## 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.

If*x* =*f*(*t*),and*y* = 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* = 2*t*^{2} and *y* = 2*t* − *t*^{3}; 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 (*x*_{1}*,y*_{1}) is

or5*x* + 4*y* = 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* = 40*t* and *y* = 40*t* − *gt*^{2}, 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 4*t*. 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* cos^{3}*θ*, *y* = *a* sin^{3} *θ*.

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

*x* = (*v*_{0} cos *θ*)*t*,

*y* = (*v*_{0} sin *θ*)*t* − *gt*^{2}.