Which of the following functions grows the fastest as x goes to infinity? (2 points)
f(x) = ln(x)
f(x) = log2(x)
f(x) = x2
f(x) = 2x
To determine which function grows the fastest as x goes to infinity, we can compare the growth rates by examining their derivatives. The function with the largest derivative at infinity will be the one that grows the fastest.
Let's find the derivatives of each function:
1. f(x) = ln(x)
The derivative of ln(x) is 1/x.
2. f(x) = log2(x)
The derivative of log2(x) is 1/(x * ln(2)).
3. f(x) = x^2
The derivative of x^2 is 2x.
4. f(x) = 2x
The derivative of 2x is 2.
Now let's evaluate the derivatives as x goes to infinity:
1. As x goes to infinity, the derivative of ln(x) goes to 0.
2. As x goes to infinity, the derivative of log2(x) goes to 0.
3. As x goes to infinity, the derivative of x^2 goes to infinity.
4. As x goes to infinity, the derivative of 2x goes to 2.
Therefore, the function f(x) = x^2 grows the fastest as x goes to infinity.
To determine which function grows the fastest as x goes to infinity, we can compare the growth rates of the given functions.
1. f(x) = ln(x)
The natural logarithm function grows slowly compared to other functions. As x goes to infinity, ln(x) approaches infinity, but at a much slower rate than exponential functions.
2. f(x) = log2(x)
Similarly, the logarithm base 2 function also grows slowly as x goes to infinity. The growth rate of log2(x) is faster than ln(x) but slower than exponential functions.
3. f(x) = x^2
This function represents a quadratic expression. As x goes to infinity, x^2 also goes to infinity, but at a faster growth rate compared to logarithmic functions. However, it is still slower than exponential functions.
4. f(x) = 2^x
The exponentiation function with base 2 grows the fastest as x goes to infinity. The growth rate of 2^x is exponential, meaning the value increases rapidly as x increases.
Therefore, the function f(x) = 2^x grows the fastest as x goes to infinity.
I will assume your third function is f(x) = x^2
you could graph or sketch each one, that will tell you
http://www.wolframalpha.com/input/?i=plot+y+%3D+log(x),+y+%3D+log(x)%2F(log2),+y+%3D+x%5E2,+y+%3D+2x,+for+0%3Cx%3C10