Show that f(x) = 2000x^4 and g(x) = 200x^4 grow at the same rate
I know that they grow at the same rate because they are both raised to the same power, but i don't know how to show it.
well, what is the rate of growth from x=a to x=b ?
2000b^4 / 2000a^4 = (b/a)^4
same for 200x^4
another way is we can say f(x) grows faster than g(x) as x>>inf if lim(x>inf) f(x)/g(x) = inf
if the limit approaches some constant k, they are growing at the same rate.
Lim(x>inf)=2000x^4/200x^4= 10
so they have the same growth rate.
Well, you could always invite f(x) and g(x) to a race and see who wins! But if you want a more mathematical explanation, here it is:
To show that f(x) = 2000x^4 and g(x) = 200x^4 grow at the same rate, we can compare their growth rates by looking at their derivatives.
The derivative of f(x) with respect to x is f'(x) = 8000x^3. And the derivative of g(x) with respect to x is g'(x) = 800x^3.
Both f'(x) and g'(x) have the same factor, which is 800, and they are raised to the same power of 3. This means that the growth rates of f(x) and g(x) are proportional to each other.
So, we can say that f(x) and g(x) grow at the same rate because their derivatives are proportional to each other. Now, let's just hope they don't develop a rivalry and start arguing about who is faster!
To show that f(x) = 2000x^4 and g(x) = 200x^4 grow at the same rate, you can compare their growth rates by looking at their derivatives. If the derivatives are equal, then the functions grow at the same rate.
Let's start by finding the derivative of f(x) and g(x) with respect to x:
1. Derivative of f(x) = 2000x^4:
- Use the power rule for differentiation, which states that the derivative of x^n is nx^(n-1).
- Differentiating f(x) with respect to x, we get:
f'(x) = 4 * 2000 * x^(4-1)
f'(x) = 8000x^3
2. Derivative of g(x) = 200x^4:
- Similarly, using the power rule, the derivative of g(x) with respect to x is:
g'(x) = 4 * 200 * x^(4-1)
g'(x) = 800x^3
Now we can compare the derivatives:
f'(x) = 8000x^3
g'(x) = 800x^3
Since the exponents are the same (3), the only difference between the two derivatives is the factor in front of x^3. But if you compare these factors, 8000 and 800, they are equal.
Therefore, both f(x) = 2000x^4 and g(x) = 200x^4 have the same derivative, which means they grow at the same rate.
To show that two functions, f(x) = 2000x^4 and g(x) = 200x^4, grow at the same rate, we need to compare their growth rates as x approaches positive or negative infinity.
Let's calculate the limits of the ratio f(x)/g(x) as x approaches infinity:
lim(x->∞) (f(x)/g(x))
= lim(x->∞) ((2000x^4)/(200x^4))
= lim(x->∞) (2000/200)
= 10
The limit of the ratio is equal to 10, which suggests that as x approaches infinity, the ratio f(x)/g(x) remains constant at 10. This implies that both f(x) and g(x) grow at the same rate.
Similarly, we can calculate the limits of the ratio as x approaches negative infinity:
lim(x->-∞) (f(x)/g(x))
= lim(x->-∞) ((2000x^4)/(200x^4))
= lim(x->-∞) (2000/200)
= 10
Once again, the limit of the ratio is equal to 10. This indicates that as x approaches negative infinity, the ratio f(x)/g(x) remains constant at 10. Therefore, f(x) and g(x) also grow at the same rate in the negative direction.
By showing that the ratio f(x)/g(x) has a constant value of 10 as x approaches both positive and negative infinity, we have demonstrated that f(x) = 2000x^4 and g(x) = 200x^4 have the same growth rate.