Suppose f(x)=ln x.

(a) Find the linear approximation of f at a=1.
L(x)=

(b) Use the linear approximation to estimate ln 1.27.
ln1.27=

y = ln x

m = dy/dx = 1/x
at x = 1, m = 1/1 = 1
so
y = x + b
at x = 1, y = ln 1 = 0
so
0 = 1*1 + b
so
b = -1
so
L(x) = x -1
and then i believe you would just plug in ln(1.27) for x. good luck

To find the linear approximation of f(x) = ln(x) at a = 1, we can use the formula for linear approximation:

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

(a) First, we need to find f'(x), the derivative of f(x).

f(x) = ln(x)

Using the logarithmic differentiation rule, we can differentiate f(x) with respect to x:

f'(x) = 1/x

Now, we substitute a = 1 into f(x) and f'(x):

f(1) = ln(1) = 0
f'(1) = 1/1 = 1

Substituting these values into the linear approximation formula:

L(x) = f(1) + f'(1) * (x - 1)
L(x) = 0 + 1 * (x - 1)
L(x) = x - 1

So, the linear approximation of f(x) at a = 1 is L(x) = x - 1.

(b) To estimate ln(1.27) using the linear approximation, we substitute x = 1.27 into the linear approximation equation:

L(x) = x - 1
L(1.27) = 1.27 - 1
L(1.27) = 0.27

Therefore, ln(1.27) is approximately equal to 0.27.

To find the linear approximation of a function at a specific point, we can use the tangent line to the function at that point. The equation of the tangent line is in the form of y = f(a) + f'(a)(x - a), where f(a) is the value of the function at a, f'(a) is the derivative of the function at a, and (x - a) represents the difference between x and a.

(a) To find the linear approximation of f(x) = ln(x) at a = 1, we need to find the value of f(1) and f'(1).

1. Finding f(1):
Substitute x = 1 into the function f(x) = ln(x):
f(1) = ln(1) = 0

2. Finding f'(1):
To find the derivative of f(x) = ln(x), we use the chain rule:
f'(x) = 1/x

Substitute x = 1 into f'(x):
f'(1) = 1/1 = 1

Now we have f(1) = 0 and f'(1) = 1.

Plugging these values into the equation of the tangent line, we get:
L(x) = f(a) + f'(a)(x - a)
L(x) = 0 + 1(x - 1)
L(x) = x - 1

Therefore, the linear approximation of f(x) = ln(x) at a = 1 is L(x) = x - 1.

(b) Using the linear approximation to estimate ln(1.27):
We can substitute x = 1.27 into the linear approximation L(x) = x - 1:
L(1.27) = 1.27 - 1
L(1.27) = 0.27

Therefore, ln(1.27) can be estimated to be approximately 0.27.