## The Calculus Primer (2011)

### Part VI. Further Applications of the Derivative

### Chapter 20. SLOPES, TANGENTS, AND NORMALS

**6—1.Slope of a Curve.** We have seen that for any curve, *y* = *f*(*x*)*,* the slope *m* of the tangent to the curve at any particular point (*x*_{1}*,y*_{1}) is given by *m* = tan ø = , where the symbol indicates the particular value which the variable expression takes when *x* = *x*_{1} and *y = y*_{1}*.* The reader is urgently cautioned not to think of as meaning the derivative of *y*_{1} with respect to *x*_{1}; such an interpretation would be meaningless since both *x*_{1} and *y*_{1} are *constants,* not variables.

EXAMPLE 1.Find the slope of the tangent to *y* = *x*^{4} + 2*x*^{3} − 5*x +* 3 at the point where *x* = 2.

*Solution.*

EXAMPLE 2.Find the slope of the tangent to *y*^{2} = 12*x* + 13 at the point (3,7).

*Solution.* Differentiating:

EXAMPLE 3.Find the slope of the curve 2*x*^{2} + *y*^{2} = 4*x* + 42 at the points where *x* = 3.

*Solution.* If *x*_{1} = 3, then *y*_{1} = ±6. Differentiating:

EXAMPLE 4.Find the slope of the curve

*x*^{2}*y*^{2} *+ x*^{3} − 2*x* − *y*^{4} − 6*y* = 0

at the point (0,0).

*Solution.* Differentiating:

**6—2.The Angle between Two Intersecting Curves.** The reader will recall that the angle between two intersecting curves is defined as the angle between the tangents to the two curves, respectively, at their point of intersection *P*(*x*_{1}*,y*_{1}). Since

it is merely necessary first to determine *m*_{1} and *m*_{2} by differentiation, then to substitute the values of *x*_{1} and *y*_{1} in each of the two derivatives so found, and then to substitute these values in the expression for tan *θ*.

EXAMPLE. Find the angle of intersection between the circle

*x*^{2} *+ y*^{2} = 25

and the parabola 4*y*^{2} = 9*x*.

*Solution.* Solving these equations simultaneously we have:

*x* = 4, *y* = ±3.

Let*x*_{1} = 4,*y*_{1} = ±3.

Differentiating:

hence, arc tan = arc tan (− 3.417) = 106°20′ (approx.)

**EXERCISE 6—1**

**1.** Find the slope of the curve

at any point.

**2.** Show that the tangent to the circle *x*^{2} + *y*^{2} + 2*x +* 4*y* = 0 at the origin is parallel to the line *x +* 2*y* = 10.

**3.** Find the slope of the curve

*y = x*(*x*^{3} *+* 7)*⅔*

at the point where *x* = 1.

**4.** At what point on the curve *y*^{2} = 3*x*^{3} is the slope equal to ?

**5.** Find the angle of intersection between the curves

*x*^{2} + *y*^{2} + 2*x* − 3 = 0and*x*^{2} *+ y*^{2} = 7.

**6.** At what points on the circle *x*^{2} + *y*^{2} = *k*^{2} is the slope of the tangent to the circle equal to – ?

**7.** Find the angle of intersection of the two parabolas *x*^{2} *=* 4*py* and *y*^{2} = 4*px*.

**8.** Prove that for all values of *k*, the equation *kx = e ^{v}* has the same slope, that is, the slope is independent of the value of

*k.*

**6—3.Equations of Tangent and Normal.** The equation of the tangent, *ST,* to a curve at a given point *P*_{1}(*x*_{1}*,y*_{1}) is easily derived.

The slope of the tangent in question is given by ; since the point of tangency lies on the tangent as well as on the curve, we may use the point-slope formula for the equation of a line having a given slope and passing through a given point. Thus:

The *normal* is the line perpendicular to a tangent at the point of contact. Hence the equation of the normal, *P*_{1}*N*, given by

EXAMPLE 1.Find the equations of the tangent and the normal to the curve 4*x*^{2} + 9y^{2} = 25 at the point where *x* = 2 and *y* is positive.

*Solution.*

At *x*_{1} = 2, *y*_{1} *=* ±1; taking *y*_{1} positive:

Hence, the equation of the tangent at (2,1) is

*y* – 1 = – (*x* – 2),or8*x* + 9*y* – 25 = 0

and the equation of the normal at (2,1) is

*y* – 1 = – (*x* – 2),or9*x* + 8*y* – 10 = 0.

EXAMPLE 2.Find the equations of the tangent and the normal to the equation 2*x*^{3} = *y*^{2} at the point where *x* = 1 and *y* is negative.

*Solution.*

At *x*_{1} = 1, *y*_{1} = ±; taking *y*_{1} negative:

Hence, the equation of the tangent is

The equation of the normal:

**6—4.Length of Subtangent and Subnormal.** Referring to the diagram in §6—3, the segment *RM* is called the *subtangent* of the point *P*_{1}; the length of the segment *MQ* is called the *subnormal* of *P*_{1}. The lengths of these segments, for any curve *f*(*x,y*) = 0, can be readily derived.

EXAMPLE 1.Find the lengths of the subtangent and the subnormal to the curve *x*^{2} = 8*y* + 4 when *x* = 6.

At *x*_{1} = 6, *y*_{1} = 4;

EXAMPLE 2.Find the lengths of the subtangent and the subnormal to *x*^{2} = 2*y*^{3} at *x* = 4.

At *x*_{1} = 4, *y*_{1} = 2; hence,

**EXERCISE 6—2**

**1.** Find the length of the subtangent to 3*x*^{2} − *y*^{2} = 12 at *x* = 4.

**2.** Find the equation of the tangent and the normal to the curve *x*^{3} = *y + xy* at the point where *x* = − 2.

**3.** Find the lengths of the subtangent and the subnormal to the curve *y*^{2} = 4*px* at any point (*x,y*).

**4.** Find the equation of the tangent, and the length of the subtangent, to the circle *x*^{2} + *y*^{2} *=* 25 at *x* = 3.

**5.** Find the equation of the tangent to the curve *a*^{2}(*x* − *y*) = *x*^{3} + *x*^{2}*y* at the origin.

**6.** Find the length of the subtangent to the curve *x = ky ^{n}* when

*x*=

*k.*

**7.** Find the length of the subtangent to the curve *y* = *k ^{x}.*

**8.** Prove that the area of the triangle formed by the coordinate axes and any tangent drawn to the curve 2*xy* = a^{2} is a constant and equal to *a*^{2}.