) for t ∈ [a, b].

Ryan Blair (CSULB) Math 123: Calculus on Parametric Curves Tuesday April 26, 2016 3 / 7 Example: Find the parametric equation for the portion of the circle of radius R in the 3rd quadrant. Give the terminal point and the initial point. Example: All graphs of functions in can be represented as a parametric curve.

Parametric Curves Parametric Curves

Curves in the plane that are not graphs of functions can often be represented by parametric curves. Definition A parametric curve in the xy-plane is given by x = f (t) and y = g(t) for t ∈ [a, b].

Example: Find the parametric equation for the unit circle in the plane.

Ryan Blair (CSULB) Math 123: Calculus on Parametric Curves Tuesday April 26, 2016 3 / 7 Example: All graphs of functions in can be represented as a parametric curve.

Parametric Curves Parametric Curves

Curves in the plane that are not graphs of functions can often be represented by parametric curves. Definition A parametric curve in the xy-plane is given by x = f (t) and y = g(t) for t ∈ [a, b].

Example: Find the parametric equation for the unit circle in the plane. Example: Find the parametric equation for the portion of the circle of radius R in the 3rd quadrant. Give the terminal point and the initial point.

Ryan Blair (CSULB) Math 123: Calculus on Parametric Curves Tuesday April 26, 2016 3 / 7 Parametric Curves Parametric Curves

Curves in the plane that are not graphs of functions can often be represented by parametric curves. Definition A parametric curve in the xy-plane is given by x = f (t) and y = g(t) for t ∈ [a, b].

Example: Find the parametric equation for the unit circle in the plane. Example: Find the parametric equation for the portion of the circle of radius R in the 3rd quadrant. Give the terminal point and the initial point. Example: All graphs of functions in can be represented as a parametric curve.

Ryan Blair (CSULB) Math 123: Calculus on Parametric Curves Tuesday April 26, 2016 3 / 7 Parametric Curves Awesome Examples

Cycloid: (t − sin(t), 1 − cos(t)) An Epitrochiod: 11 11 (11cos(t) − 6cos( t), 11sin(t) − 6sin( t)) 6 6 Wolfram Breaker: 1 1 1 1 (sin(t) + sin(5t) + cos(2.3t), cos(t) + cos(5t) + sin(2.3t)) 2 4 2 4

Ryan Blair (CSULB) Math 123: Calculus on Parametric Curves Tuesday April 26, 2016 4 / 7 Example: Derive this formula from the chain rule. Example: Find the points on the cycloid with horizontal lines.

Derivatives of parametric curves Derivatives of Parametric Curves

If y is a differentiable of x and t and x is a differentiable function of t then

dy dy = dt dx dx dt dx when dt 6= 0.

Ryan Blair (CSULB) Math 123: Calculus on Parametric Curves Tuesday April 26, 2016 5 / 7 Example: Find the points on the cycloid with horizontal tangent lines.

Derivatives of parametric curves Derivatives of Parametric Curves

If y is a differentiable function of x and t and x is a differentiable function of t then

dy dy = dt dx dx dt dx when dt 6= 0.

Example: Derive this formula from the chain rule.

Ryan Blair (CSULB) Math 123: Calculus on Parametric Curves Tuesday April 26, 2016 5 / 7 Derivatives of parametric curves Derivatives of Parametric Curves

If y is a differentiable function of x and t and x is a differentiable function of t then

dy dy = dt dx dx dt dx when dt 6= 0.

Example: Derive this formula from the chain rule. Example: Find the points on the cycloid with horizontal tangent lines.

Ryan Blair (CSULB) Math 123: Calculus on Parametric Curves Tuesday April 26, 2016 5 / 7 Example: Find the under one arch of the cycloid.

Derivatives of parametric curves Area and Parametric curves

Theorem If the graph of y = F (x) on [a, b] is parameterized by x = f (t) and y = g(t) for t ∈ [α, β] then

Z b Z β A = ydx = g(t)f 0(t)dt a α

Ryan Blair (CSULB) Math 123: Calculus on Parametric Curves Tuesday April 26, 2016 6 / 7 Derivatives of parametric curves Area and Parametric curves

Theorem If the graph of y = F (x) on [a, b] is parameterized by x = f (t) and y = g(t) for t ∈ [α, β] then

Z b Z β A = ydx = g(t)f 0(t)dt a α Example: Find the area under one arch of the cycloid.

Ryan Blair (CSULB) Math 123: Calculus on Parametric Curves Tuesday April 26, 2016 6 / 7 Example: Derive this formula from the definition of and the Pythagorean theorem. Example: Find the length of one arch of the cycloid.

Derivatives of parametric curves Arc length for parameterized curves

Theorem Given a curve C = (f (t), g(t)) with t ∈ [α, β], then the length of C is s Z β df 2 dg 2 L = + dt α dt dt

Ryan Blair (CSULB) Math 123: Calculus on Parametric Curves Tuesday April 26, 2016 7 / 7 Example: Find the length of one arch of the cycloid.

Derivatives of parametric curves Arc length for parameterized curves

Theorem Given a curve C = (f (t), g(t)) with t ∈ [α, β], then the length of C is s Z β df 2 dg 2 L = + dt α dt dt Example: Derive this formula from the definition of integral and the Pythagorean theorem.

Ryan Blair (CSULB) Math 123: Calculus on Parametric Curves Tuesday April 26, 2016 7 / 7 Derivatives of parametric curves Arc length for parameterized curves

Theorem Given a curve C = (f (t), g(t)) with t ∈ [α, β], then the length of C is s Z β df 2 dg 2 L = + dt α dt dt Example: Derive this formula from the definition of integral and the Pythagorean theorem. Example: Find the length of one arch of the cycloid.

Ryan Blair (CSULB) Math 123: Calculus on Parametric Curves Tuesday April 26, 2016 7 / 7