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MTH 173
Your 'question form' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.
** Question Form_labelMessages **
question regarding final exam
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here are some question i practiced. Can you help me with them
Question 1
The weight of a stack of logs in a fireplace is w(t), where t is the time in minutes since the fire was started. The fire starts small as the logs are lighted and ends small as the burned-out cinders smolder.
• Sketch a graph of w(t) vs. t and describe the behavior of w’(t) (e.g, where is it increasing, where is it decreasing, etc.).
• If w(0) = 28 lbs, then what do you think the maximum value of w ' (t) might be? What might w(t) be at the instant w ' (t) is a maximum? Give specific justification for each answer.
• What is w’(t) at the instant the fire goes out?
The graph is a decreasing one. Starting concave down as fire starts small, it becomes steeper and decreases faster as fire catches up. Has an inflection point about midway and then concave up for further time of fire. This is because fire further reduces and finally the w(t) function reaches a 0 value as fire goes out.
Given w(0) = 28 lbs. maximum value of w’(t) will be 0 as w’(t) is negative for all times except when fire goes out at the end. The value of w(t) at that time will be 0.
w’(t) when fire goes out is 0 as no further decrease in w(t) occurs after the time fire goes out.
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Good.
Should the question ask for the maximum of | w ' (t) |, which is possible, you would want to make an estimate according to your graph. The max of | w ' (t) | would occur at the inflection point.
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Question 2
A container holds about a gallon of cool water, which is being constantly and thoroughly stirred. A 4-pound lump of hot steel is quickly added to the container and with an initial 'halflife' of about 21 seconds gives up its thermal energy to the water. 63 seconds later the lump is quickly removed from the water and 7 seconds later a 7.5-pound chunk of cold concrete is quickly added; the lump warms up at the expense of the thermal energy of the water, with an initial 'halflife' of about 44 seconds.
Sketch a graph showing the temperature of the water as a function of clock time.
• Is this function continuous?
• Is this function differentiable?
Can you help me with the graph. I really do not understand how to proceed with this?
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When an object is added to the water it will exchange thermal energy with the water, heating up if its temperature is below that of the water, cooling down when its temperature is greater than that of the water. The exchange will cease when the water and the object reach a common temperature.
The greater the difference in the temperatures the more quickly the temperature of the object will change.
If it takes, say, 10 seconds for the object to give up half of its excess thermal energy, then it will take 10 additional seconds to give up half of what remains, and another 10 seconds to give up half of what still remains.
At the instant the object is removed from the water, the energy exchange suddenly ceases and the temperature of the water stops changing.
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Question 3
As you zoom in toward the origin some of the functions on the following list become indistinguishable. Group the functions so that all the functions in each group have graphs which cannot be distinguished from one another if we zoom in close enough to the origin:
y = arcsin x y = sin x - tan x y = x - sin x y = arctan x y = sin x / (1 + sin x)
y = x^2 / (1 + x^2) y = (1 - cos x) / cos x y = x / (1 + x^2) y = sin x / x - 1
y = - x ln x y = e^x - 1 y = x^10 / x^(1/10) y = x / (x + 1)
For each group give the equation of the function near the origin. code `t
We just need to see the value of the function as x tends to 0 and the value of the first derivative. If it is same, the graph becomes indistinguishable.
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That is correct.
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Question 4
A container consists of three uniform circular cylinders stacked one on top of the other. The first cylinder has a diameter of 21 cm, the second a diameter of 63 cm and the third a diameter of 36 cm. The height of each cylinder is equal to its radius. Water is being added to the system at a constant rate.
Sketch a graph showing the behavior of water depth vs. clock time.
• Is this function continuous?
• Is this function differentiable?
Just wanted to make sure that when the when the water changes container it will be instantaneous and the function will not be differentiable when at the time when water changes containers.
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That is correct.
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Question 5
The cost and revenue functions for a certain product are equal when the number of units produced is x = 980. The cost function has y-intercept $ 6100 while the revenue when no units are produced is naturally 0. The graphs of both functions are increasing for all x. The cost function is concave down between initial production x = 0 and final production x = 1646 . The revenue function is increasing and concave down for all x, while the cost function has a point of inflection at x = 1646 and becomes concave up for x > 1646.
Sketch a graph of cost and revenue functions which satisfy this description, and sketch a graph of the resulting profit function. Indicate the following:
• the number of units required to break even.
• the number of units, according to your graph, which will result in the greatest profit.
• the approximate values of the marginal cost and marginal revenue when the profit is maximized.
I am good with the graphs
What does it mean by the units required to break even
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To break even is to have neither a profit nor a loss.
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And can you get me hints to solve the 2nd to parts of the question
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The marginal cost is the rate at which cost changes, i.e., the derivative of the cost function.
The marginal revenue is the rate at which revenue changes, i.e., the derivative of the revenue function.
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