2. If the temperature of a gas is constant, the pressure is inversely proportional to the volume. Find the rate of change of the pressure with respect to the volume.

3. The intensity of light varies inversely as the square of the distance from the source. Find the ratio of the intensities at two points, one 2 feet from the source, the other 4 feet. If a body moves away from the source, find the rate at which the intensity changes as the distance increases. Compare the rates of change (find their ratio) at the two points given above.

4. Find the points at which tangents to the graphs of the following equations are horizontal, if any, and construct the figures.

5. Construct the graph of x2

xy

=

(c) y

x2 + 1

4, and find the points at which the

tangent line is horizontal. Has the curve any points of inflection?

6. Find the points of inflection of the graph of y = x/(x2 + 1).

7. The original amount of carbon monoxide in a mixture of formic and sulphuric acids is a, and the amount x produced in the time t is given by the equation

Find the rate at which carbon monoxide is formed.

8. Find the angle at which the graphs of y

=

1/x2 and y x intersect.

=

9. Find the distance from the origin to the point (x, y) in terms of x

and y, and show that the graph of x2 + y2 of the tangent line at any point.

=

=

16 is a circle.

Find the slope

10. Show that the graph of y +√3 is concave upward, and that of y = −√Ã3 is concave downward. Show that both graphs are tangent to the x-axis at the origin. What, then, is the form of y = x2 = ±√x3 near the origin?

100. Equations of Tangent and Normal Lines. The slope of the line tangent to a curve at any point P1(x1, y1) on it, is the value of the derivative at P1, that is, the value of Day for x = X1. Hence the equation of the line tangent at P1 may be found by the equation y − y1 = m(x − x1), given on page 66.

DEFINITION. The normal to the curve at any point on the curve is the line perpendicular to the tangent at that point.

If m is the slope of the tangent line, then the slope of the normal is 1/m, since the slope of one of two perpendicular lines is the negative reciprocal of the slope of the other (page 200).

of

EXAMPLE 1. Find the equation of the tangent and normal to the graph

y = x2

at the point for which x = 4.

The slope of the tangent line at any point is m = Dayx, and hence that

EXAMPLE 2. Find the equation of the tangent and normal to the parabola in Example 1 at any point (P1(x1, y1) on the curve. If M is the projection of Pi on the y-axis, and if the tangent and normal cut the y-axis at T and N respectively, show that O bisects TM and that MN is constant.

At P1 the slope of the tangent is m = x1/2, and hence the equation of the tangent is

Setting x = 0 in (3) and solving for y, the intercept on the y-axis is

(3)

Since P1 lies on the graph of y = x2/4, its coördinates satisfy this equation so that y1 = 12/4, and hence x12 = 4yı. Substituting in (4)

The tangent at any point P1 may therefore be constructed by laying off on the y-axis OT - y1, and joining T to P1.

The slope of the normal at P1 is m = - 2/x1. Hence the equation of the normal is

y - yi

The intercept of (5) on the y-axis is

y = y1 + 2.

Hence N lies 2 units above P1, or MN = 2, a constant.

The normal at P1 may be constructed by drawing the line through P1 and the point N which is 2 units above the projection of P1 on the y-axis.

A frequent source of error in finding the equation of the tangent or normal to a curve at any point P1 on the curve is the failure to keep in mind that x and y are variables, the coördinates of any point on the line, while x and y1 are constants, the coördinates of a fixed point. The slope of the tangent or normal line at a given point P1 is a constant.

EXERCISES

1. Obtain equations (1) and (2) above by substituting the coördinates of Pi (4, 4) in equations (3) and (5).

2. Show that the length of the line joining P1(x1, y1) and P2(x2, y2) is √(x1 − X2)2 + (Y1 — Y2)2.

=

3. Show that the coördinates of the middle point of the line PP2 are x = (x1+x2)/2, y (yı + y2)/2. Hint: If P is the middle point, the values of Ax computed for P1 and P and for P and P2 are equal; and so also are the values of Ay.

4. Find the equations of the tangent and normal to the following curves at the points indicated. Construct the figure in each case.

5. Find the equations of the tangent and normal to the parabola y = x2 at any point

P1(x, y) on the parabola.

