Calculus: Analytic Geometry and Calculus, with Vectors

176 Functions, limits, derivatives

Show that, when x 0 0,

y'(x) = -qx"-q-1 cos I + pxP-1 sin x5

Tell why this formula cannot be valid when x = 0. Then show with the aid of

the sandwich theorem 3.287 that if p > 2, then y'(0) = 0. Show that if p >

q + 1 then lim y(x) = 0. Show that if p < q + 1, then lim y(x) does not exist.

31 Before starting this problem, we make the profound observation that 0

times a number is 0 but that nobody ever tries to define the product of 0 and

something that does not exist. With this in mind, show that the first of the

formulas

(1) a lxi2 = 2x, ax x12=21xl 'x

is valid for each x and that the second is valid if, and only if, x p 0. Hint: For

the first part, observe that x12 = x2. For the second part, consider separately

the cases for which x > 0, x < 0, and x = 0. Remark: Putting f (x) = x2 and

g(x) = Jxj shows that g'(x) can fail to exist even when it is known that df (g(x))/

dx exists. The calculations in

(2) 0(x) = f(g(x)), 4'(x) = f'(g(x))g'(x),

dy dy du

dx

_

du dx

may therefore be incorrect even when 0 and f are both differentiable. In any

case, we are not doing rigorous mathematics when we start with the first of the

formulas

(3) siny=x, cosy-=1

and obtain the second without giving a thought to the question whether y is a

differentiable function of x. Congratulations can therefore be showered upon

students who, at this time, have a healthy lack of enthusiasm for problems like

Problem 20.

3.7 Rates, velocities Let f be defined over some interval a :9 x < b

and let y = f(x). When x and x + Ax both lie between a and b and

Ax 5 0, the difference quotient in

(3.71) AY=f (X + Ox) - AX)

Ox Ax

is the average rate of change of y with respect to x over the interval from the

lesser to the greater of x and x + Ax. If this average rate (which is the

difference quotient) has a limit as Ax approaches zero, then this limit

[which is the derivative dy/dx or f'(x)] is the rate of change of y with respect

to x at the given x. These are definitions which can, perhaps without