Asked by Mike
                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
            
        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
Answers
                    Answered by
            oobleck
            
    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, ...
    
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, ...
                    Answered by
            Bosnian
            
    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.
    
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.
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