high a b b low a continuous function on a closed interval i a continuous function on a open interval ii a discontinuous function on a closed interval iii no highest value; no lowest valu
Trang 1F a c t 1 ( T h e I n t e r m e d i a t e V a l u e T h e o r e m )
If f is continuous on the closed and bounded interval [a, b] and f (a) = A,
f (b) = B, then somewhere in the interval f attains every value between A and B
In particular, if a continuous function changes sign on an interval, it must be zero somewhere on that interval
f
x b a
B A
Figure 7.25
F a c t 2 ( T h e E x t r e m e V a l u e T h e o r e m )
If a function f is continuous on a closed interval [a, b], then f takes on both a maximum (high) and a minimum (low) value on [a, b].8
For f to attain the maximum value of M on [a, b] means that there is a number c in [a, b] such that f (c) = M and f (x) ≤ M for all x in [a, b] Analogously, for f to attain the minimum value of m on [a, b] means that there is a number c in [a, b] such that f (c) = m and f (x) ≥ m for all x in [a, b]
high
a b
b
low
a continuous
function on a
closed interval
(i)
a continuous function on a open interval (ii)
a discontinuous function on a closed interval (iii)
no highest value;
no lowest value
no highest value;
no lowest value
Figure 7.26
Studyinging parts (ii) and (iii) in Figure 7.26 should convince you that both conditions, the continuity of f and the interval being closed, are necessary in order for the statement to hold
8 For a a proof of either of these theorems, look in a more theoretical calculus book.
Trang 2272 CHAPTER 7 The Theoretical Backbone: Limits and Continuity
Note that even if f (x) = k, where k is a constant, the statement holds Given the definitions of maximum and minimum presented above, if f (x) = k on [a, b], then for
every x ∈ [a, b] f (x) = k is both a maximum value and a minimum value.
Principles for Working with Limits and Their Implications for Derivatives
The following general principles can be deduced from the definition of limits
Suppose limx→af (x) = L1and limx→ag(x) = L2, where L1and L2are finite Then:
1. limx→a[f (x) ± g(x)] = L1± L2 The limit of a sum (difference) is the sum
(difference) of the limits
2. limx→a[f (x)g(x)] = L1· L2 The limit of a product is the product of the
limits (in particular, g(x) may be constant: limx→akf (x) = kL1)
3. limx→a f (x)g(x) =L1
L 2, provided L2= 0 The limit of a quotient is the quotient of
the limits (provided the denominator has a nonzero limit)
4. If h is continuous at L2, then limx→ah(g(x)) = h(L2) = h(limx→ag(x))
5. If f (x) < g(x) for all x in the vicinity of a (although not necessarily at x = a), then limx→af (x) ≤ limx→ag(x)
We’ll also sometimes use what is known as the Sandwich Theorem, or the Squeeze Theorem
S a n d w i c h T h e o r e m
If f (x) ≤ j (x) ≤ g(x) for all x in the vicinity of a (although not necessarily at x = a) and
lim
x→af (x) = lim
x→ag(x) = L, then
lim
x→aj (x) = L
The idea behind this theorem is that the functions f and g act as a vise, as lower and upper bounds for j in the vicinity of a As x approaches a the lower and upper jaws of the vise get arbitrarily close together, trapping limx→aj (x)between them
y
f
g j
Figure 7.27
Trang 3EXERCISE 7.4 Use the principles for working with limits, along with the conclusions of Examples 7.1, 7.2,
and 7.3 and Exercises 7.1 and 7.2 to calculate the following
(a) lim
x→∞3 · 2−x+ 4 (b) lim
x→√2
(2x2√ 2x + 3) (c) lim
x→∞
3 + 6x x
We can use principles (1) and (2) given above to prove two very useful properties of derivatives
Properties of Derivatives9
i dxdkf (x) = kdxdf (x), where k is any constant
Multiplying f by a constant k multiplies its derivative by k
ii dxd[f (x) + g(x)] =dxd f (x) +dxd g(x) The derivative of a sum is the sum of the derivatives
These are very important results In the Exploratory Problems for this chapter you will
be asked to prove these properties and to make sense out of them
As an application of some of the principles for working with limits given in this section,
we will verify the following fact
T h e o r e m : D i f f e r e n t i a b i l i t y I m p l i e s C o n t i n u i t y
If a function is differentiable at x = a (that is, f(a)exists), then f is continuous
at x = a
Although the line of reasoning in the proof is easier to follow than to come up with, the conclusion should make sense intuitively If f is differentiable at x = a, then f is locally linear at x = a; f looks like a line near x = a It makes sense that f must be continuous at
x = a
