EXAMPLE 1 Find an equation of the tangent line to the function y = 3x at the point P(1, 3). SOLUTION We will be able to find an equation of the tangent line t as soon as we know i slope m. The difficulty is that we know only one point, P. on t, whereas we need two points to compute the slope. But observe that we can compute an approximation to m by choosing a nearby point Q(x, 3x) on the graph (as in the figure) and computing the slope mpo of the secant line PQ. [A secant lin from the Latin word secans, meaning cutting, is a line that cuts (intersects) a curve more than once.] We choose x *1 so that Q+ P. Then, x - 1 For instance, for the point Q(1.5, 6.75) we have Video Example mpo = 1-1 The tables below the values of mpg for several values of x close to 1. The closer Q is to P. the closer x is to 1 and, it appears from the tables, the closer mpg is to . This suggests that slope of the tangent line t should be m = . mpo x 3 1.5 7.5 .5 4.5 5.700 1.01 6.030 .99 5.970 1.001 6.003 .999 5.997 1.1 6.3 We say that the slope of the tangent line is the limit of the slopes of the secant lines, and we express this symbolically by writing lim mpo =m and lim 3x - 3 = . Assuming that this is indeed the slope of the tangent line, we use the point-slope form of the equation of a line (see Appendix B) to write the equation of the tangent line through (1, 3) as y- O-D *- 1) y - Ox - D- or The graphs below illustrate the limiting process that occurs in this example. As Q approaches Palong the graph, the corresponding secant lines rotate about Pand apPproach the tangent line t. QxpprechesPoerige e Pinn the le
EXAMPLE 1 Find an equation of the tangent line to the function y = 3x at the point P(1, 3). SOLUTION We will be able to find an equation of the tangent line t as soon as we know i slope m. The difficulty is that we know only one point, P. on t, whereas we need two points to compute the slope. But observe that we can compute an approximation to m by choosing a nearby point Q(x, 3x) on the graph (as in the figure) and computing the slope mpo of the secant line PQ. [A secant lin from the Latin word secans, meaning cutting, is a line that cuts (intersects) a curve more than once.] We choose x *1 so that Q+ P. Then, x - 1 For instance, for the point Q(1.5, 6.75) we have Video Example mpo = 1-1 The tables below the values of mpg for several values of x close to 1. The closer Q is to P. the closer x is to 1 and, it appears from the tables, the closer mpg is to . This suggests that slope of the tangent line t should be m = . mpo x 3 1.5 7.5 .5 4.5 5.700 1.01 6.030 .99 5.970 1.001 6.003 .999 5.997 1.1 6.3 We say that the slope of the tangent line is the limit of the slopes of the secant lines, and we express this symbolically by writing lim mpo =m and lim 3x - 3 = . Assuming that this is indeed the slope of the tangent line, we use the point-slope form of the equation of a line (see Appendix B) to write the equation of the tangent line through (1, 3) as y- O-D *- 1) y - Ox - D- or The graphs below illustrate the limiting process that occurs in this example. As Q approaches Palong the graph, the corresponding secant lines rotate about Pand apPproach the tangent line t. QxpprechesPoerige e Pinn the le
Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
6th Edition
ISBN:9781337111348
Author:Bruce Crauder, Benny Evans, Alan Noell
Publisher:Bruce Crauder, Benny Evans, Alan Noell
Chapter2: Graphical And Tabular Analysis
Section2.4: Solving Nonlinear Equations
Problem 17E: Van der Waals Equation In Exercise 18 at the end of Section 2.3, we discussed the ideal gas law,...
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![EXAMPLE 1
Find an equation of the tangent line to the function y = 3x at the point P(1, 3).
SOLUTION We will be able to find an equation of the tangent line t as soon as we know its slope m. The difficulty is that we know only one point, P, on t, whereas we need two points to compute the
slope. But observe that we can compute an approximation to m by choosing a nearby point Q(x, 3x) on the graph (as in the figure) and computing the slope mpo of the secant line PQ. [A secant line,
from the Latin word secans, meaning cutting, is a line that cuts (intersects) a curve more than once.]
We choose x*1 so that Q + P. Then,
E-E = Odu
For instance, for the point Q(1.5, 6.75) we have
Video Example )
mpQ =
The tables below show the values of meo for several values of x close to 1. The closer Q is to P, the closer x is to 1 and, it appears from the tables, the closer meo is to
. This suggests that the
slope of the tangent line t should be m =
mpo
2
9
3
1.5
7.5
.5
4.5
1.1
6.3
.9
5.700
1.01 6.030.99 5.970
|1.001 6.003 .999 5.997
We say that the slope of the tangent line is the limit of the slopes of the secant lines, and we express this symbolically by writing
3x - 3
lim
1 x-1
lim mpo =m
and
=
Assuming that this is indeed the slope of the tangent line, we use the point-slope form of the equation of a line (see Appendix B) to write the equation of the tangent line through (1, 3) as
(x - 1)
y =
x -
y -
or
The graphs below illustrate the limiting process that occurs in this example. As Q approaches P along the graph, the corresponding secant lines rotate about Pand approach the tangent line t.
Qappecaches Pkom the right
Q approaches Pfron the left](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa2574aa9-02ae-4c85-9a07-6af47bfea014%2F417412d4-2bad-46d0-a90b-6277bba89d8b%2F7e4ejqv_processed.png&w=3840&q=75)
Transcribed Image Text:EXAMPLE 1
Find an equation of the tangent line to the function y = 3x at the point P(1, 3).
SOLUTION We will be able to find an equation of the tangent line t as soon as we know its slope m. The difficulty is that we know only one point, P, on t, whereas we need two points to compute the
slope. But observe that we can compute an approximation to m by choosing a nearby point Q(x, 3x) on the graph (as in the figure) and computing the slope mpo of the secant line PQ. [A secant line,
from the Latin word secans, meaning cutting, is a line that cuts (intersects) a curve more than once.]
We choose x*1 so that Q + P. Then,
E-E = Odu
For instance, for the point Q(1.5, 6.75) we have
Video Example )
mpQ =
The tables below show the values of meo for several values of x close to 1. The closer Q is to P, the closer x is to 1 and, it appears from the tables, the closer meo is to
. This suggests that the
slope of the tangent line t should be m =
mpo
2
9
3
1.5
7.5
.5
4.5
1.1
6.3
.9
5.700
1.01 6.030.99 5.970
|1.001 6.003 .999 5.997
We say that the slope of the tangent line is the limit of the slopes of the secant lines, and we express this symbolically by writing
3x - 3
lim
1 x-1
lim mpo =m
and
=
Assuming that this is indeed the slope of the tangent line, we use the point-slope form of the equation of a line (see Appendix B) to write the equation of the tangent line through (1, 3) as
(x - 1)
y =
x -
y -
or
The graphs below illustrate the limiting process that occurs in this example. As Q approaches P along the graph, the corresponding secant lines rotate about Pand approach the tangent line t.
Qappecaches Pkom the right
Q approaches Pfron the left
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