Using a linear approximation or differentials, approximate:(26.98)^(3/4)
thank you for your help!
f(x) = x^(3/4)
df = 3/4 x^(-1/4) dx
so, if we let x=27 and dx = -0.02,
df = (3/4) 27^(-1/4) (-.02) = -.00658
So, f(26.98) ≈ f(27) - .00658
= 11.84467 - 0.00658 = 11.83809
f(26.98) is really 11.838085
To approximate the value of (26.98)^(3/4) using linear approximation or differentials, we can start by finding the linearization of the function f(x) = x^(3/4) at a "nice" point near 26.98.
The linearization of a function f(x) at the point x = a is given by:
L(x) = f(a) + f'(a)(x - a)
First, let's find the value of f(a) at an approximate value for a. Let's choose a = 27, since it is close to 26.98.
f(27) = 27^(3/4)
Next, we need to find the derivative f'(x) of the function f(x). Taking the derivative of f(x) = x^(3/4) gives us:
f'(x) = (3/4)x^(-1/4)
Now, we can find the value of f'(27):
f'(27) = (3/4)(27)^(-1/4)
Finally, we substitute these values into the linearization formula:
L(x) = f(27) + f'(27)(x - 27)
Using the linear approximation formula, we can approximate (26.98)^(3/4) as L(26.98):
L(26.98) = f(27) + f'(27)(26.98 - 27)
To find the approximate value, evaluate the expression:
L(26.98) = [27^(3/4)] + [(3/4)(27)^(-1/4)](26.98 - 27)
Simplify this expression to get the final approximation for (26.98)^(3/4).