# Using a linear approximation, estimate f(2.1), given that f(2) = 5 and f'(x) = square root 3x-1.

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### 2 Answers

Linear approximation is a way of finding the value of the function for a variable as follows: f(x) `~~` f(a) + f'(a)*(x - a)

Here we have f(2) = 5 and f'(x) = `sqrt (3x - 1)`

f(2.1) = f(2) + f'(2)*(2.1 - 2)

=> f(2.1) = 5 + f'(2)*0.1

=> f(2.1) = 5 + `sqrt(3*2 - 1)*0.1`

=> f(2.1) = 5 + 0.1*`sqrt 5`

=> f(2.1) = 5 + 0.2236

=> f(2.1) = 5.2236

**The required value of f(2.1) = 5.2236**

For any function f(x), linear approxiamtion would result the following.

f(x+dx) = f(x)+f '(x) * dx.

In this example, x = 2 and dx is 0.1.

f '(x) = sqrt(3*x -1) = sqrt(3*2 -1) = sqrt(5) = 2.236

therefore,

f(2.1) = f(2+0.1) = f(2) + f '(2)*0.1

f(2.1) = 5 + 2.236*0.1

**f(2.1) = 5.2236.**