Elements of the Differential and Integral Calculus/Chapter VI

CHAPTER VI edit

SIMPLE APPLICATIONS OF THE DERIVATIVE

64. Direction of a curve. It was shown in § 32, p. 31, that if

 

is the equation of a curve (see figure), then

 
Example of tangents to curve.

  slope of line tangent to the curve at any point P.

The direction of a curve at any point is defined to be the same as the direction of the line tangent to the curve at that point. From this it follows at once that

  slope of the curve at any point P.

At a particular point whose coördinates are known we write

  slope of the curve (or tangent) at point  .

At points such as D, F, H, where the curve (or tangent) is parallel to the axis of X;

  = 0°; therefore  .

At points such as A, B, G, where the curve (or tangent) is perpendicular to the axis of X;

  = 90°; therefore  .

At points such as E, where the curve is rising,[1]

  an acute angle; therefore   a positive number.

The curve (or tangent) has a positive slope to the left of B, between D and F, and to the right of G.

At points such as C, where the curve is falling,[1]

  an obtuse angle; therefore   a negative number.

The curve ( or tangent) has a negative slope between B and D, and between F and G.

 
Example curve.

ILLUSTRATIVE EXAMPLE 1. Given the curve   (see figure).

(a) Find   when  .

(b) Find   when  .

(c) Find the points where the curve is parallel to OX.

(d) Find the points where   = 45°.

(e) Find the points where the curve is parallel to the line   (line AB).

Solution. Differentiating,   slope at any point.

(a)  ; therefore   = 135°. Ans.

(b)  ; therefore  . Ans.

(c)   = 0°,  ; therefore  . Solving this equation, we find that   or 2, giving points C and D where the curve (or tangent) is parallel to OX.

(d)   = 45°,  ; therefore  . Solving, we get  , giving two points where the slope of the curve (or tangent) is unity.

(e) Slope of line  ; therefore  . Solving, we get  , giving points E and F where curve (or tangent) is parallel to line AB.

Since a curve at any point has the same direction as its tangent at that point, the angle between two curves at a common point will be the angle between their tangents at that point.

ILLUSTRATIVE EXAMPLE 2. Find the angle of intersection of the circles

(A)  ,

(B)  

Solution. Solving simultaneously, we find the points of intersection to be (3, 2) and (1, -2).

 
Two intersecting circles.
    from (A). By §63, p. 69
    from (B). By §63, p. 69
    slope of tangent to (A) at (3, 2).
   slope of tangent to (B) at (3, 2).

The formula for finding the angle between two lines whose slopes are   and   is

   . 55, p. 3 [§ 1]

Substituting,  ; therefore   = 45°. Ans.

This is also the angle of intersection at the point (1, -2).

EXAMPLES

The corresponding figure should be drawn in each of the following examples:

1. Find the slope of   at the origin. Ans.  .

2. What angle does the tangent to the curve   at the origin make with the axis of X? Ans.   = 135°.

3. What is the direction in which the point generating the graph of   tends to move at the instant when  ? Ans. Parallel to a line whose slope is 5.

4. Show that   (or slope) is constant for a straight line.

5. Find the points where the curve   is parallel to the axis of X. Ans. x = 3, x = -1.

6. At what point on   is the slope equal to 3? Ans. (2, 4).

7. At what points on the circle   is the slope of the tangent line equal to  ? Ans.  .

8. Where will a point moving on the parabola   be moving parallel to the line  ? Ans. (6, -3).

9. Find the points where a particle moving on the circle   moves perpendicular to the line  . Ans. (± 12, \mp 5).

10. Show that all the curves of the system   have the same slope; i.e. the slope is independent of  .

11. The path of the projectile from a mortar cannon lies on the parabola  ; the unit is 1 mile, OX being horizontal and OY vertical, and the origin being the point of projection. Find the direction of motion of the projectile

(a) at instant of projection;

(b) when it strikes a vertical cliff   miles distant.

(c) Where will the path make an inclination of 45° with the horizontal?

(d) Where will the projectile travel horizontally?

Ans. (a)  ; (b) 135°; (c)  ; (d) (1, 1).

12. If the cannon in the preceding example was situated on a hillside of inclination 45°, at what angle would a shot fired up strike the hillside? Ans. 45°.

13. At what angles does a road following the line   intersect a railway track following the parabola  . Ans.  , and  .

14. Find the angle of intersection between the parabola   and the circle  . Ans.  

15. Show that the hyperbola   and the ellipse   intersect at right angles.

16. Show that the circle   and the cissoid  

(a) are perpendicular at the origin;

(b) intersect at an angle of 45° at two other points.

