MATH 021:Z1 Lab 1

Estimating Limits Numerically and Graphically
An intro to using Mathematica for Derivatives

Show your work, and record your answers in the boxes provided.

Exercise 1

(a) Define h(θ)=((sinθ)/θ)

What is the domain of h(θ) ? Write your answer in interval notation.

Plot h(θ) for θ in the interval [-2π,2π]  
(You can try excluding value(s) where h(θ) is undefined, but it won’t really be visible on the plot.)

(b) The (lim_(θ->0)) ((sinθ)/θ) is a famous limit that we will need in calculus.

Make a table of ordered pairs {θ,h[θ]} for values of θ approaching 0 from the right and from the left, similar to the tables you made in Exercise 1.

Based on the numerical and graphical evidence, what is (lim_(θ->0)) ((sinθ)/θ)?

(c) Now evaluate the limit using L’Hospital’s Rule. Then, use L’hospital’s rule to compute the limit:
     (lim_(x->0)) ((x + tan(x))/x)="answer"

Exercise 2

As an example, let’s find the first and second derivatives of f(x)=x^100. Begin by defining f(x).

f[x_]:=x^100

Now find the first and second derivatives.

f'[x]

f''[x]

Now we can find higher order derivatives as follows.  Use the syntax D[function, {variable, order}]
Below is how we compute the 80th derivative of f(x)

D[f[x],{x,80}]

Let f(x) = x((e^x))

(a) On paper, find f’(x),  and f’’(x) and simplify your answers. Show 
your work.

(b) Check your work from part (a) by using Mathematica to find f’(x),  and f’’(x) , and 

f’’’(x) and use Simplify[] to simplify the results. 

(c) Use Mathematica to find the 150th derivative of f(x). Use Simplify to simplify the 

results.

Now let f(x) =  ((sin(x^3)e^(cos(x^2))Pi^x)/((x^3+3x)^(3/2)))

Use mathematica to find f’ and f’’.  (pretty nasty if done by hand...)

Exercise 3

The Formula for Newton’s Method
The linearization for f(x) at our initial estimate (Subscript[x, 0 ])is given by:

L(x) = f ' ((Subscript[x, 0])) (x-(Subscript[x, 0])) + f((Subscript[x, 0]))

To find the root of L(x), set it equal to 0:

 f ' ((Subscript[x, 0])) (x-(Subscript[x, 0])) + f((Subscript[x, 0])) = 0
 
Solving for x, we get the following.

x=(Subscript[x, 0])-((f((Subscript[x, 0])))/(f'((Subscript[x, 0]))))
Calling this result (Subscript[x, 1]), we have:

(Subscript[x, 1])=(Subscript[x, 0])-((f((Subscript[x, 0])))/(f'((Subscript[x, 0]))))
To find (Subscript[x, 2]), follow the exact same procedure with (Subscript[x, 1]) in place of (Subscript[x, 0]).

(Subscript[x, 2])=(Subscript[x, 1])-((f((Subscript[x, 1])))/(f'((Subscript[x, 1]))))

Similarly,

(Subscript[x, 3])=(Subscript[x, 2])-((f((Subscript[x, 2])))/(f'((Subscript[x, 2]))))

and so forth. 

To find the new x at each step, we evaluate x-((f(x))/(f'(x))) at the current x. We can define a function to do this.

new(x)  = x-((f(x))/(f'(x)))
To apply Newton’s Method, begin with your initial estimate (Subscript[x, 0]), substitute it into the “new” function to get a new x, substitute the result into the “new” function to get a new x, substitute that result into the “new” function, and so forth, until the approximations converge to the desired root of f(x). You’ll know when to stop when successive x-values are no longer changing.

Instructions
In each question, set up your Newton’s Method function to carry out calculations to 15 significant digits. That is, set up your new(x) function using N[ ,15]

For each question, use a plot to make a good initial estimate x0. Then use your new(x) function to find x1, x2, etc., naming each result as you go along. Stop when the results converge to a limit, that is, when two successive results are the same. 

Remember to enter your initial estimates as integers or fractions, not decimals. Always substitute named 
x-values into functions (for example new[x0] or f[x5]). Do not copy/paste or retype decimal numbers; doing so may cause the loss of hidden digits, which will lead to loss of precision and wrong answers.
Show all of your work. Don’t skip steps or erase one step to do the next; show everything. If there is more than one answer to a question, show the work for each answer.

The functions f(x)=9x+14cos(4x) and g(x)=235-32x 
intersect each other at one point. Plot the two functions together so that you have a good view of the point of intersection.

Then use Newton’s method to find the x-coordinate of the point of intersection, following all of the instructions at the top of the lab. 

Once you have the x-coordinate of the point of intersection, use it to find the y-coordinate. Round the (x,y) coordinates of your final answer to 10 significant digits. Do not round your final x until AFTER you use it to find y. That is, use the UNROUNDED, named x-value (x4 or x5 or whatever it is) to calculate the 
y-coordinate of the point of intersection, and THEN round both x and y to 10 significant digits.

Clear everything before you go on.

Exercise 4

(a)

Define p(x)=4(x^2)-362+27(e^(arctan(x))) 

Plot p(x) so that you have a good view of all of its roots, and use Newton’s Method to find them. Show work for all roots that you find (don’t erase one to do another).

Follow all of the instructions at the top of the lab, and round your final answers to 10 significant digits.

(b)

The function p(x) defined in part (a) has an absolute minimum. Use Newton’s Method to find the 
x-coordinate of the absolute minimum, following all of the instructions at the top of the lab. Once you have the x-coordinate of the absolute minimum, use it to calculate the y-coordinate at that point. Do not round x until AFTER you use it to find y. Once you have both x and y, round them to 10 significant digits.

If you aren't sure how to begin, it might help to plot p'(x).