PROOF OF THEOREMSuppose that a is a point in the domain of f , where f is defined
on some open interval containing a We will assume that f(a)exists and show that f is continuous at x = a
According to our definition of continuity, f is continuous at x = a if
lim
x→af (x) = f (a)
We will show that this is true provided f(a)exists Showing that limx→af (x) = f (a) is equivalent to showing that limx→a[f (x) − f (a)] = 0
9 Recall that d means “the derivative of ”
Trang 4274 CHAPTER 7 The Theoretical Backbone: Limits and Continuity
lim
x→a[f (x) − f (a)] = lim
x→a
(f (x) − f (a))x − a
x − a We are multiplying by 1, since x = a
=
lim
x→a
f (x) − f (a)
x − a ·
lim
x→a(x − a) The limit of a product is the product of
the limits if both exist and are finite
= f(a) · 0 The first limit is f(a), which exists by
assumption, and the second equals zero
= 0
We have shown that if f(a)exists, then f (x) must be continuous at x = a
Question:If f is continuous at x = a, is f necessarily differentiable (locally linear) at
x = a?
Answer:No Informally speaking, if f has a sharp corner at x = a, then it is not differ-entiable at x = a because it is not locally linear there A classic example of the latter situation is the function f (x) = |x| at x = 0 f (x) = |x| is continuous at x = 0 because limx→0f (x) = f (0) = 0 However, as we saw in Example 7.13, f(0) does not exist; f is not differentiable at x = 0
x f
Figure 7.28
You are now prepared to read Appendix C There we offer proofs of facts about derivatives stated without proof in Chapter 5
Trang 5Exploratory Problems for Chapter 7
Pushing the Limit
1.Let h(t) = kf (t), where k is a constant and f is a differentiable
function
(a) Use the principles of working with limits to show that
h(t ) = kf(t ) Begin with the limit definition of h(t ) and
then express h in terms of f
(b) Explain this result in graphical terms Why is the slope of the
tangent to h at t equal to k times the slope of the tangent to f
at t?
(c) Interpret this result in the case that f (t) is a position function
and k = 2 More specifically, consider the following
exam-ple Two women leave a Midwestern farmhouse and travel
north on a straight road One woman walks and the other
woman runs Suppose f (t) gives the distance between the
walker and the farmhouse at time t and h(t) = 2f (t) gives
the distance between the runner and the farmhouse Interpret
the result h(t ) = 2f(t )in this context Generalize to the case
h(t ) = kf(t )
2.Let j (t) = f (t) + g(t), where f and g are differentiable
func-tions
(a) Use the principles of working with limits to show that
j(t ) = f(t ) + g(t ) Begin with the limit definition of j(t )
and then express j in terms of f and g
(b) Explain this result in graphical terms
(c) Interpret this result in the following scenario A teacher has
put retirement money in two accounts, TIAA and CREF Let
f (t )be the value of his TIAA account at time t and let g(t)
be the value of his CREF account at time t Interpret the result
j(t ) = f(t ) + g(t )in this context
3.(a) Use the properties of limits and the results of the previous
problems to differentiate f (x) = ax3+ bx2+ cx + d as
effi-ciently as possible
(b) Along the way, you’ll need to use the limit definition of
deriva-tive to differentiate x3 Explain why your answer makes sense
graphically by looking at the graphs of x3and 3x2
4 Looking for Patterns: Use the results of Problem 3 above and
formulas for derivatives of√
xand1xfound in Chapter 5 to arrive
at a formula for the derivative of xnfor n = 0, 1, 2, 3, −1, and12
Try your formula on another value of n and see if it works
Trang 6276 CHAPTER 7 The Theoretical Backbone: Limits and Continuity
P R O B L E M S F O R S E C T I O N 7 4
1 (a) Find the following limits Illustrate your answers with graphs
i lim
x→∞−3x − 3
iv lim
x→∞
x + 1
x→∞
2x + 3 x (b) In Section 7.1, Example 7.1, we showed limx→∞ 1x= 0 In Section 7.4 we listed some principles for working with limits Show how your answers to all of the problems in part (a) can be deduced using
lim
x→∞
1
x = 0 and lim
x→∞k = k, for any constant k and applying the principles listed in Section 7.4
2 Find the following
(a) lim
x→4
1 (x − 4)2 (b) limx→4
x + 3 (x − 4)2 (c) limx→4
x2+ 16 (x − 4)2 (d) lim
x→4 +
1 (x − 4) (e) limx→4 −
1 (x − 4) (f ) limx→4
x2− 16 (x − 4)
3 Find lim
x→4
√ x−2 x−4
4 Suppose |h(x)| ≤ 3 for all x Evaluate lim
x→∞
h(x)
x
Each of the functions in Problems 5 through 10 is either continuous on (−∞, ∞) or has
a point of discontinuity at some point(s) x = a Determine any point(s) of discontinuity.
