(,) 3x™ -gx + ,xm Q/™ f) 4.9 3x™?

h ) (.x1” + x-“˜)(4x™ 3J;),

g) (2 + 1)(x2 + 3x + 2),

k) (x™ 3x™)˜,

m) (2 + 2.$(4x + S)?.

2.12 Find the equation of the tangent line to the graph of the given function for the

specified value of x. [Hint: Given a point on a line and the slope of the line, one can

˜onstru˜f the equation of the line.]

u) f(x) = x2. x,1 = 3: b) f(x) = x/(x™ + 2), .x,, = 1

2.13 Prove parts (i and h of Theorcm 2.4,

In Theorem 2.3. WC proved that the derivative ofI = xi is y™ = k.& for all positive

2.14

integers k. Use the Quotient Rule, Theorem 2.4d. to extend this result to negative

intrgrrs k.

2.5 DIFFERENTIABILITY AND CONTINUITY

As we saw in Section 2.3, a function f is differcntiahle at xg if, geometrically

speaking. its graph has a tangent line at (.x0, f(.˜,˜)), or analytically speaking. the

limit

exists and is the same for every sequence {h,,} which converges to 0. If a function

is differentiable at every point x,, in its domain D, we say that the function is

differentiable. Only functions whnsc graphs are “smooth curves” have tangent

lines everywhere; in fact. mathematicians commonly use the word “smooth” in

place of the word “differentiable.”

Figure

2.12 The graph off(x) = 1x1.

A Nondifferentiable Function

As an example of a function which is not differentiable everywhere, consider the

graph of the absolute value function f(x) = 1 1 I” F™g ure 2.12. This graph has a

x I

sharp comer at the origin. There is no natural tangent line to this graph at (O,(l).

Alternatively, as Figure 2.13 indicates, there are infinitely many lines through

(0. 0) which lie on one side of the graph and hence would bc candidates for the

tangent lint. Since the graph of l.x has no well-defined tangent line at ,r = 0. the

function /xl is not differentiable at x = 0.

To see why the ar˜alyric definition (8) of the derivative does not work for 1x1.

substitute into (8) each of the following two sequences which converge to zero:

h,, = {+.l. +.Ol, f.001,. +(.I)“, .)

k,, = {-.I, -.Ol, -.OOl,..., -(.l)” ,... }.

12.51 DIFFEKENTIABII.ITY 31

AND C O N T I N U I T Y

Substituting these sequences into the definition (8) of the derivative, we compute

The first sequence is {I, 1,. , 1,. .}, which clearly converges to + I; the second

sequence is {- I, 1,. , 1, .}, which clearly converges to - I. Since different

sequences which converge to 0 yield different limits in (K), the function 1x1 does

nor have a derivative at x = 0.

A property of functions more fundamental than differentiability is that of conti-

nuity. From a geometric point of view, a function is continuous if its graph has

no breaks. Even though it is uot differentiable at n = 0. the function f(r) = 1x1 is

still continuous. On the other hand, the function

whose graph is pictured in Figure 2.14, is not continuous at x = 0. In this cast.

we call the point x = 0 a discontinuity of 6, It should bc clear that the graph of

a function cannot have a tangent line at a point of discontinuity. In other words,

in urdcr for a function tu bc diffcrentiahle. it must al lcast be continuous. For

functions dcscribcd by concrete formulas. discontinuiiics arise when the function

is dehncd by dit™fcrcnt Surmulas on different parts of the number line and when the

values of these two formulas arc different at the point where the formula changes.

for example. at the point x = 0 in (9).

Figure

2.14

The break in the graph of ,T at the origin in Figure 2.11 mans that there are

points on the x-axis on either side ofrero which arc arbitrarily close to each other.

hut whose values under s are not close to each other. Even though (p. I)” and

(+. I)” are arbitrarily close to each other, g(( -.l)“) is close to -I whileg((+. I)“)

is clusc to + I. As x c˜˜sscs 0, the value of the function suddenly changes by two

units. Small changes in ,Y do not lead to small changes in g(x). This leads to the

CoIlowing more analytic detinition of continuity.

The function g(x) defined in (9) does not satisfy this definition at x = 0

hccause

&f((-.I)“) = -I, h u t f(O) = +l.

If f has a second derivative everywhere, then f” is a well-defined function of x.

We will see later that the second derivative has a rich geometric meaning in terms

of the shape of the graph off. If f” is itself a continuous function of x, then WC say

thal f is twice continuously differentiable. or C” for short. Every polynomial is

a C™ function.

This process continues. If f is C™, so that x - f"(x) is a continuous function,

WC can ask whether f” has a derivative st xII. If it does. we write this derivative as

3

f yr”) or f”˜(X,,) or g&Y”).

For example, for the cubic polynomial f(x) in Example 2.9, f”˜(x) = 6. If f”˜(x)

exists for all x and if f”˜(n) is itself a continuous function of x, then we say that

the original function f is c”.

This process continues for all positive integers. If f(x) has derivatives of order

I, 2,. , k and if the kth derivative of /,

fl™l@) = ,,w

d”f

is itself a continuous function. WC say that f is C™. If f has a continuous derivative

of every order. that is, if f is C™ fbr every positive integer k. then we say that f is

C“ or “intinilelv differcntiahle.” All polynomials are C” functions.

2.7 APPROXIMATION BY DIFFERENTIALS

This completes OUT inrruduction tu the fundamental concepts and calculations of

CRICUIUS. We turn now to the task of using the derivative to shed light on func-

lions. In the next chapter. the derivative will be used 10 understand functions murr

completely. to graph functions more efficiently. to solve optimization prohlemh.

and 10 characterize the maximizer or minimizer of a function, especially in cco-

nwnic xttings. W C begin our discussion of the uses of calculus hy showing how

Ihe definition of the derivative Icads naturally 10 Ihe construction of the linear