The linearization is .
The approximation is .
In this lesson we will use the tangent line to approximate the value of a function near the point of tangency.
Given a function , the equation of the tangent line at the point where is given by or The main idea of this section is that if we let then and for values of close to . The function is called the linearization of at . The advantage of working with is that values of a linear function are usually easy to compute. In a typical linear approximation problem, we are trying to approximate a value of . We need to choose and create . Once we have accomplished this, our solution is There are two keys to choosing . First should be close to the -value of interest and second, we must be able to compute the exact value of .
example 1 Approximate using a linear approximation.
Let and since 4 is near 5 and , we let . To create we also need to compute . Since we have
Let and since 4 is near 5 and , we let . To create we also need to compute . Since we have
The linearization of at is now given by
Finally, if is near 4, then , so
example 2 Approximate using linear approximation.
Let and since 100 is near 99 and , we let . To create we also need to compute . Since we have
Let and since 100 is near 99 and , we let . To create we also need to compute . Since we have
The linearization of at is now given by
Finally, if is near 100, then , so(problem 2a) Find the linearization of at and use it to approximate .
(problem 2b) Find the linearization of at and use it to approximate .
The linearization is .
The approximation is .
example 3 Approximate using linear approximation.
Let and since 10 is near 8 and , we let . To create we also need to compute . Since we have
Let and since 10 is near 8 and , we let . To create we also need to compute . Since we have
Next,
and finally,(problem 3) Find the linearization of at and use it to approximate .
The linearization is .
The approximation is .
example 4 We can approximate using linear approximation as follows. Let and since
0 is near and , we let . To create we also need to compute . Since we have
Next,
and finally,We conclude this example by computing relative error. Expressed as a percentage, the relative error is given by For the actual value, we will use the value of from a calculator. We have
(problem 4) Find the linearization of at and use it to approximate . Also, find the
relative error.
The linearization is .
The approximation is .
The relative error in this approximation is (to two decimal places) .
example 5 Approximate using linear approximation.
Let and since 1 is near 1.1 and , we let . To create we also need to compute . Since we have
Let and since 1 is near 1.1 and , we let . To create we also need to compute . Since we have
Next,
and finally,(problem 5) Find the linearization of at and use it to approximate .
The linearization is .
The approximation is .
example 6 Approximate using linear approximation.
Let and since 2 is near 1.9 and , we let . To create we also need to compute . Since we have
Let and since 2 is near 1.9 and , we let . To create we also need to compute . Since we have
Next,
and finally,(problem 6) Find the linearization of at and use it to approximate .
The linearization is .
The approximation is .
example 7 Approximate using linear approximation.
In the formula it is understood that the angle is measured in radians. Therefore, in order to use our linear approximation formula we need to restate our problem in radians as:
In the formula it is understood that the angle is measured in radians. Therefore, in order to use our linear approximation formula we need to restate our problem in radians as:
We let and since 0 is near and , we let . To create we also need to compute . Since we have
Next,
and finally,
Here is a detailed, lecture style video on linear approximation:
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