Show that f(x) = x^3 and g(x) = 200x^3 grow at the same rate.
To show that f(x) = x^3 and g(x) = 200x^3 grow at the same rate, we need to compare their growth rates as x approaches infinity.
First, let's consider the function f(x) = x^3. As x approaches infinity, the growth rate of f(x) can be determined by looking at the behavior of the function. Since x^3 is a cubic function, as x becomes larger and larger, the dominant term in the function is the x^3 term. As a result, the growth rate of f(x) as x approaches infinity is determined by the coefficient of the x^3 term, which is 1. So, f(x) grows at a rate of 1.
Now let's consider the function g(x) = 200x^3. As x approaches infinity, the growth rate of g(x) can be determined in the same way as we did for f(x). In this case, the dominant term is the 200x^3 term, which means the growth rate is determined by the coefficient of this term, which is 200. So, g(x) grows at a rate of 200.
Comparing the growth rates of f(x) and g(x), we can see that f(x) grows at a rate of 1, while g(x) grows at a rate of 200. Since 1 is not equal to 200, we can conclude that f(x) = x^3 and g(x) = 200x^3 do not grow at the same rate.
To show that two functions grow at the same rate, we need to compare their growth rates by examining their limits as the input value approaches infinity.
Let's calculate the limits of the functions f(x) = x^3 and g(x) = 200x^3 as x tends to infinity:
For f(x) = x^3:
lim(x->∞) x^3
We can use L'Hôpital's rule to calculate this limit. By taking the derivative of both the numerator and the denominator, we have:
lim(x->∞) 3x^2
Since the limit as x tends to infinity of 3x^2 is also infinity, we can conclude that f(x) = x^3 grows at the rate of infinity.
Now let's analyze the limit of g(x) = 200x^3 as x tends to infinity:
lim(x->∞) 200x^3
By factoring out the common constant 200:
200 * lim(x->∞) x^3
We can see that the limit of x^3 as x approaches infinity is also infinity, but since it is multiplied by the constant 200, the overall growth rate is 200 times greater compared to f(x) = x^3.
Therefore, we can conclude that g(x) = 200x^3 grows at a faster rate than f(x) = x^3 since the limit of g(x) as x tends to infinity is 200 times larger than the limit of f(x) as x tends to infinity.
If by "same" you mean the same percentage per unit time, then
f'/f = 3x^2/x^3
g'/g = 600x^2/200x^3 = 3x^2/x^3
Naturally, since g started out with 200 times as much, its actual growth rate is 200 times as much as f's.