Fact: Where f is increasing, the slope of its graph is nonnegative; where f is decreasing, its slope is nonpositive.9 9 You may wonder why we say that f is increasing implies that the sl
Trang 1Where f(x) = 0 the graph of f has a horizontal tangent line.
2x + 3 = 0 2x = −3
x = −1.5 Below we graph f and f Notice that the answers make sense in terms of our intuitive ideas about slope
f '
f '(x)=2x+3
f
f(x)=x +3x2
–3 (–1.5, –2.25)
–1.5
Figure 5.15
EXERCISE 5.2 Use the limit definition of derivative to find the derivatives of x2and 3x Conclude that the
derivative of x2+ 3x is the sum of the derivatives of x2and 3x
Notice that so far the strategy that has been working for computing derivatives is to simplify the difference quotient to the point that we can factor an h from the numerator
to cancel with the h in the denominator (assuming that h = 0) and, once done, the limit becomes apparent Sometimes the simplification allowing cancellation is a bit more com-plicated, as illustrated in the next problem You should compute some simpler derivatives
on your own before reading the next example It is designed for your second reading of the
chapter
EXAMPLE 5.5 Let f (x) =√x Find f(x)
SOLUTION
f(x) = lim
h→0
f (x + h) − f (x) h
= lim h→0
√
h
We have a dilemma here; to make progress we must cancel the h in the denominator This means getting rid of the square roots We’re working with an expression, so our options are to multiply by 1 (multiply numerator and denominator by the same nonzero quantity)
or to add zero The former will be most productive Multiplying the expression√
by (√
A +√B)will eliminate the square roots.8(√
A −√B)is called the conjugate of
(√
(√
8 Multiplying ( √
A +√B) by itself does not eliminate the square roots ( √
A +√B)( √
A +√B) = A + 2√AB + B, which
Trang 2192 CHAPTER 5 The Derivative Function
This algebraic maneuver is worth stashing away in your mind for future reference
f(x) = lim
h→0
√
x + h +√x
= lim h→0
(√
x + h −√x)
√
x + h +√x) (√
x + h +√x)
= lim h→0
x + h − x h(√
x + h +√x)
= lim h→0
h h(√
= lim h→0
1
√
2√x
So the derivative of x1/2is 12x−1/2
Answers to Selected Exercises Answers to Exercise 5.1
(a) f(x) = lim
h→0
f (x + h) − f (x) h
= lim h→0
m(x + h) + b − (mx + b)
h
= lim h→0
mx + mh + b − mx − b
h
= lim h→0
mh h
= m (b) k = mx + k where m = 0, so the derivative of k is zero
P R O B L E M S F O R S E C T I O N 5 2
1 Use the limit definition of derivative to show that the derivative of the linear function
f (x) = ax + b is a Why is this exactly what you would expect? You have shown that the derivative of a constant is zero Explain, and explain why this is exactly what you would expect
Trang 32 Use the limit definition of derivative to find the derivative of f (x) = kx2.
3 Let f (x) = x2 Find the point at which the line tangent to f (x) at x = 2 intersects the line tangent to f (x) at x = −1
4 Use the limit definition of derivative to find the derivative of f (x) = x3
5 Using the limit definition of the derivative, find f(x)if f (x) = (x − 1)2
6 Let g(x) =2x+5x Using the limit definition of derivative, find g(x)
For Problems 7 through 13, find f(x), f(0), f(2), and f(−1).
7 f (x) = 3x + 5
8 f (x) = πx −√3
9 f (x) =2x−53
10 f (x) = π(x + 7) − 2
11 f (x) = x2
12 f (x) = 1x
13 f (x) = x+π
2
14 Suppose f (x) = x2 For what value(s) of x is the instantaneous rate of change of f
at x equal to the average rate of change of f on the specified interval? Illustrate your answers with graphs
(a) the interval [0, 3] (b) the interval [1, 4]
15 Let g(x) = 1x For what value(s) of x is the slope of the tangent line to g equal to the average rate of change of g on the interval indicated? Illustrate your answer to parts (a) and (b) with pictures
(a) [12, 2] (b) [1, 4] (c) [c, d] d > c >0
16 Let f (x) =x12 Let P and Q be points on the graph of f with coordinates (x, f (x)) and (x + x, f (x + x)), respectively
(a) Find the slope of the secant line through P and Q Simplify your answer as much
as possible
(b) By calculating the appropriate limit, find the slope of the tangent line to f (x) at point P
17 We showed that the derivative of √x
(or x1) is 12√1
x (or 12x−1) Here we focus on
f (x) =√x − 1
Trang 4194 CHAPTER 5 The Derivative Function
(a) How is the graph of√
x − 1 related to that of√x? (b) How is the graph of the derivative of√
x − 1 related to that of the derivative of
√
x? Illustrate with a rough sketch
(c) Given your answer to part (b) explain why d
dx
√ x
x=4= dxd √x − 1
x=5 In other words, explain why the derivative of√x
at x = 4 is equal to the derivative of
√
x − 1 evaluated at x = 5
(d) Show that f(5) =14using the limit definition of derivative:
f(5) = lim
x→5
f (x) − f (5)
(You’ll need to rationalize the numerator.)
