84 3. LIMITS, DERIVATIVES, RULES, AND THE MEANING OF THE DERIVATIVE
Problem 3.15 Using the notation that an empty circle indicates a missing point on a graph,
graph the following functions on the indicated interval.
1. f .x/ D
x
2
1
x C 1
on Œ2; 2
2. g.x/ D
x
3
8
x 2
on Œ3; 3
3. h.x/ D
x
4
1
x
2
1
on Œ2; 2
4. r.x/ D
x
2
25
x C 5
on Œ8; 2
5. s.x/ D
x
2
3
x
p
3
on Œ0; 2
6. q.x/ D
x
3
1
x 1
on Œ1; 3
3.2 DERIVATIVES
We mentioned earlier that the central question of this chapter is: What is the tangent line to a
function at a point? A derivative is the slope of that line. In order to compute the derivative,
we need to use what we learned about limits in Section 3.1. So, what is a tangent line?
A tangent line is a line that touches a curve at exactly one point—at least near that point.
e point is called the point of tangency. If the curve has a complex shape, then the tangent
line may intersect the curve somewhere else as well. But, in a neighborhood of the point of
tangency, it brushes the curve only once. e gray line in Figure 3.2 shows a line tangent to a
curve.
A secant line is a line through two points on a curve. Figure 3.3 shows examples of several
secant lines, all of which share one point—the point of tangency in the other picture.
is picture helps us to understand why we need limits to compute slopes of tangent lines. e
slopes of the secant lines are all computed based on the two points they pass through. e slope
of the tangent line is based on a single point—not possible to find using the slope formula for
lines. If we think of the slope of the tangent line as the limit of the slopes of secant lines from a
moving point to the point of tangency, then the limit as the moving point approaches the point
of tangency will be the slope of the tangent line.
3.2. DERIVATIVES 85
5
-1
-3 3
Figure 3.2: A function and a tangent line.
5
-1
-3 3
Figure 3.3: A function and several secant
lines.
f(x).
5
-1
-3 3
c
c+h
f(c)
f(c+h)
secant line
Figure 3.4: A function and a general secant line.
Suppose that the point of tangency is .c; f .c//, and that we examine the secant line through
that point and a point “just a little” to the right—the distance to the right being h. en,
the second point on the secant line is .c C h; f .c C h//, giving the situation shown in the
Figure 3.4. If we take the limit as h ! 0, then that limit should be the slope of the tangent
line. Applying the formula for the slope of a line using two points, we get that the slope of the
86 3. LIMITS, DERIVATIVES, RULES, AND THE MEANING OF THE DERIVATIVE
tangent line to f .x/ at x D c is:
lim
h!0
f .c C h/ f .c/
.c C h/ c
D lim
h!0
f .c C h/ f .c/
h
is formula is called the definition of the derivative and we have a special way of denoting it:
f
0
.c/.
Knowledge Box 3.1
e slope of the tangent line to f .x/ at the point .c; f .c// is
f
0
.c/ D lim
h!0
f .c C h/ f .c/
h
In the next example, we compute the slope of a tangent line and find the formula for that tangent
line.
Example 3.16 Find the tangent line to f .x/ D x
2
at the point (1,1).
Solution:
For this problem we have c D 1. To get the formula for the line we need a point and a
slope. We have the point (1,1) on the tangent line, so all we need to calculate is the slope.
lim
h!0
f .1 C h/ f .1/
h
D lim
h!0
.1 C h/
2
1
2
h
D lim
h!0
1 C 2h C h
2
1
h
D lim
h!0
2h C h
2
h
D lim
h!0
h.2 C h/
h
D lim
h!0
h.2 C h/
h
D lim
h!0
2 C h D 2
3.2. DERIVATIVES 87
Notice that this limit is one that requires algebraic manipulation to resolve. We could not just
plug h D 0 into
2h C h
2
h
because that yields
0
0
. All tangent-slope calculations yield limits that
require algebraic manipulation—explaining the emphasis on this type of limit in the previous
section. We now have the point .1; 1/ and a slope of m D 2. e line is thus y 1 D 2.x 1/
or y D 2x 1.
Let’s conclude by graphing the function and its tangent line at c D 1.
7
-1
-3.0 3.0
Tangent to f .x/ D x
2
at (1,1).
˙
We now know how to find the slopes of tangent lines at specific values x D c. It would be nice
to have a general function for the derivative. We define the general derivative of a function as
follows.
Knowledge Box 3.2
e general derivative (or derivative) of f .x/ is
f
0
.x/ D lim
h!0
f .x C h/ f .x/
h
88 3. LIMITS, DERIVATIVES, RULES, AND THE MEANING OF THE DERIVATIVE
Example 3.17 Compute the derivative of f .x/ D
1
x
.
Solution:
f
0
.x/ D lim
h!0
f .x C h/ f .x/
h
D lim
h!0
1
x C h
1
x
h
D lim
h!0
x
x.x C h/
x C h
x.x C h/
h
D lim
h!0
x .x C h/
x.x C h/
h
D lim
h!0
h
x.x C h/
h
D lim
h!0
h
x.x C h/
h
D lim
h!0
1
x.x C h/
D
1
x
2
So, for f .x/ D
1
x
we have that f
0
.x/ D
1
x
2
.
˙
e quantity
f .x C h/ f .x/
h
is called the difference quotient for f .x/. We can say that the derivative of a function is the
limit of the difference quotient as h ! 0.
Example 3.18 Find the derivative of f .x/ D x
n
.
..................Content has been hidden....................
You can't read the all page of ebook, please click here login for view all page.