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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 Explain the process of sketching a plane curve given by parametric equations. What is meant by the orientation of the curve?

To determine

The process of sketching a plane curve given by parametric equations and the meaning of orientation of the curve.

Explanation

To sketch a plane curve, plot points found by substituting different values of the parameter into the provided equations.

For every value of parameter, a unique point can be obtained.

The point (x,y) in the plane is determine by the plane curve x=f(t) and y=f(t). Now, put the values of t, to get the point (x,y).

By plotting the resulting points in order of increasing values of t, the curve is traced out in a specific direction. This is called the orientation of the curve.

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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