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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095

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BuyFindarrow_forward

Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095
Textbook Problem

Plane Curve State the definition of a plane curve given by parametric equations.

To determine

The definition of a plane curve given by parametric equation.

Explanation

If f and g are continuous functions of t on an interval I, then the equations x=f(t) and y=g(t) are the parametric equations and t is the parameter. The set of points (x,y) obtained as t varies over the interval I is the graph of the parametric equations.

Taken together, the parametric equations and the graph are a plane curve, denoted by C.

For example:

Consider the path followed by an object that is propelled into the air at an angle of 45.

For an initial velocity of 48 feet per second, the object travels the parabolic path given by:

y=x272+x

This equation conveys where the object has been, but doesn’t provide any information about the position with relation of time.

To determine this time, introduce a third variable t called a parameter.

By writing both x and y as functions of t, the parametric equations will be:

x=242t

And,

y=16t2+242t.

From these set of equations, position of particle at time t can be determined.

Let determine the curve at the points t=0 and t=1.

At t=0,

The value of x is;

x=242(0)=0

The value of y is;

y=16(0)2+242(0)=0

So, the point (x,y) at t=0 is (0,0)

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