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Parametric Curves & Calculus

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Parametric Curves & Calculus - Complete Interactive Lesson

Part 1: Parametric Derivatives

Parametric Curves & Calculus

Part 1 of 7 — Parametric Equations

What Are Parametric Equations?

Instead of y=f(x)y = f(x), we express both coordinates in terms of a parameter tt:

x=f(t),quady=g(t)x = f(t), quad y = g(t)

First Derivative

rac{dy}{dx} = rac{dy/dt}{dx/dt} = rac{g'(t)}{f'(t)}

Worked Example

x=t2x = t^2, y=t3y = t^3

rac{dx}{dt} = 2t, rac{dy}{dt} = 3t^2

rac{dy}{dx} = rac{3t^2}{2t} = rac{3t}{2}

At t=2t = 2: slope =3= 3.

Parametric Derivatives 🎯

Key Takeaways — Part 1

  1. dydx=dy/dtdx/dt\frac{dy}{dx} = \frac{dy/dt}{dx/dt}
  2. Horizontal tangent when dydt=0\frac{dy}{dt} = 0 (and dxdt≠0\frac{dx}{dt} \neq 0)
  3. Vertical tangent when dxdt=0\frac{dx}{dt} = 0

Part 2: Second Derivatives

Parametric Curves

Part 2 of 7 — Second Derivative

Second Derivative for Parametric Curves

ight)}{ rac{dx}{dt}}$$ Take the derivative **with respect to** $t$ of $ rac{dy}{dx}$, then divide by $ rac{dx}{dt}$. ### Example $x = t^2$, $y = t^3$. We found $ rac{dy}{dx} = rac{3t}{2}$. $ rac{d}{dt}left( rac{3t}{2} ight) = rac{3}{2}$ $$ rac{d^2y}{dx^2} = rac{3/2}{2t} = rac{3}{4t}$$

Second Derivative 🎯

Key Takeaways — Part 2

The second derivative formula: differentiate dy/dxdy/dx with respect to tt, divide by dx/dtdx/dt.

Part 3: Arc Length

Parametric Curves

Part 3 of 7 — Arc Length (Parametric)

Parametric Arc Length Formula

ight)^2 + left( rac{dy}{dt} ight)^2},dt$$ ### Example: Circle $x = 3cos t$, $y = 3sin t$, $0 leq t leq 2pi$ $ rac{dx}{dt} = -3sin t$, $ rac{dy}{dt} = 3cos t$ $L = int_0^{2pi}sqrt{9sin^2 t + 9cos^2 t},dt = int_0^{2pi} 3,dt = 6pi$

Arc Length 🎯

Key Takeaways — Part 3

Arc length: (dx/dt)2+(dy/dt)2 dt\sqrt{(dx/dt)^2 + (dy/dt)^2}\,dt — Pythagorean theorem in the parameter space.

Part 4: Speed

Parametric Curves

Part 4 of 7 — Area Under Parametric Curves

Area Formula

A = int_a^b y(t), rac{dx}{dt},dt

or equivalently A=intabg(t)cdotf′(t),dtA = int_a^b g(t) cdot f'(t),dt

Speed of a Parametric Curve

ight)^2 + left( rac{dy}{dt} ight)^2}$$ ### Distance Traveled $$ ext{Distance} = int_a^b sqrt{left( rac{dx}{dt} ight)^2 + left( rac{dy}{dt} ight)^2},dt$$

Area & Speed 🎯

Key Takeaways — Part 4

Speed is the magnitude of the velocity vector.

Part 5: Area Under Parametric Curves

Parametric Curves

Part 5 of 7 — Eliminating the Parameter

Converting to Rectangular

Sometimes useful for understanding the shape:

ParametricRectangular
x=t,y=t2x = t, y = t^2y=x2y = x^2
x=cos⁡t,y=sin⁡tx = \cos t, y = \sin tx2+y2=1x^2 + y^2 = 1
x=2cos⁡t,y=3sin⁡tx = 2\cos t, y = 3\sin tx24+y29=1\frac{x^2}{4} + \frac{y^2}{9} = 1

Eliminating Parameter 🎯

Key Takeaways — Part 5

Eliminate tt by solving one equation for tt and substituting.

Part 6: Problem-Solving Workshop

Parametric Curves

Part 6 of 7 — Practice Workshop

Mixed Practice 🎯

Workshop Complete!

Part 7: Review & Applications

Parametric Curves — Review

Part 7 of 7 — Final Assessment

Final Assessment 🎯

Parametric Curves — Complete! ✅

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