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Math 107 #3

MATH 107, QUIZ 3 Due Tuesday, September 14, 2021

NAME: _______________________________

I have completed this assignment myself, working independently and not consulting anyone except the instructor.

INSTRUCTIONS

· There are 6 problems (on 6 pages). The quiz is worth 100 points, equivalent to 9% of your final course grade. This quiz is
open book
and
open notes
. This means that you may refer to your e-text/textbook, notes, and online classroom materials, but
you must work independently and may not consult anyone
(and confirm this with your submission). You may take as much time as you wish, provided you turn in your quiz no later than 11:59 PM (US Eastern Time Zone) Tuesday, September 14, 2021.

·
Show work/explanation. Answers without any work may earn little, if any, credit.
You may type or write your work in your copy of the quiz, or if you prefer, create a document containing your work. Scanned work is acceptable also; a single file in pdf format, with your name and quiz number in the file name, is preferred. In your document, be sure to include your name and the assertion of independence of work.

· If you have any question, please post it in “Ask the Professor” discussion on LEO if the answer to your question would benefit others in class; otherwise, please contact me privately via e-mail.

PLEASE READ THE QUESTIONS/PROBLEMS CAREFULLY; SHOW ALL YOUR WORK AND REASONING,
Just the answer, without supporting work, will receive no credit.

1. [27 points] Draw the curved graph of a function y = h(x) such that

(a) it begins with a small unfilled circle (an “open” (exclusive) beginning) somewhere in the left side of the y-axis, has one “valley” and one “peak,” in an order that you choose, has three x-intercepts, does not pass through the origin, and has a “closed” (inclusive) ending somewhere in the right side of the y-axis, It should be noted that the graph asked for can be any graph drawn that would meet the general requirements stated. There is no need to look for a mathematical form for h(x); just focus on the behavior of the function described. A typical graph, meeting the problem requirements, is shown below; however, you may draw your own graph differently, e.g., reversing the order of the “peak” and “valley” (specific extremum) and incorporating specific x- and y-intercepts, but your graph must meet the general requirements stated.

(b) Label the axes and identify the informative points on the graph by showing each as an ordered pair on the graph.

(c) Write “y = h(x)” somewhere next to its graph for identification.

(d) Find the zeros (the x-coordinates of the x-intercepts) of the function drawn and write them using correct mathematical notations.

(e) Find the x-intercepts of the graph of h(x) and write each as an ordered pair.

(f) Find the y-intercept of the graph of h(x) and write it as an ordered pair.

(g) Write the mathematical interval representing the domain of the function drawn.

(h) Write the mathematical interval representing the range of the function drawn.

(i) State, in mathematical notations (interval(s)), where the function would be decreasing.

(j) State, in mathematical notations (interval(s)), where the function would be increasing.

(k) Write the coordinates of the absolute minimum of the function, if any.

(l) Write the coordinates of the absolute maximum of the function, if any.

(m) Write the coordinates of the relative minimum of the function, if any.

(n) Write the coordinates of the relative maximum of the function, if any.

(o) If y = t(x) is a transformation of y = h(x), such that t(x) = 0.5 h(x+2), draw the graph of t(x), preferably on the same coordinate system (x-y plane) that you used for the graph of h(x), or separately. You may use a “thicker pen,” a different color, or simply a dashed curve (instead of a solid curve) while drawing t(x). Write “y = t(x)” somewhere next to its graph for identification. Identify the coordinates of the informative points on t(x).

(p) What transformations did the graph of function h(x) go through to become the graph of function t(x)?

(q) Find the “average rate of change” of h(x) over the closed interval bounded by the lowest point of the “valley” and the highest point of the “peak” of the function. Hint: See Definition 2.3 of the eBook.

