Show at least two different ways to prove that the equation x = 2−x has

exactly one real solution.

- (10 points) Suppose f ∈ C[a, b], that x1 ≤ x2 . . . ≤ xn are in [a, b]. Show that there

exists a number ξ between x1 and xn with, f(ξ) = 1

n

Xn

i=1

f(xi). - (10 points) Suppose function f has a continuous third derivative. Show that:

−3f(x) + 4f(x + h) − f(x + 2h)

2h

− f

0

(x)

≤ ch2

. - (10 points) As h → 0, find the rate of convergence of the function

F(h) =

sin h − h +

h

3

6

h

5

. - (25 points) Consider the function f(x) = ln(x).

(a) Find the Taylor polynomial of degree n about x0 = 1. Write the simplified

expressions for the polynomial approximation Pn(x) and the remainder Rn(x).

Write a computer program (in MATLAB or PYTHON) to approximate f(x) by

the polynomial approximation for n terms. Include in your code a plot of the

true function f(x) compared to the linear, quadratic and cubic approximations.

Attach a copy of the code and output.

(b) Find the degree n that will guarantee an accuracy of 10−3 when ln(1.5) is approximated by Pn(1.5) using the result from part(a). - (25 points) Consider the sequence {xk} defined by xk+1 =

x

2

k + 9

2xk

, k = 0, 1, 2, . . . ,.

(a) Show that for the initial guess x0 = 4, the sequence has a limit x

∗ = 3.

(b) Show that the convergence of the sequence to the limit x

∗ = 3 is quadratic.

(c) Write a computer program (in MATLAB or PYTHON) that will implement the

recursive relation to compute the first 10 terms of the sequence and print them.

Attach a copy of the code and output. - (25 points) Consider finding the integral: I(x) = Z x

0

sin(t

2

) dt. While this integral

cannot be evaluated in terms of elementary functions, the following approximating

technique may however be used.

(a) Derive a Taylor Series expansion about x = 0 for I(x).

(b) Write a computer program (in MATLAB or PYTHON) to approximate I(x) by

the approximation in part (a) for n terms. Use the program to plot the approximation of I(x) for 2 terms, for 5 terms and for 10 terms. Plot the three approximate

functions respectively by plotting over the domain [0, 1]. Attach a copy of the

code and output.

Sample Solution

A few people get irritated over hiccuping again and again. There are even individuals who have constant hiccuping. Truth be told, "how to dispose of hiccups" is one of the most-looked through inquiries on the web. All things considered, there are numerous things you can do. So as to dispose of hiccups, you can finish breathing and stance methods, use pressure focuses to further your potential benefit, eat and drink certain things, and utilize some exceptional cures. These arrangements will be talked about in the accompanying passages. Right off the bat, there are many breathing and posing procedures you can use to dispose of hiccups. The vast majority of the procedures are anything but difficult to do. As per Healthline, there are six physical techniques for vanquishing your next hiccup: rehearsing moderate breathing, holding your breath in for some time, breathing into a paper pack, embracing your knees for two minutes while plunking down, compacting your chest by inclining or twisting forward, and finishing the Valsava move. As expressed by Healthline, "To do this move, attempt to breathe out while squeezing your nose and keeping your mouth shut" ("How to Get Rid of Hiccups: 26 Remedies That Can Actually Help"). You can't do every one of these procedures at the same time, yet you can attempt them individually and see which ones work best for you. Likewise, you can attempt to utilize your body's weight focuses to further your potential benefit to scatter hiccups. In spite of the fact that weight focuses appear to be an exclusive science, standard western medication suggests utilizing them. As per Medical News Today, these three methods that manage these focuses can help: "[1] Pull on the tongue – hold the finish of the tongue in the fingers and pull. [2] Press on the stomach delicately. [3] Place delicate weight on each side of the nose while gulping" (Nordqvist, Christian). Despite the fact that these strategies may appear to be bizarre from the outset, they have a specific measure of adequacy that warrants their proposal. Another normal method to dispose of hiccups is to expend certain nourishments and fluids. There are a lot of things we can eat and drink to pursue away a dreadful scene of hiccuping. As referenced by Healthline, it is acceptable to "drink ice water, gradually drink a glass of warm water ceaselessly to inhale, drink water through a fabric or paper towel, suck on an ice 3D square, swish ice water, eat a spoonful of nectar or nutty spread, eat some sugar, suck on a lemon, put a drop of vinegar on your tongue" ("How to Get Rid of Hiccups: 26 Remedies That Can Actually Help") and that's just the beginning. By and by, try out which nourishment and drink methods work best, rather than attempting numerous simultaneously. And afterward, there are some abnormal procedures that have been prescribed by researchers and different masters. One of these hypotheses expresses that hiccups can be eased by having a climax (Peleg, R.). Another examination referenced that a rectal back rub can help in disposing of hiccups (Odeh, M, et al). Other elective techniques incorporate tapping or scouring the rear of one's neck, jabbing the rear of the throat with a Q-tip, and diverting yourself with something captivating ("How to Get Rid of Hiccups: 26 Remedies That Can Actually Help"). The last one is even a respected convention in specific nations. Hiccups can be an irritation, particularly on the off chance that they transform into an interminable condition. Fortunately, there are numerous moves we can make to clear up this issue: you can finish breathing and stance procedures, utilize pressure focuses, eat and drink specific substances, and utilize some elective cures. It is ideal to discover what works best for you out of the numerous decisions talked about in this exposition, as each body is unique. Works Cited "The most effective method to Get Rid of Hiccups: 26 Remedies That Can Actually Help." Healthline, Healthline Media, www.healthline.com/wellbeing/how-to-dispose of-hiccups#other-cures. Nordqvist, Christian. "Hiccups: How to Get Rid of Hiccups." Medical News Today, MediLexicon International, 20 June 2017, www.medicalnewstoday.com/articles/9896.php.>

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