Find the linearization L(x) of the function at a.
f(x) = x3/4, a = 81
To find the linearization of a function at a given point, we need to use the concept of tangent lines. The linearization of a function f(x) at a point a is given by the equation:
L(x) = f(a) + f'(a)(x - a)
where f(a) is the value of the function at a, and f'(a) is the derivative of the function evaluated at a.
In this case, the function is f(x) = x^(3/4), and the point of interest is a = 81.
Step 1: Calculate f(a)
To find f(a), substitute the value of a into the function:
f(a) = a^(3/4)
= 81^(3/4)
Step 2: Calculate f'(a)
To find f'(a), we need to differentiate the function with respect to x and then evaluate it at a.
Let's find the derivative of f(x) = x^(3/4):
f'(x) = (3/4)x^(-1/4)
Now, evaluate f'(a):
f'(a) = (3/4)(a^(-1/4))
= (3/4)(81^(-1/4))
Step 3: Substitute the values into the linearization formula
Using the linearization formula, substitute the values of f(a) and f'(a):
L(x) = f(a) + f'(a)(x - a)
= 81^(3/4) + (3/4)(81^(-1/4))(x - 81)
Simplifying the equation further may require evaluating the values of 81^(3/4) and 81^(-1/4) if needed.
To find the linearization of a function at a particular point, we can use the formula:
L(x) = f(a) + f'(a)(x-a)
1. Find f(a):
Plug the value of a into the function f(x) = x^(3/4):
f(a) = a^(3/4)
Substitute a with 81:
f(a) = 81^(3/4)
2. Find f'(x):
Differentiate the function f(x) = x^(3/4) with respect to x:
f'(x) = (3/4)x^(-1/4)
3. Find f'(a):
Plug the value of a into the derivative f'(x):
f'(a) = (3/4)a^(-1/4)
Substitute a with 81:
f'(a) = (3/4)(81)^(-1/4)
4. Write the linearization equation:
L(x) = f(a) + f'(a)(x-a)
Substitute the values we found:
L(x) = 81^(3/4) + (3/4)(81)^(-1/4)(x-81)
Simplify the expression if necessary.