2. [17 points] Linear relationship/Mathematical Modeling. Under specific physical/chemical conditions, water freezes at 32 degrees Fahrenheit (F) on the Fahrenheit temperature scale or at 0 degree Celsius (0) on the Celsius temperature scale, and boils at 212 F or 100 C.
Showing your work
,

(a) Graph the freezing and boiling points of water on “C-F plane,” in place of an “x-y plane.”

(b) Find the slope of the line F = f(C), passing through the freezing point and the boiling point provided above. Keep working with simplified fractions, not decimals.

(c) Briefly explain what the interpretation of your finding for part (b) above is. Give a simple example.

(d) Derive the linear relationship between the Fahrenheit and Celsius temperature scales in slope-intercept form as F = mC + b.

(e) Showing your work, check the function/equation that you derived for Part (d) above, using a point residing on the line.

(f) Derive (set up) the linear relationship between the Fahrenheit and Celsius temperature scales in point-slope form, keeping in mind that C and F have replaced x and y, respectively.

(g) Based on the linear relationship found for Part (d) or for Part (f) above, if, on the Fahrenheit temperature scale, the normal/natural body temperature of a healthy person is 98.6 F, what would be the body temperature in C of a patient who is running a fever 2.2 F above normal? Report your answer to the tenth of a degree.

(h) At what temperature do both temperature scales show the same value? (Hint: F = C, then, replace C with F (or vice versa) in one of the C-F relationships that you found above; then solve the equation for the unknown.)

3. [12 points] Mathematical modeling

The length of a rectangular tablecloth is 5 3/5 (five and three fifths) feet longer than its width. If the tablecloth is W feet wide and L feet long,

(a) Express the perimeter of the tablecloth (P) as a function of its width, i.e., develop the model P = f(W).

NOTE: In developing your mathematical model (the function), you are not to work with specific numerical values for the length and the width. Work with L, W, and their relationship, knowing that the perimeter of a rectangle is the sum of its four sides.

(b) Knowing that length times width would be the area (A) of the rectangular tablecloth, develop a model that would give the area of the tablecloth as a function of its length, i.e., develop the model A = g(L).

(c) To demonstrate evaluation of functions at some values of their arguments (independent variables), use your mathematical models (functions) developed in Parts (a) and (b) above to determine the perimeter and the area of the tablecloth when its width is 6 2/5 (six and two fifth) feet. Report the answers to two decimal places. Do not forget the units for your answers.

4. [12 points] Considering the fundamental definition of |x|, provided in Section 2.2, and piecewise-defined functions,

Graph the following function, using the x-y coordinates provided or the ones that you would draw:

|x|, for x < – 3


f(x) = , for – 3 ≤ x ≤ 4

–x, for x > 4

5. [21 points] Given g(x) = 2 – |x – 4|,

(a) Noting the two separate applications of Definition 2.4 on page 173 of the eBook and reviewing Example 2.2.1 on page 174 of the eBook, rewrite the function without absolute value (write g(x) as a piecewise-defined function).

(b) Graph the function g(x), either using its piecewise form developed for Part (a) above or considering the characteristic “V” shape of the graph of f(x) = |x| subjected to the transformations leading to g(x).

(c) List the transformations that the graph of function f(x) = |x| would go through to become the graph of function g(x) above.

(d) Find the zero(s) of the function analytically (solve g(x) = 0), using the original form of the function or its piecewise form developed for Part (a) above.

(e) Find the x-intercept(s), to be reported as ordered pair.

(f) Find the y-intercept, to be reported as ordered pair.

(g) Determine the domain and the range of the function.

(h) State, using interval notation, where the function is decreasing and where the function is increasing.

6. [11 points]

(a) Find the graph of g(x) following the information provided in the graph of f(x) below.

(b) Support your answer briefly by stating what transformation(s) the function f(x) has gone through to become g(x).

(c) Having found g(x), write the domain and the range of g(x) in interval format.


End of quiz: please do not forget to write and sign (or type) the required statement reflected after “NAME:” at the top of page 1 of the quiz in case what you post does not contain the said page reflecting your signature and date.

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