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EG 10565 Name Calculus on Parametric Equations
Since parametric curves are often used to describe the motion of an object, the main concerns in calculus of parametric equations are: rates of change and distance traveled. In other words:
First, we need to define a new(ish) term: a smooth curve :
Now, we can talk about the parametric form of the derivative,
dy dx
In “regular” curves we’ve worked with, a horizontal tangent line occurs when the slope is
and a vertical tangent line occurs when the slope is.
In parametrics, then, a smooth curve, C has:
A horizontal tangent line at t if
A vertical tangent line at t if
Consider the plane curve C defined by the parametric equations x t ( ) 3 t^2 12 , y t ( ) 2 t.
a. Find an equation of the lines tangent to C when t = 1.
b. Find all the points on the curve C at which the tangent line is vertical.
[Physics-type substance] A projectile is fired from the ground at an angle θ , 0 < θ < π /2, to the horizontal with an initial speed of v 0 m/s. If we ignore air resistance, the position of the projectile after t seconds is given by the parametric equations:
a. Find the slope of the tangent line to the motion of the projectile as a function of t.
b. When does the projectile reach its maximum height? What is that maximum height?
If we wanted to know how far the projectile actually traveled along its (parabolic) path, how would we determine that?
Area Under Curves Defined Parametrically
Let’s start by recalling how we write an integral for the area “under” a curve in the Cartesian system.
Given a function y f ( ) x on an interval [ a , b ], the area under that curve can be found with:
Analogously, if we have a parametric curve, the area under that curve can be found with:
Find the area enclosed by the parametric curves x t^3 1 , y 2 t t^2 and the x -axis.
