Approximate the following value using derivatives (without a calculator)
(1.1)^4 - (1.1)^2
I only know how to do linear approximation when using square roots. How do I do this type of problem?
The final answer would be 1.21
First subtract:
(1.1^4) - (1.1^2) = 1.1^2
then solve:
(1.1)^2 = 1.21
How it works:
If you have x^4 - x^2 and the value of x is the same, just subtract the exponents. x^4 - x^2 = x^2
Google search exponent and go to mathisfun for more info and practice.
In this case 1.1^4 is just 1.1 * 1.1 * 1.1 * 1.1
so you can subtract 1.1^2 which is 1.1 * 1.1 from that, and get 1.1 * 1.1 .
1.1 * 1.1 = 1.21
Good Luck!
P.S. Can't edit here..
The exponent of a number says how many times to use the number in a multiplication of itself.
To approximate the value of an expression using derivatives, you can use Taylor's theorem. However, for this specific problem, you don't need to go that far. We can simply use linear approximation, similar to what you've mentioned for square roots.
We can start by finding the derivative of the function f(x) = x^4 - x^2. Using the power rule, the derivative is f'(x) = 4x^3 - 2x.
Next, we need to choose a value close to 1.1 to use for the approximation. Let's choose x = 1. If we substitute x = 1 into f(x) and f'(x), we get f(1) = 1^4 - 1^2 = 0 and f'(1) = 4(1)^3 - 2(1) = 2.
Now, we can use the linear approximation formula:
Approximation = f(1.1) ≈ f(1) + f'(1) * (1.1 - 1)
Substituting the known values, we have:
f(1.1) ≈ 0 + 2 * (1.1 - 1)
Simplifying, we get:
f(1.1) ≈ 2 * 0.1
f(1.1) ≈ 0.2
So, approximately, (1.1)^4 - (1.1)^2 is equal to 0.2.