I'm confused on how to do this.
1.) Show that f(x)=x^4 -2x +8 has nonzero constant fourth-order differences.
SIMPLIFY THE EXPRESSION: (HOW?)
2.) 8^log(8)^x
eight log root 8 ^x (here it is written if you can't understand the actual expression. It is definitely hard typing this out cuz I do not know if you could understand this/ to whomever answers.)
1)
x y d1 d2 d3 d4
0 8
1 7 -1
2 20 13 14
3 83 63 50 36 24
4 256 173 110 60 24
5 623 367 194 84 24
6 1292 669 302 108 24
Note how the 4th difference column is all 24 - 4th derivative of x^4 = 24
8^log_8(x) = x
exponentiation and logs are inverse operations.
b^log_b(x) = log_b(b^x) = x
the definition of log(N) is the power you need to get N, whatever the base.
and that's "to whoever answers". "whoever answers" is a noun phrase, used (as a unit) as the object of the preposition. A common mistake.
1.) To show that f(x)=x^4 - 2x + 8 has nonzero constant fourth-order differences, you need to compute the differences of the function several times and check if they are all nonzero.
Here is the step-by-step process to compute the fourth-order differences:
Step 1: Compute the first-order differences.
Take the differences between consecutive terms of the function f(x). This can be done by subtracting each term from the term that follows it. For example:
f(x) = x^4 - 2x + 8
f(x+1) = (x+1)^4 - 2(x+1) + 8
Now, compute the first-order differences by subtracting f(x) from f(x+1):
First-order differences = f(x+1) - f(x) = [(x+1)^4 - 2(x+1) + 8] - [x^4 - 2x + 8]
Step 2: Compute the second-order differences.
Take the differences between consecutive terms of the first-order differences. This can be done by subtracting each term from the term that follows it.
Second-order differences = First-order differences (x+1) - First-order differences (x)
Step 3: Repeat the process for the third and fourth-order differences.
Continue taking the differences between consecutive terms of the previous order differences until reaching the desired order (in this case, the fourth-order differences).
If all the fourth-order differences are nonzero, then the function f(x) has nonzero constant fourth-order differences.
2.) The expression 8^log(8)^x is slightly unclear. Let's break it down step by step to better understand it.
First, let's address the part log(8)^x. Since there are no parentheses, we'll assume that the base of the logarithm is 10. In that case, log(8)^x means taking the logarithm of 8 (base 10) and then raising it to the power of x. So log(8)^x can be written as (log10(8))^x.
Now, we have the expression 8^((log10(8))^x). To simplify this expression, we can simplify the inner part first. Since log10(8) is a constant, let's denote log10(8) as a.
So the expression becomes 8^(a^x).
To simplify further, we need to use the properties of exponents. One important property is that (a^b)^c is equal to a^(b*c).
Applying this property to our expression, 8^(a^x) can be rewritten as (8^a)^x.
Now, let's substitute log10(8) back in for a. We can use a calculator to find the value of log10(8) and then substitute it in.
Finally, we have the simplified expression: (8^(log10(8)))^x.
Using a calculator, we can find that log10(8) is approximately 0.9031.
Thus, the simplified expression is (8^0.9031)^x.
Keep in mind that without parentheses to clarify the base of the logarithm, the interpretation of the expression might change. In any case, the steps I provided should help you understand how to simplify expressions involving logarithms and exponential functions.