Question

Approximate 19 ¹/⁴ using the Linear Approximation L(x) of f(x)=x¹/⁴ at a=16. (Use symbolic notation and fractions where needed.) approximation based on linearization:

Answer 1

To approximate 19 ¹/⁴ using the linear approximation L(x) of f(x)=x¹/⁴ at a=16, we can use the formula for linear approximation: L(x) = f(a) + f'(a)(x-a). Plugging in the values, the approximation is approximately 2.328.

Step-by-step explanation:

To approximate 19 ¹/⁴ using the linear approximation L(x) of f(x)=x¹/⁴ at a=16, we can use the formula for linear approximation:

L(x) = f(a) + f'(a)(x-a)

First, let's find f(a), which is f(16). Since f(x) = x¹/⁴, f(16) = 16¹/⁴ = 2.

Next, let's find f'(a), the derivative of f(x). The derivative of x¹/⁴ is (1/4)x^(-3/4). Evaluating at x=a=16, we get f'(16) = (1/4)(16)^(-3/4) = 1/32.

Now we can plug these values into the formula: L(x) = 2 + (1/32)(x-16). To approximate 19 ¹/⁴, substitute x=19 ¹/⁴ into L(x): L(19 ¹/⁴) = 2 + (1/32)(19 ¹/⁴ - 16).

Calculating the above expression, we get approximately 2.328.