Answer:
y = (1/4)x + 7/4
Explanation:
To find the linearization of f(x) = √(x+3) at a=1, we need to follow these steps:
Find the first derivative of f(x) with respect to x:
f'(x) = 1 / (2√(x+3))
Evaluate f(1) to find the y-coordinate of the point where we want to find the linearization:
f(1) = √4 = 2
Evaluate f'(1) to find the slope of the tangent line at the point (1, f(1)):
f'(1) = 1 / (2√4) = 1/4
Use the point-slope form of the equation of a line to write the equation of the tangent line at (1, 2):
y - 2 = (1/4)(x - 1)
Simplify the equation of the tangent line:
y = (1/4)x + 7/4
This is the linearization of f(x) = √(x+3) at a=1.