#### To determine

**(a)**

**To find:**

Linear approximation of f(x)=lnx near x=1

#### Answer

The approximation for the function f(x)=lnx near x=1 is given by y=x−1.

#### Explanation

We may approximate f(x)=lnx nearby its tangent line. This is linear approximation.

Clearly when x=1, we have f(1)=ln1=0

Differentiating f(x) with respect to x, we get f′(x)=1x (Since ddx(lnx)=1x)

Equation of the tangent line using point slope form is given by

y−y1=m(x−x1)

Here x1=1,y1=0,m=f′(1)=11=1

Therefore y−0=1(x−1)

Hence y=x−1

#### To determine

**(b)**

**To illustrate:**

Part (a) by graphing f and its linearization.

#### Answer

Graph of the function f(x)=lnx and its linear approximation y=x−1 is given below

#### Explanation

Graph of the function f(x)=lnx and its linear approximation y=x−1 is given below

#### To determine

**(c)**

**To state:**

The values of x for which the linear approximation is accurate within 0.1.

#### Answer

0.61 <x< 1.51

#### Explanation

From the diagram in b) we get to see that linear approximation is accurate within 0.1 when x is between 0.61 and 1.51.

y=x−1 is represented in green, y=lnx−0.1 in red and y=lnx+0.1 in blue