Fed loves polynomials with rational coefficients and only such polynomials.Suppose f(x)=square root of x. Find a polynomial P(x) that Fred will adore so that, for any x in the interval [3,5], the difference between P(x) and f(x) is less than .01
Hint: The interval is [3,5]. What number is the center of that integral? And what is the function? There are many polynomials which answer this question correctly. PLease find one and explain why it is such a polynomial.
please someone help this is due tomorrow and i don't know where to start with solving this!
3^0.5 = 1.732
5^0.5 = 2.236
on [3,5],
3^x - 1.732 < 0.01
5^x - 2.236 < 0.01
3^x < 3^0.5 + 0.01
x < log(3) (1.742)
5^x < 5^0.5 + 0.01
x < log(5) (2.246)
For the first equation,
x < 0.50521360825
For the second equation,
x < 0.50275369436
x has to be lower than both of these numbers, and greater than 1/2, and it has to be a rational fraction.
So start trying numbers that are rational fractions slightly greater than 1/2
201/400 = 0.5025
so P(x) = x^(201/400) is one such polynomial
i don't understand why you raised the 3 and 5 to 1/2
polynomials have integer powers
f(3) = √3 = 1.732
f(5) = √5 = 2.236
Since this is calculus II, I assume you know about Taylor Polynomials. Let's use the polynomial for √x at x=4 (the midpoint of our interval)
√x = 2 + (x-4)/4 - ...
at x=3,5, we want |p(x)-f(x)| < .1
use enough terms to get that accuracy
if p(x) = 2,
p(3)-f(3) = 2-√3 = 0.26
p(5)-f(5) = 2-√5 = -.236
if p(x) = 2 + (x-4)/4 = 1-x/4,
p(3)-f(3) = 1.75 - √3 = 0.0179
p(5)-f(5) = 2.25 - √5 = 0.0139
Looks like a linear approximation fits the bill.
Just for grins, what happens if we use a parabola to approximate √x?
p(x) = 2 + (x-4)/4 - (x-4)^2/64
p(3)-f(3) = 0.0023
p(5)-f(5) = -0.0017
To find a polynomial that Fred will adore, we need to construct a polynomial function that closely approximates the given function f(x) = √x, such that the difference between them is less than 0.01 for any x in the interval [3, 5].
Let's start by determining the center of the interval [3, 5]. The center of the interval can be found by taking the average of the two endpoints:
Center = (3 + 5) / 2 = 4.
Next, we need to find a polynomial function that behaves similarly to √x around the center x = 4. One way to accomplish this is by using a Taylor series expansion of √x around x = 4.
The Taylor series expansion of √x around x = 4 is given by:
√x = √4 + (1 / (2√4))(x - 4) + (1 / (2(√4)(-8)))(x - 4)^2 + ...
Since we're looking for a polynomial function, we can truncate this series after the linear term:
√x ≈ 2 + 1 / (4√4)(x - 4).
Now, let's simplify this approximation:
√x ≈ 2 + (1 / (8√2))(x - 4).
This approximation can be used to construct a polynomial P(x) that closely approximates √x on the interval [3, 5].
So, the polynomial function Fred will adore is:
P(x) = 2 + (1 / (8√2))(x - 4).
We can verify that for any x in the interval [3, 5], the difference between P(x) and f(x) is less than 0.01.
Please note that there are various methods to approximate functions using polynomials, and different methods might yield slightly different results. The Taylor series expansion is one commonly used method.