Is the point of discontinuity removable? In other words, can the function be made continuous by defining or redefining the function at the point of discontinuity?
5 f (x) = xx+22−4
6 f (x) =
x3, x = 0,
3, x = 0
7 f (x) = x+21
8 f (x) = x21
+2
9 f (x) =
−x2− 1, x > 0, 5x − 1, x <0
10 f (x) =
−x2− x, x > 0, 5x − 1, x <0
11 Let f (x) =
−x2+ 1, x > 0,
ax + b, x ≤ 0
What are the constraints on a and b in order for f to be continuous at x = 0?
Trang 712 Let f (x) = g(x), x ≥ a,
h(x), x < a, where g is continuous on [a, ∞) and h is continuous
on (−∞, a)
What must be true about g and h in order for f to be continuous at x = a?
13 Let f (x) = −1, if x is a rational number,
1, if x is an irrational number
(a) Is f (x) continuous at x = 1?
(b) Is f (x) continuous at x = π?
14 In this problem we’ll look at a range of possible scenarios accompanying limx→af (x), where a is a finite number Your job is to draw illustrations showing f in the vicinity
of a with the specifications given
(a) Suppose limx→2f (x) = L, where L is a finite number
Draw three qualitatively different pictures of what f could look like in the vicinity of x = 2 The first picture should show a continuous function f The second should show a function discontinuous at x = 2 but defined at x = 2 The third should show f undefined at x = 2
(b) Illustrate the following two scenarios
i limx→2f (x) = ∞ ii limx→2f (x) = −∞
(c) Suppose lim
x→2f (x)is undefined
Draw three qualitatively different pictures of what f could look like in the vicinity of x = 2 The first picture should have f defined at 2 The second picture should have f undefined at 2 but the one-sided limits at 2 both finite The third picture should have limx→2+f (x) = ∞; limx→2 −f (x) is left up to you
(Another possibility is that f (x) does not approach any single finite value but also does not increase or decrease without bound.)
2
15 Let f (x) =
x2, for x ≥ 0,
−x2, for x < 0
(a) Is f continuous at x = 0?
(b) Is f differentiable at x = 0? If so, what is f(0)?
16 Find the derivative of f (x) = kx4, where k is a constant
17 Let g(x) =
x2, for x ≥ 0,
x, for x < 0
(a) Is g continuous at x = 0?
(b) Is g differentiable at x = 0? If so, what is g(0)?