17. Find the angle of intersection of the parabola   and the witch  . Ans.   = 71°33'.9.

18. Show that the tangents to the folium of Descartes   at the points where it meets the parabola   are parallel to the axis of Y.

19. At how many points will a particle moving on the curve   be moving parallel to the axis of X? What are the points? Ans. Two; at (1, - 4) and ( ).

20. Find the angle at which the parabolas   and   intersect. Ans.  .

21. Find the relation between the coefficients of the conics   and   when they intersect at right angles. Ans.  .

65. Equations of tangent and normal, lengths of subtangent and subnormal. Rectangular coördinates. The equation of a straight line passing through the point ( ) and having the slope   is

 
Line tangent to curve AB.
 . 54, (c), p. 3 [§ 1]

If this line is tangent to the curve AB at the point P( ), then from § 64, p. 73,

 [2]

Hence at point of contact   the equation of the tangent line TP1 is

(1)  .

The normal being perpendicular to tangent, its slope is

   . By 55, p. 3 [§ 1]

And since it also passes through the point of contact  , we have for the equation of the normal  

(2)  .

That portion of the tangent which is intercepted between the point of contact and OX is called the length of the tangent ( ), and its projection on the axis of X is called the length of the sub tangent (= TM). Similarly, we have the length of the normal ( ) and the length of the subnormal (= MN).

In the triangle  ; therefore

(3)  [3]   length of subtangent.

In the triangle  ; therefore

(4)  [4]   length of subnormal.

The length of tangent ( ) and the length of normal ( ) may then be found directly from the figure, each being the hypotenuse of a right triangle having the two legs known. Thus

     
(5)     = length of tangent.
     
(6)     = length of normal.

The student is advised to get the lengths of the tangent and of the normal directly from the figure rather than by using (5) and (6).

When the length of subtangent or subnormal at a point on a curve is determined, the tangent and normal may be easily constructed.

EXAMPLES

1. Find the equations of tangent and normal, lengths of subtangent, subnormal tangent, and normal at the point   on the cissoid  .

 
Graph of cissoid
Solution.    .
Hence     slope of tangent.
Substituting in (1) gives
     , equation of tangent.
Substituting in (2) gives
     , equation of normal.
Substituting in (3) gives
      length of subtangent.
Substituting in (4) gives
      length of subnormal.

Also   length of tangent.

and   length of normal.

2. Find equations of tangent and normal to the ellipse   at the points where  .

Ans. At  .
At  .

3. Find equations of tangent and normal, lengths of subtangent and subnormal at the point   on the circle  .[5]

Ans.  .

4. Show that the subtangent to the parabola   is bisected at the vertex, and that the subnormal is constant and equal to  .

5. Find the equation of the tangent at   to the ellipse  .

Ans.  .

6. Find equations of tangent and normal to the witch   as at the point where  .

Ans.  .

7. Prove that at any point on the catenary   the lengths of subnormal and normal are   respectively.

8. Find equations of tangent and normal, lengths of subtangent and subnormal, to each of the following curves at the points indicated:

(a)   at  . (e)   at  .
(b)   at  . (f)   where  .
(c)   where  . (g)  ,  .
(d)   at  . (h)   at  .

9. Prove that the length of subtangent to   is constant and equal to  .

10. Get the equation of tangent to the parabola   which makes an angle of 45° with the axis of X.

Ans.  .

HINT. First find point of contact by method of Illustrative Example 1, (d), p. 74 [§ 64].

11. Find equations of tangents to the circle   which are parallel to the line  .

Ans.  .

12. Find equations of tangents to the hyperbola   which are perpendicular to the line  .

Ans.  .

13. Show that in the equilateral hyperbola   the area of the triangle formed by a tangent and the coördinate axes is constant and equal to  .

14. Find equations of tangents and normals to the curve   at the points where  .

Ans. At  .
At  .

15. Show that the sum of the intercepts of the tangent to the parabola

 

on the coordinate axes is constant and equal to  .

16. Find the equation of tangent to the curve   at the origin.

Ans.  .

17. Show that for the hypocycloid   that portion of the tangent included between the coördinate axes is constant and equal to  .

18. Show that the curve   has a constant subtangent.

66. Parametric equations of a curve. Let the equation of a curve be

(A)  .

If   is given as a function of a third variable,   say, called a parameter, then by virtue of (A)   is also a function of  , and the same functional relation (A) between   and   may generally be expressed by means of equations in the form

(B)  

 
Graph of circle showing parameters.

each value of   giving a value of   and a value of  . Equations (B) are called parametric equations of the curve. If we eliminate   between equations (B), it is evident that the relation (A) must result. For example, take equation of circle

      or  .
Let    ; then
     , and we have
(C)    
 

as parametric equations of the circle in the figure,   being the parameter.

