Exam 1: Functions, graphs, and Limits

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Minimum cost.From a tract of land,a developer plans to fence a rectangular region and then divide it into two identical rectangular lots by putting a fence down the middle.Suppose that the fence for the outside boundary costs $7 per foot and the fence for the middle costs $4 per foot.If each lot contains 5000 square feet,find the dimensions of each lot that yield the minimum cost for the fence. ​

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​The average time between incoming calls at a switchboard is 3 minutes.If a call has just come in,the probability that the next call will come within the next t minutes is ​The average time between incoming calls at a switchboard is 3 minutes.If a call has just come in,the probability that the next call will come within the next t minutes is   Find the probability that the next call will come within the next   minute.Round your answer to two decimal places. Find the probability that the next call will come within the next ​The average time between incoming calls at a switchboard is 3 minutes.If a call has just come in,the probability that the next call will come within the next t minutes is   Find the probability that the next call will come within the next   minute.Round your answer to two decimal places. minute.Round your answer to two decimal places.

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​Use analytic methods to find the limit as ​Use analytic methods to find the limit as   for the given function. ​ ​   ​ for the given function. ​ ​ ​Use analytic methods to find the limit as   for the given function. ​ ​   ​ ​

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​The red figure is translated to a new position in the plane to form the blue figure.Find the vertices of the transformed figure from the following graph.(In case your exam is printed in black and white - the red figure has one vertex at (0,0)). ​ ​ ​The red figure is translated to a new position in the plane to form the blue figure.Find the vertices of the transformed figure from the following graph.(In case your exam is printed in black and white - the red figure has one vertex at (0,0)). ​ ​   ​ ​

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​Find the limit. ​ ​ ​Find the limit. ​ ​   ​ ​

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​Find the second derivative for the function ​Find the second derivative for the function   and solve the equation   ​ and solve the equation ​Find the second derivative for the function   and solve the equation   ​ ​

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​The cumulative sales (in thousands of units)of a new product after it has been on the market for t years may be modeled by ​The cumulative sales (in thousands of units)of a new product after it has been on the market for t years may be modeled by   During the first year,5000 units were sold.What is the saturation point for this product? How many units will be sold after 5 years? ​ During the first year,5000 units were sold.What is the saturation point for this product? How many units will be sold after 5 years? ​

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​Locate any relative extrema and inflection points of the function ​Locate any relative extrema and inflection points of the function   Use a graphing utility to confirm your results. ​ Use a graphing utility to confirm your results. ​

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​What percent of a present amount of radioactive radium ​What percent of a present amount of radioactive radium   will remain after 900 years? The half-life of radium is 1,600 years. ​ will remain after 900 years? The half-life of radium is 1,600 years. ​

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Sketch the graph of the equation. Sketch the graph of the equation.   ? ​ ? ​

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​Given ​Given   and   ,find   ​ and ​Given   and   ,find   ​ ,find ​Given   and   ,find   ​ ​

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​Find the limit. ​ ​ ​Find the limit. ​ ​   ​ ​

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​Determine whether the statement is true or false.If it is false,explain why or give an example that shows it is false. ​ If ​Determine whether the statement is true or false.If it is false,explain why or give an example that shows it is false. ​ If   ,then   ​ ,then ​Determine whether the statement is true or false.If it is false,explain why or give an example that shows it is false. ​ If   ,then   ​ ​

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​Find ​Find   for the equation   ​ for the equation ​Find   for the equation   ​ ​

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​A population of bacteria is introduced into a culture.The number of bacteria P can be modeled by ​A population of bacteria is introduced into a culture.The number of bacteria P can be modeled by   where t is the time (in hours).Find the rate of change of the population when   ​ where t is the time (in hours).Find the rate of change of the population when ​A population of bacteria is introduced into a culture.The number of bacteria P can be modeled by   where t is the time (in hours).Find the rate of change of the population when   ​ ​

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Analytically determine the location(s)of any horizontal asymptote(s). ​ ​ Analytically determine the location(s)of any horizontal asymptote(s). ​ ​   ​ ​

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The radius r of a sphere is increasing at a rate of 5 inches per minute.Find the rate of change of volume when The radius r of a sphere is increasing at a rate of 5 inches per minute.Find the rate of change of volume when   inches.Round your answer to one decimal place. ​ inches.Round your answer to one decimal place. ​

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​Medication.The number of milligrams x of a medication in the bloodstream t hours after a dose is taken can be modeled by ​Medication.The number of milligrams x of a medication in the bloodstream t hours after a dose is taken can be modeled by     Find the maximum value of x.Round your answer to two decimal places. ​ ​Medication.The number of milligrams x of a medication in the bloodstream t hours after a dose is taken can be modeled by     Find the maximum value of x.Round your answer to two decimal places. ​ Find the maximum value of x.Round your answer to two decimal places. ​

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Find the x-values (if any)at which f(x)is not continuous and identify whether they are removable or nonremovable. ​ Find the x-values (if any)at which f(x)is not continuous and identify whether they are removable or nonremovable. ​   ​​ ​​

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​Find the second derivative of the function. ​ ​Find the second derivative of the function. ​   ​ ​

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