Which of the following functions grows the slowest?
j(t)=1/4 ln(t^200)
a(t)=t^5/2
i(t)=ln(t^100)
g(t)=3t^2-t
b(t)=t^4-3t+9
The growth of a function is its rate of change, which is found by taking its derivative.
Evaluate the derivatives of the given functions and identify the smallest.
j(t) = 1 / 4 ln ( t^200 )
ln ( t^200 ) = 200 ln
so
j(t) = 1 / 4 * 200 ln t
j(t) = 50 ln t
j'(t) = 50 * 1 / t
j'(t) = 50 / t
a(t) = t ^ 5 / 2
a'(t) = 5 / 2 * t ^ ( 5 / 2 - 1 )
a'(t) = 5 / 2 * t ^ ( 5 / 2 - 2 / 2 )
a'(t) = 5 / 2 t ^ ( 3 / 2 )
i(t) = ln ( t^100 )
i'(t) = 1 / t * 100
i'(t) = 100 / t
g(t) = 3 t ^ 2 - t
g'(t) = 3 * 2 * t - 1
g'(t) = 6 t - 1
b(t) = t ^ 4 - 3 t + 9
b'(t) = 4 t ^ 3 - 3
We can rule out the equations:
a(t) = t ^ 5 / 2
g(t) = 3 t ^ 2 - t
and
b(t) = t ^ 4 - 3 t + 9
as their growth is directly related to the variable t.
Meaning that as it gets larger, the functions growth increases.
That leaves equations:
i(t) = ln ( t^100 )
and
j(t) = 1 / 4 ln ( t^200 )
whose growth is inversely related to variable t.
We can see that i'(t) is twice j'(t), so j(t) has the smallest growth.
Well, these functions are all quite fast, but if we're looking for the slowest one, it's like trying to find the least tall clown in a circus. They're all still pretty tall!
But if I had to pick one, I would say "g(t) = 3t^2 - t" grows the slowest. It's like the clown who tripped on his big shoes and slowed down a bit.
To determine which function grows the slowest, we can compare their growth rates. The growth rate of a function can be determined by looking at the exponents or power of the variable. The higher the power, the faster the function grows.
Let's analyze each function:
1. j(t) = 1/4 ln(t^200)
The function j(t) has a logarithmic growth rate since it contains a natural logarithm. Logarithmic growth is generally slower than polynomial growth.
2. a(t) = t^(5/2)
The function a(t) has a power of 5/2, which is a fractional exponent. This represents a square root of t raised to the power of 5, indicating polynomial growth.
3. i(t) = ln(t^100)
Similar to j(t), i(t) also contains a natural logarithm. Hence, it has logarithmic growth.
4. g(t) = 3t^2 - t
The function g(t) has a polynomial growth rate with a power of 2, which is larger than the growth rate of a logarithmic function.
5. b(t) = t^4 - 3t + 9
Similar to g(t), b(t) has a polynomial growth rate with a power of 4, which is also higher than logarithmic growth.
Comparing the growth rates, we can determine that the function j(t) = 1/4 ln(t^200) grows the slowest since it has a logarithmic growth rate.
To determine which function grows the slowest, we need to compare the growth rates of the functions as the input (t) gets larger.
Let's evaluate the growth rates of each function by looking at their exponents.
1. For j(t) = 1/4 ln(t^200):
The exponent of t is 200. Since the growth rate is determined by the exponent, this function grows slower than any function with a larger exponent.
2. For a(t) = t^(5/2):
The exponent of t is 5/2. This function has a smaller exponent than j(t) but a larger exponent than g(t), b(t), and i(t).
3. For i(t) = ln(t^100):
The exponent of t is 100. Similar to j(t), this function's growth rate is dependent on the exponent. Since 100 is larger than 5/2, i(t) grows slower than a(t).
4. For g(t) = 3t^2 - t:
This function is a polynomial of degree 2, so the exponent of t is 2. It has a smaller exponent than j(t), a(t), and i(t), but a larger exponent than b(t).
5. For b(t) = t^4 - 3t + 9:
This function is a polynomial of degree 4, so the exponent of t is 4. It has the largest exponent among all the given functions.
Therefore, the function that grows the slowest is j(t) = 1/4 ln(t^200).
in order, slowest to fastest:
log
polynomial
exponential
so, look at the logs
1/4 ln(t^200) = 1/4 * 200 ln t = 50 ln t
ln(t^100) = 100 ln t
so, ...