If we eliminate   between equations (C) by squaring and adding the results, we have

 ,

the rectangular equation of the circle. It is evident that if   varies from 0 to  , the point   will describe a complete circumference.

In § 71 we shall discuss the motion of a point P, which motion is defined by equations such as

 .

We call these the parametric equations of the path, the time   being the parameter. Thus in Ex. 2, p. 93, we see that

 

are really the parametric equations of the trajectory of a projectile, the time   being the parameter. The elimination of   gives the rectangular equation of the trajectory

 .

Since from (B)   is given as a function of  , and   as a function of  , we have

      by XXV
   ; by XXVI
that is,
(D)    .

Hence, if the parametric equations of a curve are given, we can find equations of tangent and normal, lengths of subtangent and subnormal at a given point on the curve, by first finding the value of   at that point from (D) and then substituting in formulas (1), (2), (3), (4) of the last section.

ILLUSTRATIVE EXAMPLE 1. Find equations of tangent and normal, lengths of subtangent and subnormal to the ellipse

(E)  [6]

at the point where  .

Solution. The parameter being  ,  ,
   .

Substituting   in the given equations (E), we get   as the point of contact. Hence

     .
Substituting in (1), p. 76,    ,
or,    , equation of tangent.
Substituting in (2), p. 76,    ,
or,    , equation of normal.
Substituting in (3) and (4), p. 77,
      length of subnormal.
      length of subtangent.

ILLUSTRATIVE EXAMPLE 2. Given equation of the cycloid[7] in parametric form

 

θ being the variable parameter; find lengths of subtangent, subnormal, tangent, and normal at the point where  .

   .
Substituting in (D), p. 80,     slope at any point.

Since  , the point of contact is  , and  .

Substituting in (3), (4), (5), (6) of the last section, we get

length of subtangent  , length of subnormal  ,
length of tangent  , length of normal  . Ans.

EXAMPLES

Find equations of tangent and normal, lengths of sub tangent and subnormal to each of the following curves at the point indicated:

  Tangent Normal Subt. Subn.
1.  .  ,  , 2,  .
2.  .  ,  ,  , 96.
3.  .  ,  ,  ,  .
4.  .  ,  , -2,  .
5.  .  ,  ,  , -1

6.  .

7.  .

8.  .

9.  .

10.  .

11.  .

12.  .

13.  .

14.  .

15.  .

In the following curves find lengths of (a) subtangent, (b) subnormal, (c) tangent, (d) normal, at any point:

16. The curve  
  Ans. (a)  , (b)  , (c)  , (d)  .
17. The hypocycloid (astroid)  
  Ans. (a)  , (b)  , (c)  , (d)  .
18. The circle  
19. The cardioid  
20. The folium  
21. The hyperbolic spiral  

  1. 1.0 1.1 When moving from left to right on curve.
  2. By this notation is meant that we should first find  , then in the result substitute   for   and   for  . The student is warned against interpreting the symbol   to mean the derivative of   with respect to  , for that has no meaning whatever, since   and   are both constants.
  3. If subtangent extends to the right of T, we consider it positive; if to the left, negative.
  4. If subnormal extends to the right of M, we consider it positive; if to the left, negative.
  5. In Exs. 3 and 5 the student should notice that if we drop the subscripts in equations of tangents, they reduce to the equations of the curves themselves.
  6. As in the figure draw the major and minor auxiliary circles of the ellipse. Through two points B and C on the same radius draw lines parallel to the axes of coördinates. These lines will intersect in a point   on the ellipse, because
     
    Major and minor auxiliary circles of the ellipse.
       
    and  ,
    or,   and  .

    Now squaring and adding, we get

     ,

    the rectangular equation of the ellipse.   is sometimes called the eccentric angle of the ellipse at the point P.

  7.  
    Plot of a cycloid.
    The path described by a point on the circumference of a circle which rolls without sliding on a fixed straight line is called the cycloid. Let the radius of the rolling circle be  , P the generating point, and M the point of contact with the fixed line OX, which is called the base. If arc PM equals OM in length, then P will touch at O if the circle is rolled to the left. We have, denoting angle POM by θ,
     ,
     ,

    the parametric equations of the cycloid, the angle θ through which the rolling circle turns being the parameter.   is called the base of one arch of the cycloid, and the point V is called the vertex. Eliminating θ, we get the rectangular equation