assignment week 11.sagews

# Your name here
# ASHLEY GARVEY

An exponential function $f(x) = b^x$, where $b > 1$, is "eventually bigger than" a power function $g(x) = x^p$, where $p>0$.

(1) Define the exponential function $f(x) = 2^x$ and the power function $g(x) = x^9$ in Sage for use throughout this assignment.

f(x) = 2^x

g(x) = x^9

#2 Im sorry I could not figure out how to copy and paste the question from the PDF
# As both f(x) and g(x) approach infinity the limit for both is also 0. Once plugging in x= 1,2 & 3 to both equations you soon realize that both funtions limits are infinity and that g(x) grows much faster than f(x).

# f(1)=2 f(2)=4 f(3)=8
# g(1)=1 g(2)=512 g(3)=19383

(3) Which function is larger at $x=0$? Use Sage to support your claim. Explain your work.

f(0)

1

g(0)

# The function f(0) is larger at x=0. f(0)=1 and g(0)= 0. The reason for this is because anything raised to the 0 power is equal to 1 and 0 raised to any power is equal to 0

(4) At some $x$-value between 1 and 2, the exponential function is overtaken by the power function. Use Sage to support this claim. Use comments to explain your work and conclusions.

(5) At some $x$-value between 51 and 52 the functions intersect each other again. Use Sage to support this claim. Use comments to explain your work and conclusions.

solve(f(x)=g(x))

# We want the function to print out when f(x) = g(x). Therefore we want to know when x^9-2**x=0.