The graph of y is concave upward (from the left to E and from D to the right) or downward (from E to D) according as the graph of Dam lies above or below the x-axis.

What further relations exist between the graphs of Dry and D2m?

97. Velocity and Acceleration. The velocity of a body moving along a line is the rate of change with respect to the time of its distance s measured along the line from a certain station (Section 32, page 93). Hence,

The acceleration a of a moving body is defined to be the rate of change of its velocity v, and therefore

EXAMPLE. The position of a body moving on a line at the time t is given by

s = t3 - 5t2 + 2t + 8.

(3) Find the velocity and acceleration at any time. Determine the position, velocity, and acceleration when t - 1. When is the body at rest? Differentiating, we get

=

and

=

0, v = 15, and a = - 16.

(4)

(5)

When t - 1, we find that s = Hence at that time the body passed through the station, moving in the positive direction at the rate of 15 feet per second, which was decreasing at the rate of 16 feet per second per second.

The body will be at rest at any instant at which v

=

0; that is when (6)

=

0.2 or 3.2 seconds.

2. Differentiate y = (x2 − 3)2: (a) by applying (5), Section 94, regarding u = x2 - 3; (b) by first removing the parentheses.

=

4. Find the slope of the line tangent to the graph of y x2 4x at any point P(x, y); at the point for which x 1. Plot the graph, and draw the tangent at the latter point.

5. The distance fallen in t seconds by a ball thrown downward with a velocity of 48 feet per second is s = 48t16t2. Find the velocity at any time. How fast will it be moving when t = 3?

6. A billiard ball rolling down a smooth plane inclined at a little less than half a degree moves according to the law

S=112.

Find the velocity at any time. Plot the graphs of s and v on the same axes, using a large scale, from t = 0 to t 5. From the graphs answer the following:

=

(a) What is the position and velocity of the ball after 4.5 seconds? (b) With what velocity will the ball be moving after it has rolled 3 feet? (c) How far must it roll to acquire a velocity of 2 feet per second? (d) When is the distance equal to the velocity? How many solutions? (e) Draw a tangent to the graph of s and measure its slope. Measure the ordinate on the point on the graph of v which has the same abscissa as the point of tangency. How do the two compare?

7. Find the points at which the tangents to the graph of y are horizontal. Construct the figure.

=

=

x3- 27x

8. The height after t seconds of a body thrown vertically upward with a velocity of 96 feet per second is s 96t16t2. Find the velocity at any time. When will the velocity be zero? How high will the body rise? 9. Find the angles which the graph of y the x-axis.

=

x3 + 5x2 - 6x makes with

=

4x cut the graph of y

=

x2?

10. At what angles does the line y 11. Oil dropped on the floor spreads out in a circle. Find the rate at which the circumference increases with respect to the radius, and the rate at which the area increases with respect to the radius.

12. The kinetic energy of a body of mass m moving with a velocity v is given by

K

==

mv2.

Find the rate at which the kinetic energy changes with respect to the velocity.

13. If P is the pressure of a body on a surface and F the friction between them, what does the derivative of F with respect to P represent?

14. If s is a quadratic function of t, show that the acceleration is uni. form (constant).

15. Find the points of maxima, minima, and inflection on the graphs of each of the equations below. On the same axes, plot the graphs of y. m = Dлy, and Dm, and discuss the relations between them.

16. The distance from the starting point, after t seconds, to a ball rolled up a plane inclined at a little less than half a degree with an initial velocity of 8 feet per second is

8 = 112 +8t.

How far will it roll up the plane, and how fast will it be moving when it returns to the starting point?

17. The position of a body moving on a line is given by one of the equations following. Find the velocity and acceleration at any time. On the same axes plot the graphs of s, v, and a, and discuss the motion with reference to each of them.

18. Find the largest possible number of horizontal tangents to the graph of a polynomial of degree n; the largest possible number of points of inflection.

19. If the graph of a function y is concave toward the x-axis, show that y and Dam have opposite signs.

20. If y = ax3 + dx2 + cx + d, show that the abscissas of the points on the graph at which the tangent line is horizontal satisfy a quadratic equation. Find the condition that the number of horizontal tangents is two, one, or zero. Apply this condition to determine the number of horizontal tangents in 15, b, c, d.

98. Derivative of a Rational Function. In order to differentiate any rational function we need but one more rule.

Theorem. The derivative of a fraction is a fraction whose numerator is the denominator times the derivative of the numerator less the numerator times the derivative of the denominator, and whose denominator is the square of the denominator. Symbolically, и v Dzu - uDzv

Let

D2

Finding the limit as Ax approaches zero, we get

v (v + Av)

(3x 4) Dx(x2+5)

(x2 + 5)2

After one fixes the rule in mind, it becomes easy to write down the second fraction on the right without taking the time to write out the first.

EXAMPLE 2. Find the derivative of y = 1/u", where u is a function of x. We have

Dx

1

un

=

un Dx1 - 1 Dxun

u2n

Since the result obtained in Example 2 may be written in the form

D2(un) = - nu-n-1Dzu,

we see that Theorem 5, Section 94, holds for negative as well as positive, integral values of the exponent n.

99. Derivative of an Irrational Function. In order to differentiate an irrational function it is sufficient to show that Theorem 5, Section 94, holds also for a fractional exponent, n = p/q.

As y and u are merely different notations for the same function of x, their derivatives are the same.

Hence

Multiplying (5) by (1), and dividing by 9,

(3)

(4)

(5)

Theorem. If y = u", where in is a fraction, then Dây = nu2¬1DÂu.

EXAMPLE. Differentiate y

Since

and since we may regard x2

=

a2 as a function u, we have, by the theorem, Dxy = (x2 - a2)−12x.