Approximate using derivatives. (Must show work using derivatives!) Do not use a calculator:
(1.1)^4 - (1.1)^2
let f(x) = x^4 - x^2
df = (4x^3 - 2x) dx
let x=1, dx=0.1
df ≈ (4-2)(0.1) = 0.2
f(x+dx) ≈ f + df = 0 + 0.2
real value: 0.2541
To approximate the value of the expression (1.1)^4 - (1.1)^2 using derivatives, we can use the concept of the derivative as the instantaneous rate of change of a function.
Let's start by considering the function f(x) = x^4 - x^2. We want to find the value of f(1.1)^4 - f(1.1)^2, which means we need to evaluate f(x) at x = 1.1.
First, let's find the derivative of f(x). The derivative of x^4 with respect to x is 4x^3, and the derivative of x^2 is 2x. Therefore, the derivative of f(x) = x^4 - x^2 is f'(x) = 4x^3 - 2x.
Next, we can calculate the value of the derivative at x = 1.1. Plugging in x = 1.1 into f'(x), we get:
f'(1.1) = 4(1.1)^3 - 2(1.1)
= 4(1.331) - 2(1.1)
= 5.324 - 2.2
= 3.124
Now, we can use the derivative to approximate the change in the function f(x) over a small interval around x = 1.1. This can be done using the linear approximation formula:
Δf ≈ f'(1.1) * Δx
Since we want to approximate the value of f(1.1)^4 - f(1.1)^2, Δf corresponds to the change in the function f(x) over a small interval around x = 1.1, and Δx represents this small interval.
To choose an appropriate Δx, we can let Δx be a small change in x, such as 0.1. Therefore, Δx = 0.1.
Now, we can calculate the approximate change in f(x):
Δf ≈ f'(1.1) * Δx
≈ 3.124 * 0.1
≈ 0.3124
Finally, to approximate the value of (1.1)^4 - (1.1)^2 using derivatives, we can find an initial approximation by evaluating f(1.1), and then adjusting it by adding the approximate change we calculated:
Approximation ≈ f(1.1) + Δf
≈ (1.1)^4 - (1.1)^2 + 0.3124
Evaluating this expression gives us the approximate value of (1.1)^4 - (1.1)^2 using derivatives.