Show that f(x) = x3 and g(x) = 200x3 grow at the same rate
To show that two functions, f(x) = x^3 and g(x) = 200x^3, grow at the same rate, we need to prove that their growth rates are equal.
Let's start by finding the derivative of both functions.
The derivative of f(x) = x^3 can be found using the power rule of differentiation. According to the power rule, if we have a function f(x) = x^n, where n is a constant, the derivative is given by:
f'(x) = n * x^(n-1)
Applying the power rule to f(x) = x^3, we get:
f'(x) = 3 * x^(3-1)
= 3 * x^2
Similarly, find the derivative of g(x) = 200x^3:
g'(x) = 3 * 200x^(3-1)
= 600x^2
Now, we need to compare the derivatives, f'(x) and g'(x), to check if they are equal.
f'(x) = 3 * x^2
g'(x) = 600x^2
To compare them, we set the two derivatives equal to each other and solve for x:
3 * x^2 = 600x^2
Dividing both sides by x^2 (assuming x ≠ 0) gives us:
3 = 600
This equation is not possible since 3 ≠ 600. Therefore, the equation does not have a solution.
Hence, we can conclude that f(x) = x^3 and g(x) = 200x^3 do not grow at the same rate. The growth rate of g(x) = 200x^3 is 200 times greater than the growth rate of f(x) = x^3.
To show that f(x) = x^3 and g(x) = 200x^3 grow at the same rate, we need to compare the growth rates of the two functions.
The growth rate of a function can be determined by examining its derivative. If two functions have the same derivative, then they have the same growth rate.
So, let's take the derivative of both functions:
For f(x) = x^3:
f'(x) = 3x^2
For g(x) = 200x^3:
g'(x) = 600x^2
Now, let's compare the two derivatives.
We can see that f'(x) = 3x^2 and g'(x) = 600x^2 are not equal. This means that f(x) = x^3 and g(x) = 200x^3 do not have the same growth rate.
In fact, g(x) = 200x^3 grows much faster than f(x) = x^3 because the coefficient 200 in front of x^3 amplifies the growth rate of g(x) compared to f(x).
In conclusion, f(x) = x^3 and g(x) = 200x^3 do not grow at the same rate.