exercise week 12.sagews

# Your name here
Ashley Garvey

(1.) Define the functions ${\displaystyle f(x) = \tan\left(\frac{\pi x^2}{4}\right)}$ and $g(x) = \pi x + 1 - \pi$ in Sage for use throughout this exercise.

plot(tan((pi*x**2)/4))

f(x)=tan((pi*x**2)/4)

plot(pi*x+1-pi)

g(x)=pi*x+1-pi

(2.) Plot both functions on the same set of axes. Choose a scaling that lets you see the key features of each graph.

plot(tan(pi*x**2)/4) + plot(pi*x+1-pi)

(3.) Find an $x$-value where the two graphs intersect. You may identify this value visually. Make sure you also establish mathematically that you have found an $x$-value where the graphs intersect.

f(x)=tan((pi*x**2)/4)

g(x)=pi*x+1-pi

find_root(tan((pi*x**2)/4)==pi*x+1-pi)

var(x)

solve(tan((pi*x**2)/4)==pi*x+1-pi)

(4.) Find the corresponding $y$-value where the intersect. You may identify this value visually. Make sure you also establish mathematically that you have found a point $(x, y)$ where the graphs intersect.

(5.) One of the graphs is a line. What is its slope? In Sage, we can calculate the derivative of $f(x)$ using the following command: $\texttt{derivative(f(x), x)}$. Use Sage to find the derivative of $f(x)$. Use Sage to find the value of the derivative of $f(x)$ at $x = 1$. Explain how this value compares with the slope of the line you identified in this question. From a calculus point of view, what is the significance of how $f'(1)$ compares to the slope? Refer to your graph in your answer.

#the slope of g(x) is pi

derivative(f(x), x)

︠ba55f18c-1aa8-4281-8688-f2a8de97de07︠
derivative(f(1), 1)

#first when we found the derivative of the f(x) we were finding the derivative of the entire function. secondly when we found the derivative of the funtion we were only finding the derivative 0f the function at the particular point where x=1.