18 Show that dxd √
x + 8 =2√1
x+8 using the limit definition of derivative You’ll use different versions of the definition in parts (a) and (b) In both cases it will be necessary
to rationalize the numerator in order to evaluate the limit
(a) f(x) = limh→0 f (x+h)−f (x)h (b) f(x) = limb→x f (b)−f (x)b−x
19 Let f (x) = x−1 Use the limit definition of derivative to show that f(x) = −12x−3
Key Notions
In this section and the following, we’ll interpret the derivative function fas the slope function Assertions will be made that should make sense intuitively However, we will not prove these assertions and “facts” here; proofs will be delayed until after Chapter 7
Fact:
Where f is increasing, the slope of its graph is nonnegative; where f is decreasing, its slope
is nonpositive.9
9 You may wonder why we say that f is increasing implies that the slope is nonnegative rather than just saying that the slope
is positive Consider a function like f (x) = x 3
It is increasing everywhere, yet its slope at x = 0 is zero; locally it looks like a horizontal line around (0, 0).
y y=x 3
x x
y
magnified picture
Trang 5Where f is increasing, f≥ 0; where f is decreasing, f≤ 0.
It follows that
where fis positive, the graph of f is increasing (from left to right);
where fis negative, the graph of f is decreasing (from left to right);
where fis zero, the graph of f locally looks like a horizontal line
Concavity is determined by whether fis increasing or decreasing
Where f>0 and fis increasing, the graph of f looks like Figure 5.16(a)
Where f>0 and fis decreasing, the graph of f looks like Figure 5.16(b):
Where f<0 and fis increasing, the graph of f looks like Figure 5.16(c):
Where f<0 and fis decreasing, the graph of f looks like Figure 5.16(d):10
Figure 5.16
Where the slope of f is increasing, we say the graph of f is concave up In other words,
where fis increasing the graph of f is concave up Figures 5.16(a) and (c) are examples
of concave-up graphs
concave up
Figure 5.17
Where the slope of f is decreasing, we say the graph of f is concave down In other
words, where fis decreasing, the graph of f is concave down Figures 5.16(b) and (d) are examples of concave-down graphs.11
concave down
Figure 5.18
10 If f < 0 and f is decreasing, then f is becoming increasingly negative as the independent variable increases 11
Trang 6196 CHAPTER 5 The Derivative Function
COMMON ERRORConsider the graph in Figure 5.16(c) A common error is to think that the slope is decreasing, because as x increases the drop is becoming more gentle This is incorrect
As the slope changes from, say, −2 to −1, the slope is increasing (becoming less
negative); similarly, if the temperature goes from −10 degrees to −5 degrees, we say that
the temperature is increasing As the slope changes from −2 to −1, the steepness of the line decreases, but the slope of the line increases Should you ever become confused, label the
slopes at a few points and put these slopes in order on a number line
Interpreting the Derivative Function Graphically
Let’s look qualitatively at the relationship between a function and its derivative function
Sketching the Derivative Function Given the Graph of f
The problem of sketching fwhen we are given the graph of f is equivalent to the problem
of sketching the velocity graph for a trip when we know the graph of position versus time When sketching the derivative function, begin with the most fundamental questions:
Where is fzero? undefined? positive? negative?
