Local Linearization
(Tangent Line at a point)
When the derivative of a function y=f(x) at a point x=a
exists, it guarantees the existence of the tangent line at that
point.
Nearby the point of tangency, the graph of f(x) looks like the
tangent line at the point (a, f(a)).
This result is expressed by saying that nearby x=a, the
values of f(x) are approximately the same as the values of
the tangent line at x=a, or that
f (x)
Lim = "1"
x®a Tangent Line at x = a
For points nearby x = a, f (x) » YTangent Line at x = a
2
Graphs of f(x) and Its Tangent Line
nearby x=2
In the following slides look at the difference between the
values of f(x) and the values of the tangent line at x=2 for
values of x “close” to 2
3
What is the largest distance between the function and its
tangent line? __________
What is the largest error made if the tangent line is used to
estimate values of the function? _______
The estimates using the tangent line are under or overestimate?
_____________ How do you know?
4
8
6
4
2
0
1.0
1.5
2.0
2.5
3.0
What is the largest error made if the tangent line is
used to estimate values of the function? _______
5
4.5
4.0
3.5
1.8
1.9
2.0
2.1
2.2
What is the largest error made if the tangent line is
used to estimate values of the function? _______
6
4.2
4.1
4.0
3.9
3.8
1.96
1.98
2.00
2.02
2.04
What happens with the error made if the tangent
line is used to estimate values of the function?
7
Estimating Values Using the Tangent
Line
Use linearization of y=x2 to estimate
o 3.32
o 2.52
o 4.72
8
For 3.32 use the the tangent line at the point (3,9)
Equation of tangent line at (3,9) is _____________________
Estimate the value of the tangent line at x=3.3
Is it an overestimate/underestimate? Use the sign of y”(3) to determine
whether the tangent line is above or below y=x2 nearby x=3
Do the other estimates
9
Exercise
Estimate the value of 2.83 using linear
approximations
• Choose a function to work (from the basic ones)
• Choose the point at which you want to find the
tangent line line (easy to work with)
• Find the linearization of the function at that
point and use it to estimate your answer
• Is your answer an over/underestimate?
10