MATA31H3 Midterm: MATA31 Midterm 2013 Winter

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turquoisegnu128

16 Oct 2018

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Mata31 calculus i for computer & mathematical sciences. You will lose 2 marks if you don"t write the tutorial number. Do not open this booklet until instructed to do so. Notes: there are 17 numbered pages in the test including the blank pages 6, there are 17 numbered pages in the test including the blank pages 6, It is your responsibility to ensure that, at the start of the exam, this booklet has all its pages and that all pages are handed in: simple scienti c calculators allowed, justify your answers. Find the domain of the function (cid:114) f (x) = x3 9x x2 1 ln x x2 + 1 page 1. 1. (b) (5 marks) determine whether the following functions are even, odd or neither. (a) (b) f (x) = x2 + Solve the equation ln(ln x) = 1 for x.

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Answer question numbers #6,#12,#14,#16,#20 include full steps and solutions with explation of your answers.

please use the following information to answer questions 1-6

Imagine you have graduated and have just gotten a job in a fictionalized version of Paducah, KY, home of the world famous Dippin' Dots (Links to an external site.)Links to an external site.. You got the job by pitching a truly marvelous idea: The Grand Cone-yan, an ice cream festival as grand as any Court Days (Links to an external site.)Links to an external site. or Apple Festival (Links to an external site.)Links to an external site..

The centerpiece of the festival is a 10 foot tall ice cream cone filled with local ice cream dots.

The day before the festival the contractors have brought the truck of ice cream, and begin to fill the cone. After only a single minute, there is a foot of ice cream in your frozen glass cone. Since the entire cone is only 10 feet tall, you decide to come back in 10 minutes to witness the final filling.

You are shocked when you return to find the cone has only been filled to the 2 foot mark! (The worker politely corrects you to say it is actually 2 feet and 2 inches.) You give it another 10 minutes, but the cone is only 2 foot 9 inches filled.

Not knowing what else to do, you make a spreadsheet of the progress.

Time (minutes)	0	1	10	20	30	60	120
Height (feet, inches)	0' 0"	1' 0"	2' 2"	2' 9"	3' 1"	3' 11"	4' 11"

You wonder why they have slowed down so much, so you keep checking in every hour.

Time (hours)	1	2	3	4	5	6
Height (feet, inches)	3' 11"	4' 11"	5' 8"	6' 3"	6' 6"	6' 11"

After six hours you are worried the entire festival might fail, but you won't give up! You keep spreadsheeting.

Time (hours)	7	8	9	10	11	12
Height (feet, inches)	7' 4"	7' 8"	7' 11"	8' 3"	8' 6"	8' 9"

After 12 hours, you take a quick trip home to change clothes, and return to your spreadsheeting.

Time (hours)	13	14	15	16	17	18
Height (feet, inches)	8' 11"	9' 2"	9' 5"	9' 7"	9' 9"	10' 0"

Finally, late in the night, around 3am, it is complete. Your masterpiece. The Grand Cone-yan. But why did it take so long? It took a lot longer than 10 minutes. Why did the workers slow down so much? The first minute finished the first foot, but three feet took 30 minutes instead of 3 minutes. And now it's 3am!

The workers say that they filled at a steady rate of 13 gallons per hour, and that any faster would run the risk of drain freeze.

Your responses

1. What is wrong with the following argument?

If you really did 13 gallons per hour, then the 10 foot cone should have only taken 10 / 13 = 0.7 hours, less than an hour tops.

2. What is wrong the following question?

How do you convert gallons into feet?

3. Find an authoritative source for the volume of a cone. Give the formula and explain what each part of the formula means in this problem.

4. Most formulas will have both radius and height. Find a reasonable source for the relationship between the radius and height of an ice cream cone. Give an expression like

R = H

5. Calculate the volume of the cone using your formula. At a constant rate of 13 gallons per hour, how long should it have taken to fill up?

You may use the approximate formula V=0.02908882 h3 if the formula you discovered does not match at all.

6. There should still be a discrepancy between the V=0.02908882 h3 and the spreadsheet. Can you explain what likely happened?

Sketch using these guidelines

pul aill al thes information together to sketch graphs that reveal the important You might ask: Why don't we just use a graphing calculator or computer to graph a It's true that modern technology is capable of producing very accurate graphs. But K 21+1+2 features of functions curve? Why do we need to use calculus? even the best graphing devices have to be used intelligently. It is easy to arrive at a misleading graph, or to miss important details of a curve, when relying solely on tech- nology. (See "Graphing Calculators and Computers" at www.stewartcalcelus.com, espe 4 discover the most interesting aspects of graphs and in many cases to and minimum points and inflection points exocily instead of approximately cially Examples 1, 3.4, and 5. See also Section 4.6.) The use of calculus enables us to calculate maximum FIGURE 1 For instance, Figure I shows the gr aph of /(x)-è¨-21x, + 18x + 2.At first glance it seems reasonable: It has the same shape as cubic curves like y x and it appears to have no maximum or minimum point. But if you compute the derivative, you will see that there is a maximum when x-0.75 and a minimum when x 1. Indeed if we zoom in to this portion of the graph, we see that behavior exhibited in Pigure 2 Without calculus, we could easily have overlooked it. 6a, FIGURE 2 In the next section we will graph functions by using the interaction between calculus and graphing devices. In this section we draw graphs by first considering the foilowing information. We don't assume that you have a graphing device, but if you do have one you should use it as a check on your work. â Guidelines for Sketching a Curve every item is relevant to every function. (For instance, a to make a sketch that displays the most important aspects of the function. The following checklist is intended as a guide to sketching a curve y- f(x) by hand. Not given curve might not have an asymptote or possess symmetry.) But the guidelines provide all the information you need A. Domain It's often useful to start by determining the domain D of f, that is, the set B. Intercepts The y-intercept is f(o) and this tells us where the curve intersects the of values of x for which f(x) is defined y-axis. To find the x-intercepts, we set y-0and solve for xã(You can omit this step if the equation is difficult to solve.) ry (a) Even fusction: reflectional symmetry all x in D, that is, the equation of the curve is unchanged when x is replacea by -x, then f is an even function and the curve is symmetric about the y-axis. This means that our work is cut in half. If we know what the curve looks like for 0, then we need only reflect about the y-axis to obtain the com- plete curve [see Figure 3(a). Here are some examples: y-ey-ry x). and for all x in D, then f is an odd function and the curve is symmetric about the origin. Again we can obtain the complete curve if we know what it looks like for x0. [Rotate 180Â° about the origin; see Figure 3(b).] Some b) Oud function rotational symmetry FIGURE 3 simple examples of odd functions are y-xy-x'y - and y-sin

MATA31H3 Midterm: MATA31 Midterm 2013 Winter

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