233k views
5 votes
Find the linear approximation

L(x)
to
y = f(x)
near
x = a
for the function.
f(x) =
1
x
, a = 3

User TheRusskiy
by
7.8k points

1 Answer

7 votes

Answer: Down below is a answer that is contributed to by many online sources which you can check out at chegg-.com

Explanation:

To find the linear approximation, L(x), to the function f(x) = 1/x near x = a = 3, we can use the concept of linearization. The linear approximation can be represented as:

L(x) = f(a) + f'(a)(x - a)

First, let's find f(a) and f'(a).

f(a) = 1/a = 1/3

To find f'(a), we need to differentiate f(x) = 1/x. Using the power rule, we have:

f'(x) = -1/x^2

Substituting x = a = 3, we find:

f'(a) = -1/(3^2) = -1/9

Now, we can construct the linear approximation:

L(x) = f(a) + f'(a)(x - a)

= 1/3 - (1/9)(x - 3)

= 1/3 - (1/9)x + 1/3

Simplifying, we get:

L(x) = (2/3) - (1/9)x

Therefore, the linear approximation L(x) to f(x) = 1/x near x = 3 is given by L(x) = (2/3) - (1/9)x.

User Joe Sloan
by
8.5k points

No related questions found