9.1 Defining and Differentiating Parametric Equations

• What is the variable typically used as the parameter in parametric equations?

  t
• The parameter $t$ allows you to trace a curve by varying its value.


  True
• In parametric equations, the function $f(t)$ maps values of $t$ to the x-coordinates, while $g(t)$ maps values of $t$ to the y-coordinates
• Steps to graph parametric equations

  1️⃣ Choose a range of values for $t$

  2️⃣ Plug each value of $t$ into the equations to find the corresponding $x$ and $y$ coordinates

  3️⃣ Plot the points $(x, y)$ to trace out the curve
• What do you vary in parametric equations to trace the curve?

  Parameter t
• When graphing parametric equations, the parameter t is varied to trace the curve
• What is the key difference between graphing parametric and standard equations?

  Use of parameter t
• What are parametric equations used to define?

  Curve or path
• In parametric equations, $f(t)$ defines the y-coordinates of the curve.


  False
• To graph parametric equations, plug each value of t into the equations to find the corresponding coordinates
• What formula is used to find $\frac{dy}{dx}$ in parametric form when $x =$$f(t)$ and $y =$$g(t)$?


  $\frac{dy}{dx} =$$\frac{\frac{dy}{dt}}{\frac{dx}{dt}}$
• The second step to find \frac{dy}{dx}</latex> in parametric form is to find $\frac{dx}{dt}$ by differentiating $x =$$f(t)$ with respect to t
• Match the type of equation with its derivative formula:

  Parametric Equations ↔️ $\frac{dy}{dx} =$$\frac{\frac{dy}{dt}}{\frac{dx}{dt}}$

  Standard Functions ↔️ $\frac{dy}{dx} =$$F'(x)$
• Given x = t^{2}</latex> and $y =$$2t^{3}$, what is $\frac{dx}{dt}$?


  $2t$
• The second derivative $\frac{d^{2}y}{dx^{2}}$ in parametric form is found by differentiating \frac{dy}{dx}</latex> with respect to $x$ directly.


  False
• In parametric equations, $x =$$f(t)$ and y = g(t)</latex> define the x and y coordinates in terms of the parameter t
• What do parametric equations define in terms of the parameter t</latex>?

  x and y coordinates
• What is the first step to graph parametric equations?

  Choose a range for t
• Graphing parametric equations uses a parameter $t$ to define the curve, unlike standard equations.


  True
• What is the parameter used in parametric equations to define a curve?

  t
• To graph parametric equations, you directly relate the x and y coordinates.

  False
• Steps to graph a curve defined by parametric equations

  1️⃣ Choose a range of values for the parameter t.

  2️⃣ Plug each value of t into the equations to find the corresponding x and y coordinates.

  3️⃣ Plot the points (x, y) to trace out the curve as t is varied.
• The parameter t in parametric equations allows you to trace the curve by varying its value
• What does $g(t)$ define in parametric equations?


  y-coordinates
• What formula is used to find $\frac{dy}{dx}$ in parametric form?


  $\frac{dy}{dx} =$$\frac{\frac{dy}{dt}}{\frac{dx}{dt}}$
• To find $\frac{dy}{dx}$ in parametric form, the first step is to find $\frac{dy}{dt}$ by differentiating $y =$$g(t)$ with respect to t
• What is the main difference between finding $\frac{dy}{dx}$ in parametric equations and standard functions?


  Use of parameter $t$
• Given $x =$$t^{2}$ and $y =$$2t^{3}$, what is $\frac{dy}{dt}$?


  $6t^{2}$
• Given $x =$$t^{2}$ and $y =$$2t^{3}$, what is $\frac{dy}{dx}$?


  $3t$
• What is the formula for $\frac{d^{2}y}{dx^{2}}$ in parametric form?


  $\frac{d^{2}y}{dx^{2}} =$$\frac{\frac{d}{dt} \left( \frac{dy}{dx} \right)}{\frac{dx}{dt}}$
• When finding the second derivative in parametric form, the second step is to differentiate $\frac{dy}{dx}$ with respect to t
• Match the type of equation with its second derivative formula:

  Parametric Equations ↔️ $\frac{d^{2}y}{dx^{2}} =$$\frac{\frac{d}{dt} \left( \frac{dy}{dx} \right)}{\frac{dx}{dt}}$

  Standard Functions ↔️ $\frac{d^{2}y}{dx^{2}} =$$F''(x)$
• Given $x =$$t^{2}$ and $y =$$2t^{3}$, what is $\frac{dy}{dx}$?


  $3t$
• Given \frac{d}{dt} \left( \frac{dy}{dx} \right) = 3</latex>, what is $\frac{d^{2}y}{dx^{2}}$?


  $\frac{3}{2t}$
• Steps to find $\frac{d^{2}y}{dx^{2}}$ in parametric form:


  1️⃣ Calculate $\frac{dy}{dx} =$$\frac{\frac{dy}{dt}}{\frac{dx}{dt}}$

  2️⃣ Differentiate $\frac{dy}{dx}$ with respect to $t$

  3️⃣ Divide the result by $\frac{dx}{dt}$
• What is the formula for $\frac{dy}{dx}$ in parametric form?


  $\frac{dy}{dx} =$$\frac{\frac{dy}{dt}}{\frac{dx}{dt}}$
• Given $\frac{dy}{dx} =$$3t$, what is $\frac{d}{dt} \left( \frac{dy}{dx} \right)$?


  3