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

10 pts

#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

Remember to restart your worksheet and re-execute from the top before considering your work complete. That way if your work disappears, you will know you need to use undo and copy and paste your work into a new cell.

Allow me to execute the Sage commands as follows, rather than putting them in a comment:

f(1)

2

f(2)

4

f(3)

8

g(1)

1

g(2)

512

g(3)

19683

Some things you say are true: As both f(x) and g(x) approach infinity the limit for both is also 0 (false). Once plugging in x= 1,2 & 3 to both equations you soon realize that both funtions limits are infinity (true) and that g(x) grows much faster than f(x) (true on the interval $1 \le x \le 3$; see assignment goals).

8.5 pts

f(0)

1

g(0)

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

10 pts

(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.

No work was provided.

(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))

Error in lines 1-1 Traceback (most recent call last): File "/cocalc/lib/python3.8/site-packages/smc_sagews/sage_server.py", line 1231, in execute compile(block + '\n', File "<string>", line 1 SyntaxError: expression cannot contain assignment, perhaps you meant "=="?

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

Good statement of the goal. The syntax for $\texttt{solve}$ is $\texttt{solve(f(x)==g(x), x)}$; please review it. In this case the command does not produce a meaningful solution.

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

[x == 2^(1/9*x)*e^(2/9*I*pi), x == 2^(1/9*x)*e^(4/9*I*pi), x == 2^(1/9*x - 1)*(I*sqrt(3) - 1), x == 2^(1/9*x)*e^(8/9*I*pi), x == 2^(1/9*x)*e^(-8/9*I*pi), x == 2^(1/9*x - 1)*(-I*sqrt(3) - 1), x == 2^(1/9*x)*e^(-4/9*I*pi), x == 2^(1/9*x)*e^(-2/9*I*pi), x == 2^(1/9*x)]

4 pts