﻿ ﻿PARAMETRIC EQUATIONS - Further Applications of the Derivative - The Calculus Primer

## The Calculus Primer (2011)

### Chapter 22. PARAMETRIC EQUATIONS

6—7.Derivatives of Equations in Parametric Form. The reader will recall that the relation of two variables may be stated in terms of their relation to a third variable, called the parameter. Equations in parametric form may be differentiated as follows.

Ifx =f(t),andy = ø(t),

To find , we proceed as shown:

The numerator of the third member of equation [2] may be expanded; thus

Equation [3] may also be written in the form:

With these expressions for and , the slopes, maxima and minima, points of inflection, equations of tangents and normals, lengths of sub-tangents and subnormals, etc., for equations in parametric form are readily obtained.

EXAMPLE 1.Find and for the curve whose parametric equations are x = 2t2 and y = 2tt3; find the equation of the tangent to the curve at the point for which t = 2.

Solution.

When t = 2, x = 8, y = −4.

Equation of tangent at point (x1,y1) is

or5x + 4y = 24.

EXAMPLE 2.Given the parametric equations x = e2t and y = et+1. Find (a) the values of and ; (b) the length of the subtangent and the subnormal at the point for which t = 0.

EXAMPLE 3.Given the parametric equations of the cycloid

x = a(θ − sin θ), y = a(l − cos θ).

Find the lengths of the subtangent and the subnormal to the point on the curve for which find also the value of at this point.

Solution.

Further:

EXERCISE 6—5

1. Find (a) the equation of the tangent to the curve whose parametric equations are x = 40t and y = 40tgt2, at the point on the curve given by t = 1; (b) find the rate at which the slope is changing at any point on the curve.

2. Given the parametric equations x = 4 + t and y = log 4t. Find (a) the equation of the tangent at the point on the curve given by t = (b) find the value of when t = 3.

3. In the parametric equations of the circle x = r cos ø and y = r sin ø, prove analytically (a) that the equation of the tangent at the point on the circle given by is y = r; (b) that the length of the subnormal to the point given by is equal to

4. Find for the ellipse: x = a cos θ, y = b cos θ.

5. Find for the hypocycloid of four cusps: x = a cos3θ, y = a sin3 θ.

6. Find the acceleration of the projectile whose path is given by:

x = (v0 cos θ)t,

y = (v0 sin θ)tgt2.

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