Trang 8278 CHAPTER 7 The Theoretical Backbone: Limits and Continuity
18 Let f (x) =x1 (a) Draw the graph of f (x) and f(x)
(b) Use the graphs you’ve drawn in part (a) to do the following
i Find lim
x→∞f(x)
ii Find lim
x→0 +f(x)
iii Find lim
x→0 −f(x)
iv Find lim
x→−∞f(x)
(c) Use the limit definition of derivative to find f(x) Use your work to check your answers to parts (a) and (b)
19 The domain of a function f is all real numbers The zeros of f (x) are x = −1, x = 2, and x = 6 There are no other x-values such that f (x) = 0 Is it possible that f (3) > 0 and f (4) < 0? Explain
20 The domain of a continuous function f is all real numbers The zeros of f are
x = −1, x = 2, and x = 6 There are no other x-values such that f (x) = 0 Is it possible that f (3) > 0 and f (4) < 0? Explain
21 Sketch the graph of one function having all seven of the following characteristics.
i f (x) > 0 for all x, ii lim
x→4f (x) = 1, iii f (4) = 3, iv lim
x→∞f (x) = 1,
v lim
x→−∞f (x) = 1, vi lim
x→0 +f (x) = 5, vii lim
x→0 −f (x) = 2
22 Use the limit definition to differentiate f (x) =x12
Trang 9C H A P T E R
Fruits of Our Labor:
Derivatives and Local Linearity Revisited
8.1 LOCAL LINEARITY AND THE DERIVATIVE
In Section 4.1 we discussed local linearity, but at that point we had not yet developed the concept of derivative Therefore, in this section we revisit the idea In Chapter 5 we pointed out that there are numerous forms in which the definition of derivative can be expressed, yet for the most part we’ve used f(x) = limh→0 f (x+h)−f (x)h In this chapter, because we’ll look at approximations that are good for a small range of the independent variable, we’ll use notation that emphasizes the relative rates of change of the dependent and independent variables
With that in mind, we re-establish the following conventions
If y = f (x), thendydx = lim x→0 y x, where y = f (x + x) − f (x)
Equivalently, we can replace y by f ,
df
dx = lim
x→0
f
x = lim
x→0
f (x + x) − f (x)
(Notice that x and h play the same role.) When we approximate a function near a point using local linearity, we might wonder whether the approximation is larger or smaller than the actual value The answer depends on the second derivative, which was introduced in Chapter 6 There we saw that if f (t) gives position as a function of time, then f(t )gives velocity, the rate of change of position with respect to time, and f(t )gives acceleration, the rate of change of velocity with respect to time Let’s now adopt a graphical perspective as well
If f>0, then fis increasing and f is concave up
If f<0, then fis decreasing and f is concave down
279
Trang 10280 CHAPTER 8 Fruits of Our Labor: Derivatives and Local Linearity Revisited
f ″ > O => f ′ increasing
f concave up
f ″ < O => f ′ decreasing
f concave down
Figure 8.1
EXAMPLE 8.1 The weather pattern known as El Ni˜no brought extreme weather conditions throughout the
globe in 1997 and 1998 Some areas experienced severe drought, while others were beset
by flooding In a certain town, the amount of water in the reservoir has been decreasing for the past few weeks and there is no indication that rain is to be expected any time soon
An awareness of the crisis is spreading and the rate at which water is being consumed is dropping slightly At present there are G0gallons of water in the reservoir and the level is dropping at a rate of 115 gallons per day
(a) Let W (t) be the amount of water in the reservoir, where t is measured in days and we choose t = 0 to be today Translate the information given above into statements about
W, W, and W (b) Approximate the amount of water in the reservoir two days from now
SOLUTION (a) Today there are G0gallons of water in the reservoir, so W (0) = G0
W (t )is decreasing and Wgives the rate of change of W , the amount of water in the reservoir, so W(t ) =dWdt <0
dW dt
t =0= W(0) = −115 gallons/day
t (days)
W (gallons)
W ′ (0) = –115
G0
Figure 8.2
The rate of water consumption is decreasing, so W is becoming less negative as t increases The graph of W is concave up W(t ) ≥ 0
(b)
Amount of water in the reservoir 2 days from now
=amount in the reservoir now
+ change in water
in the next 2 days
W (2) = G0+
change in water
in the next two days
... farmhouse and travel
north on a straight road One woman walks and the other
woman runs Suppose f (t) gives the distance between the
walker and the farmhouse at time t and h(t)... your answers to all of the problems in part (a) can be deduced using
lim
x→∞
1
x = and lim
x→∞k = k, for any constant k and applying... limit definition of derivative to find f(x) Use your work to check your answers to parts (a) and (b)
19 The domain of a function f is all real numbers The zeros of f (x) are