The linear approximation at x=0 to f(x)= sqrt (7+6x) is y=

for f(x)=sqrt(7+6x)

Calculate the slope at x0=0, i.e.
m=f'(x0)=f'(0)
and the value of y0=f(x0) to give the tangent point (x0, f(x0))=(0,f(0))

The linear approximation at x=0 is given by
y = m(x-x0)+y0
If you have difficulties finding the derivative of f(x), post for more information.

To find the linear approximation of the function f(x) = √(7+6x) at x=0, we can follow these steps:

Step 1: Find the derivative of f(x).
The derivative of f(x) = √(7+6x) can be found using the chain rule. Let's denote g(x) = 7 + 6x, then f(x) = √g(x).
Applying the chain rule:
f'(x) = [(1/2) * (g(x))^(-1/2)] * g'(x) = (1/2) * (7+6x)^(-1/2) * 6

Step 2: Evaluate the derivative at x=0.
Substitute x=0 into the derivative we found in step 1:
f'(0) = (1/2) * (7+6*0)^(-1/2) * 6 = (1/2) * 7^(-1/2) * 6 = 3/2√7

Step 3: Write the equation of the linear approximation.
The linear approximation of f(x) at x=0 can be written using the point-slope form as:
y - f(0) = f'(0) * (x - 0)

Since x=0, f(0) can be calculated by substituting it into the original function:
f(0) = √(7+6*0) = √7

Rewriting the equation of the linear approximation:
y - √7 = (3/2√7) * x

Therefore, the linear approximation at x=0 to f(x) = √(7+6x) is y - √7 = (3/2√7) * x.