FinaExamMiniProject.ipynb

Kernel: SageMath (stable)

# Final Exam

### Name: Fangyao Su


### Score: 

Rules for writing in this notebook

• Insert your name in the first cell of this notebook

• Answer every part of the mini project in a different cell.

• To insert a cell use the menu item Insert

• State the input(s) and output(s) before each of your functions

• Give your functions suggestive names

• Restrict the size of figures that you create

## Mini project 3: Analysis of a function with many characteristic features

__Problem formulation__
f(x)=cos(30/(1-x))-(x+2)/x
Consider a function $ .$

1. Manually find the domain of the function. Find the limits of the function as $x\rightarrow\pm\infty$. Show your work.
2. Plot the function. Your graph should include point(s) of discontinuity and illustrate the behavior of the function when $x\rightarrow\pm\infty$ reasonably well. Since the function is unbounded, restrict the range of the y-values in the plotting command to get a nice picture.
3. Find the smallest negative x-intercept of the function.
4. Find the largest local minimum on the interval $[2,5].$ Find the smallest local maximum on the interval $[-5,-1].$

1. lim_(x->infinity)=0 lim_(x->-infinity)=0

f(x)=cos(30/(1-x))-(x+2)/x
p=plot(f(x),-7,20,ymin=-10,ymax=10);p.show(figsize=[5,4])

3. Find the smallest negative x-intercept of the function

p=plot(f(x),-5,-4.2);p.show(figsize=[5,4])

4. Find the largest local minimum on the interval $[2,5].$ Find the smallest local maximum on the interval $[-5,-1].$

deriv_plot=plot(derivative(f(x),x),2,5);deriv_plot.show(figsize=[5,4],ymin=-20,ymax=20)
print "Largest local minimum: "
find_root(derivative(f(x),x),2,5)

Largest local minimum: 

4.170136668619548

deriv_plot=plot(derivative(f(x),x),-5,-1);deriv_plot.show(figsize=[5,4],ymin=-7,ymax=7)
print "smallest local maximum: "
find_root(derivative(f(x),x),-5,-1)

smallest local maximum: 

-3.6940715495180108