Parametric Graphs Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

hard
Find the slope of the tangent line to the curve x=t2+1x = t^2 + 1, y=t3βˆ’3ty = t^3 - 3t at the point where t=2t = 2.

Solution

  1. 1
    The slope is dydx=dy/dtdx/dt\frac{dy}{dx} = \frac{dy/dt}{dx/dt}. Compute: dxdt=2t\frac{dx}{dt} = 2t and dydt=3t2βˆ’3\frac{dy}{dt} = 3t^2 - 3.
  2. 2
    At t=2t = 2: dydx=3(4)βˆ’32(2)=94\frac{dy}{dx} = \frac{3(4) - 3}{2(2)} = \frac{9}{4}. The point is (5,2)(5, 2) and the tangent slope is 94\frac{9}{4}.

Answer

dydx=94Β atΒ theΒ pointΒ (5,2)\frac{dy}{dx} = \frac{9}{4} \text{ at the point } (5, 2)
For parametric curves, the slope of the tangent line is found using the chain rule: dydx=dy/dtdx/dt\frac{dy}{dx} = \frac{dy/dt}{dx/dt}, provided dxdt≠0\frac{dx}{dt} \neq 0. This avoids needing to eliminate the parameter first, which may be difficult or impossible for complex parametric curves.

About Parametric Graphs

Plotting and analyzing curves defined by parametric equations x=f(t)x = f(t), y=g(t)y = g(t), including eliminating the parameter, determining direction of motion, and finding tangent lines.

Learn more about Parametric Graphs β†’

More Parametric Graphs Examples

Example 1 easy

Sketch the direction of motion for the parametric curve [formula], [formula] as [formula] increases

Example 2 medium

Describe the graph of [formula], [formula] for [formula], including shape, direction, and starting p

Example 3 medium

What is the difference between the graphs of (a) [formula], [formula] and (b) [formula], [formula]?

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