Any short explanation for things I got wrong would be great, too, if possible! Thanks in advanced! :)
8. Which of the following functions grows the fastest?
***b(t)=t^4-3t+9
f(t)=2^t-t^3
h(t)=5^t+t^5
c(t)=sqrt(t^2-5t)
d(t)=(1.1)^t
9. Which of the following functions grows the fastest?
f(t)=2^t-t^3
a(t)=t^5/2
e(t)=e
g(t)=3t^2-t
***b(t)=t^4-3t+9
10. Which of the following functions grows the fastest?
***g(t)=3t^2-t
i(t)=1m(t^100)
e(t)=e
c(t)=sqrt(t^2-5t)
a(t)=t^5/2
11. Which of the following functions grows the slowest?
b(t)=t^4-3t+9
f(t)=2^t-t^3
h(t)=5^t+t^5
***c(t)=sqrt(t^2-5t)
d(t)=(1.1)^t
12. Which of the following functions grows the least?
g(t)=3t^2-t
i(t)=1n(t^100)
e(t)=e
c(t)=sqrt(t^2-5t)
***a(t)=t^5/2
13. Which of the following functions grows the slowest?
j(t)=1/4 1n(t^200)
a(t)=t^5/2
***i(t)=1n(t^100)
g(t)=3t^2-t
b(t)=t^4-3t+9
Steve your awnsers for 8 and 9 are wrong btw
Wow... That, just made so much more sense to me with the examples you provided. I'm going to do some more practice questions knowing what you told me. My lesson didn't teach it like you just did!
I just did some more practice problems, and got them right! Thank you so much! You made my day. :)
8. The function that grows the fastest is h(t)=5^t+t^5. It has an exponential term (5^t) which grows much faster than the other functions.
9. The function that grows the fastest is b(t)=t^4-3t+9. It has a higher degree polynomial term (t^4) which grows faster than the other functions.
10. The function that grows the fastest is g(t)=3t^2-t. It has a quadratic term (3t^2) which grows faster than the other functions.
11. The function that grows the slowest is c(t)=sqrt(t^2-5t). As the square root function grows slower than polynomial or exponential functions, this function has the slowest growth.
12. The function that grows the least is a(t)=t^5/2. Although it is a polynomial function, it has a lower degree than the other functions, resulting in slower growth.
13. The function that grows the slowest is i(t)=1n(t^100). The logarithmic function (1n) grows much slower compared to other polynomial or exponential functions, making this function have the slowest growth.
8. To determine which function grows the fastest, we can compare the growth rates of each function. One way to do this is to calculate the limit of each function as t approaches infinity. The function with the highest limit will grow the fastest.
For b(t), as t approaches infinity, the dominant term is t^4, so the function grows as t^4.
For f(t), as t approaches infinity, the dominant term is 2^t, so the function grows exponentially.
For h(t), as t approaches infinity, both terms t^5 and 5^t grow exponentially, but the exponential growth of 5^t dominates.
For c(t), as t approaches infinity, the dominant term is sqrt(t^2), which simplifies to t. So the function grows linearly.
For d(t), as t approaches infinity, the dominant term is (1.1)^t, which grows exponentially but at a slower rate than 2^t.
Based on this analysis, the function that grows fastest is h(t) = 5^t + t^5.
9. Using the same approach as in the previous question:
For f(t), as t approaches infinity, the dominant term is 2^t.
For a(t), as t approaches infinity, the dominant term is t^(5/2), which grows faster than 2^t.
For e(t), the function is a constant, so it does not grow.
For g(t), as t approaches infinity, the dominant term is t^2.
For b(t), as t approaches infinity, the dominant term is t^4.
Based on this analysis, the function that grows the fastest is b(t) = t^4 - 3t + 9.
10. Using the same approach as in the previous questions:
For g(t), as t approaches infinity, the dominant term is 3t^2.
For i(t), the function involves a polynomial with a high exponent, but compared to g(t), it grows slower.
For e(t), the function is a constant, so it does not grow.
For c(t), as t approaches infinity, the dominant term is sqrt(t^2), which simplifies to t.
For a(t), as t approaches infinity, the dominant term is t^(5/2), which grows slower than 3t^2.
Based on this analysis, the function that grows the fastest is g(t) = 3t^2 - t.
11. To determine the function that grows the slowest, we can compare the growth rates of each function. The slowest growing function will have the smallest limit as t approaches infinity.
For b(t), as t approaches infinity, the dominant term is t^4.
For f(t), as t approaches infinity, the dominant term is 2^t.
For h(t), as t approaches infinity, the dominant term is 5^t.
For c(t), as t approaches infinity, the dominant term is t.
For d(t), as t approaches infinity, the dominant term is (1.1)^t.
Based on this analysis, the function that grows the slowest is c(t) = sqrt(t^2 - 5t).
12. Using the same approach as in the previous question:
For g(t), as t approaches infinity, the dominant term is 3t^2.
For i(t), the function involves a logarithm, which grows slower than any polynomial.
For e(t), the function is a constant, so it does not grow.
For c(t), as t approaches infinity, the dominant term is t.
For a(t), as t approaches infinity, the dominant term is t^(5/2).
Based on this analysis, the function that grows the least is a(t) = t^(5/2).
13. Using the same approach as in the previous questions:
For j(t), the function involves a logarithm, which grows slower than any polynomial.
For a(t), as t approaches infinity, the dominant term is t^(5/2).
For i(t), the function involves a logarithm.
For g(t), as t approaches infinity, the dominant term is 3t^2.
For b(t), as t approaches infinity, the dominant term is t^4.
Based on this analysis, the function that grows the slowest is i(t) = ln(t^100).
Glad to help.
exponentials grow faster than powers, and constants don't grow at all(2^x > x^4 > x^5/2 > 10)
logs grow even slower than roots (since exponentials grow faster than powers): log(x) < √x
So, in order of increasing growth rate, we have
#8: c,b,d,f,h
#9: e,g,a,b,f
#10: e,i,c,g,a
#11: c,b,d,f,h
#12: e,i,c,g,a
#13: j,a,i,g,b