# What is the local linearization of #e^sin(x)# near x=1?

##### 1 Answer

# f(x) ~~ 1.2534x+1.0664 #

#### Explanation:

Let:

# f(x) = e^(sinx) #

The linear approximation of a function **linear terms** of the Taylor Series about the point

# f(x) ~~ f(a) + f'(a)(x-a) #

Differentiating

# f'(x) = e^(sinx) (cosx) #

So with

# f(1) = e^(sin1) #

# " " = 2.3197768 ... #

# f'(1) = e^(sin1) (cos1) #

# " " = 1.2533897 ... #

Hence, the linear approximation near

# f(x) ~~ 2.3198 + 1.2534(x-1) #

# " " = 2.3198 + 1.2534x-1.2534 #

# " " = 1.2534x+1.0664 #

Where we have rounded to 4dp.

**Example:**

Consider the case

# f(1.01) = e^(sin(1.01)) #

# " " = 2.332246 ... #

And the linear approximation gives us:

# f(1.01) ~~ 1.2534(1.01)+1.0664 #

# " " = 1.265934+1.0664 #

# " " = 2.332334 #

Which is correct within