Note that, like any other function, fcannot change sign without passing through a point
at which it is either zero or undefined
COMMON ERRORA standard mistake of the novice derivative sketcher is to get mesmerized
by an internal conversation about where the slope is increasing and decreasing without first considering the question of whether the slope is positive or negative Keep your priorities
in order! Observations about the concavity of f are fine tuning and are not the first order
of business
On the following page are several worked examples Where f has a horizontal tangent line, fis zero Where f is discontinuous or has a sharp corner, fis undefined We mark the values where fis zero or undefined on a number line At the bottom we have tracked the sign of f; above the number line we have used arrows to indicate where the graph of f
is increasing and decreasing Where f is increasing, f≥ 0; the region above the x-axis on the graph of fis shaded to indicate this Where f is decreasing, f≤ 0; the region below the x-axis on the graph of fis shaded The graph of fmust lie in the shaded regions
Trang 7EXAMPLE 5.6 Given the graph of f , produce a rough sketch of f.
f
x
_
0
+
0
_
0
+
0
sign of f ' graph of f
f '
x
rough sketch
of f '
f '=0 at x=a, b, c, d
EXAMPLE 5.7 Given the graph of f , produce a rough sketch of f
f
sign of f '
graph of f
undefined
f '
rough sketch
of f '
f ' undefined
at x=a
Trang 8
198 CHAPTER 5 The Derivative Function
EXAMPLE 5.8 Given the graph of f , produce a rough graph of f
f
graph of f
undefined _
f '
x rough sketch of f '
f '=0 at x=a, c
f 'undefined at x=0=b
Getting Information About f Given the Graph of f
The problem of sketching f given the graph of fis equivalent to the problem of sketching the position versus time graph for a trip given the graph of velocity as a function of time Unless we know either where the trip began or the location at some particular time, we cannot produce the actual graph of the trip; we can only get the general shape of the trip graph without information as to position at any time
For instance, if we are told that a cyclist travels 12 miles per hour for 2 hours, we know that he has traveled 24 miles, but we have no idea at all as to where he started or finished his trip If we are given his position at some particular time, then we can piece together a picture of his position throughout the trip
Analogously, because fgives information only about the slope of f , we can simply obtain information about the shape of f , not its vertical position
To make a rough sketch of the shape of f given f, begin again with the most important features of f:
Where is fzero? undefined? positive? negative?
Where fis positive, the graph of f is increasing; where fis negative, the graph of f is decreasing; where fis zero, the graph of f locally looks like a horizontal line
Keep clear in your mind the difference between fbeing positive and being increasing When you look at a graph of fit is easy to get swept away by the shape of the graph Hold back! Sometimes the graph of fgives you more information than you can easily process
We suggest organizing the sign information on a number line Begin by asking yourself (in
a calm voice) “Where is fzero? Where is fpositive?”
Trang 9As pointed out in Chapter 2, sign information is a fundamental characteristic of a function; we see now that the sign of f gives vital information Begin by constructing
a number line highlighting this information This method is illustrated in the following example
EXAMPLE 5.9 Use the graph of fto answer the following questions
i Where on the interval [−3, 10] is the value of f the largest?
ii Which is larger, f (2) or f (5)?
iii Where on the interval [−3, 10] is f increasing most rapidly?
iv Where on the interval [−3, 10] is the graph of f concave up?
f ' (NOT f)
Figure 5.19
SOLUTION i Begin by constructing a number line with information about the sign of f and its
implications for the graph of f
–2
sign of f '
graph of f
7
x
Figure 5.20
From this analysis we see that f is largest at either x = −3 or x = 7
Since fis positive on [−2, 7] the graph of f is increasing on the entire interval The increase on the interval (−2, 7) is substantially greater than the decrease on (−3, −2)
so f (7) > f (−3)
f is largest at x = 7
ii f (5) > f (2) since f is increasing on [−2, 7]
iii f is increasing most rapidly where fis greatest This is at x = 3
Trang 10200 CHAPTER 5 The Derivative Function
iv f is concave up where f is increasing This is on the intervals [−3, 3], [5, 6], and
EXAMPLE 5.10 For each of the following, given the graph of f, sketch possible graphs of f Because f
gives us only information about the slope of f , there are infinitely many choices for f ; each
is a vertical translate of graphs given
The most constructive way to “read” this example is to do so actively Do the problems
on your own and then look at the solutions
x
1 2
SOLUTION
x
f '
x
f
graph of f sign of f ' +
2
(a)
x
graph of f sign of f ' _ 0 +
1 1
(b)
x x
graph of f sign of f ' + (c)