FIG. 163.

6. Find the equations of the tangent and normal to the hyperbola

=

ax2

7. Find the equations of the tangent and normal to the parabola y at any point P1(x1, y1). If these lines cut the y-axis at T and N respectively, and if M is the projection of P1 on the y-axis, show that the origin O bisects TM, and that MN is constant. State a rule for constructing the tangent and normal at any point on the parabola.

8. Find the coördinates of the point at which the tangent to the parabola y ax2 at Pi cuts the x-axis. If the tangent cuts the x-axis at R, and the y-axis at T, prove that R is the middle point of P1T. Find the equation of the line through R perpendicular to the tangent. Show that this line always cuts the y-axis at the same point F, no matter what point on the curve P1 is.

DEFINITION. The point F(0, 1/4a) is called the focus of the parabola y = ax2. The focus is the point at which the line in Exercise 8 cuts the y-axis.

9. In the figure, P1A is parallel to the y-axis. By means of Exercise 8, and methods of plane geometry, show that angle FP1N = angle NP1A, the notation being the same as in

У

N

F

T

R

FIG. 164.

R

S

x

Exercises 7 and 8.

NOTE: A parabolic reflector, such as is used in a headlight of an automobile, is formed by revolving a parabola about its axis of symmetry. If a source of light is placed at F, Exercise 9 shows that the rays will be reflected in parallel lines.

10. Find the equations of the lines tangent to the parabola y = ax2 at the two points for which y = 1/4a ̧ Where do they intersect? At what angle?

11. Find the equation of the line through the focus F(0, 1/4a) perpendicular to the tangent to the

parabola at P1(x1, y1). Show that it intersects the line x =

12. Given the parabola y

X1 on the line

=

ax2 and a point P1(xı, y1) on it, show that the distance from the focus F(0, 1/4a) to P1is FP1 = y1 + 1/4a. Show from this that any point on a parabola is equidistant from the focus and the directrix.

13. Find the equation of the tangent to the graph of y = ax3 at any point P1. If it cuts the y-axis at T, and if M is the projection of P1 on the y-axis, show that the origin trisects TM.

14. Find the equation of the line tangent to the parabola y2 = x at any point P1 on the curve. Show that its intercept on the y-axis is half the ordinate of the point of contact and that it is perpendicular to the line joining the point of intersection with the y-axis to the focus.

15. Let F be a fixed point on the x-axis and T any point on the y-axis. Through T draw the line perpendicular to TF. Choosing different positions for T, draw a number of such lines, enough so that the form of the parabola to which they are tangent becomes apparent.

=

16. If P1 is a point on the equilateral hyperbola xy a, M its projection on the y-axis, and if the tangent line cuts the y-axis at T, then M is the middle point of OT. How can this be used to construct the line tangent at a given point on the curve?

17. Show that the point of contact of a line tangent to the equilateral hyperbola xy a is the middle point of the segment included between the axes. 18. Find the area included between the axes and the line tangent to the equilateral hyperbola xy = a at any point P1. State the result as a

theorem.

19. If a normal is drawn to the equilateral hyperbola xy = a at a point P1 on it, then P1 is the middle point of the segment included between the lines bisecting the coördinate axes.

101. Problems in Maxima and Minima. To solve a problem involving maximum and minimum values, it is necessary first to express the variable v which is to be a maximum or minimum in terms of a single independent variable x. The quantity which is to be made a maximum or minimum is usually apparent from the statement of the problem, but there is frequently some choice in the selection of the independent variable. In many problems, as in Example 2 below, the variable v may be expressed at once in terms of two variables, one of which may be eliminated by means of a relation between them. Having found the function v = f(x), one then differentiates v with respect to x, sets the derivative equal to zero, and solves for x. It must then be shown that at least one of the values of x so found makes v a maximum or minimum, and this maximum or minimum value can be determined by substitution in v = f(x).

<4-2x

FIG. 165.

EXAMPLE 1. A box is to be made out of a square piece of cardboard, four inches on a side, by cutting out equal squares from the corners and then turning up the sides. Find the dimensions of the largest box that can be made in this way.

If the squares cut out are small, the box will have a large base and a shallow depth. If the squares are large, the base will be small and the