The Calculus Primer (2011)

Part VI. Further Applications of the Derivative

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 = 2t − t³; 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 = e^2t and y = e^t+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 = 40t − gt², 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)t − gt².