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Wize AP Calculus (BC) Textbook > Parametric & Polar Curves

Calculus w/ Parametric Curves

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Tangents of Parametric Curves (Derivatives)

Consider a parametric curve with equations (x(t), y(t))\left(x\left(t\right),\ y\left(t\right)\right) for a≤t≤ba\le t\le b.

First Derivative

(a.k.a. slope of the tangent to the curve)
dydx=y′(t)x′(t)\frac{dy}{dx}=\frac{y'\left(t\right)}{x'\left(t\right)} if x′(t)≠0x'\left(t\right)\ne0
  • Find the derivatives of y and x in terms of t
  • Divide the two expressions

Second Derivative

d2ydx2=ddx(dydx)=ddt(dydx)x′(t)\frac{d^2y}{dx^2}=\frac{d}{dx}\left(\frac{dy}{dx}\right)=\frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{x'\left(t\right)}
  • Find the derivative of the first derivative dydx\frac{dy}{dx} in terms of t
  • Find the derivative of x in terms of t
  • Divide the two expressions

The parametric curve has
  • A Horizontal Tangent when y′(t)=0y'\left(t\right)=0 and x′(t)≠0x'\left(t\right)\ne0
  • A Vertical Tangent when x′(t)=0x'\left(t\right)=0 and y′(t)≠0y'\left(t\right)\ne0
*If both y′(t)=0 y'\left(t\right)=0\ and x′(t)=0x'\left(t\right)=0, use L'Hospital's rule

Practice Question

If x and y are described with parametric equations x=sin⁡t+cos⁡tx = \sin t + \cos t and y=tan⁡ty = \tan t, find   ⁣dy  ⁣dx\frac{\de{y}}{ \de{x}} at t=π/3 t = \pi/3.

Practice: Tangents

Given the parametric curves x=t2+1,  y=t3−3t,  t∈I ⁣R.x = t^2 + 1, \space \space y = t^3 - 3t,\space \space t \in \R. ,
a.) Find the equation of the tangent line to the curve at the point (x, y)=(5, 2)\left(x,\ y\right)=\left(5,\ 2\right)
b.) Identify all points where the tangent is horizontal
c.) Identify all points where the tangent is vertical

Practice Question

Given the cycloid x=2(θ−sin⁡θ)x=2\left(\theta-\sin\theta\right) and y=2(1−cos⁡θ)y=2\left(1-\cos\theta\right), find all points where the slope of the tangent is horizontal or vertical

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Applications of Parametric Curves

Consider the parametric curve defined by {x(t), y(t): t∈[a,b]}\left\{x\left(t\right),\ y\left(t\right):\ t\in\left[a,b\right]\right\} where x′(t)x'\left(t\right) and y′(t)y'\left(t\right) are continuous on [a,b]\left[a,b\right], and the curve is traversed exactly once on [a,b]\left[a,b\right].

Arc Length

L=∫ab[x′(t)]2 + [y′(t)]2  ⁣dtL={\displaystyle\int_a^b}\sqrt{ \left[x'(t)\right]^2\ + \ \left[y'(t)\right]^2}\de{t}


Area under the Curve

A=∫aby(t)x′(t)dtA= {\displaystyle \int_a^b} y\left(t\right)x'\left(t\right)dt


Surface Area

Obtained by revolving the curve about the x-axis (or al ine parallel to the x-axis):

S=∫ab2πy(t)[x′(t)]2+[y′(t)]2dtS= {\displaystyle \int_a^b} 2\pi y\left(t\right)\sqrt{\left[x'\left(t\right)\right]^2+\left[y'\left(t\right)\right]^2}dt

Practice Question

Write down the integral representing the length of the curve parameterized by x=t3+t+1,y=t2−1x = t^3+t+1, y = t^2 - 1, from the point (1, -1) to the point (3, 0).

Practice Question

Find the area under the cruve {x=t2, y=cos⁡t: t∈[0,π2]}\left\{x=t^2,\ y=\cos t:\ t\in\left[0,\frac{\pi}{2}\right]\right\}.

Practice: Surface Area

Find the area of the surface obtained by revolving the curve (x(t), y(t))=(2−t3, 3t)\left(x\left(t\right),\ y\left(t\right)\right)=\left(2-t^3,\ 3t\right) on t∈[0,2]t\in\left[0,2\right] about the x-axis.

Practice Question

Find the integral that represents the area of the surface of revolution obtained by rotating the curve defined by {x=6ln⁡t, y=3t+3t: t∈[1,e]}\left\{x=6\ln t,\ y=3t+\frac{3}{t}:\ t\in\left[1,e\right]\right\} about the x-axis.