MATH 271 The University of Oklahoma Limits and Continuity Business Math Questions

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MAC2233 Business Calculus Summer 2020 Instructor: Stephen Lappano Limits and Continuity: Limits and Continuity are mathematical concepts that have meaning in the business realm too. We give a couple of examples in this worksheet. Example: Mr. Snow opened a local eatery in Tampa in January 2018, and then he expanded his operations by opening another location in October 2018. The graph of the right shows the monthly revenue 𝑅 from his operations, where 𝑡 is the number of months since January 2018. The dotted line shows future projections. (a) Mathematically, why is there a discontinuity at 𝑡 = 10? Is this discontinuity removable? (b) Practically (i.e. considering this specific example) why is there a discontinuity at 𝑡 = 10? (c) What is the value lim−(𝑅). What is the meaning of the value in this context? 𝑡→10 (d) What is the value lim+(𝑅). What is the meaning of the value in this context? 𝑡→10 (e) How much additional revenue does the new location add to Mr. Snows revenue immediately? (f) Would you expect to see the same kind of behavior in the graph if we were considering Mr. Snows PROFIT instead of his revenue? MAC2233 Business Calculus Summer 2020 Instructor: Stephen Lappano Rates of Change Graphically: We encounter rates of change in our everyday life. For example, speed is a rate of change of distance over time. In terms of the graph, rates of change are all about slope. Example: The monthly revenue R (in thousands of dollars) of a bakery store 𝑡 months after opening is pictured below. (Note: You should show your work using the graph.) (a) Estimate the average rate of change over the interval [0,3]. Be careful with the units! It’s a rate! (b) Estimate the average rate of change over the interval [3,8]. You should be able to explain how this is related to the above graph! (c) Sketch the tangent line to the graph at the point corresponding to 𝑡 = 3. Then do the same for 𝑡 = 6. (Google it if you’re not sure what a tangent line is!). You should be drawing 2 lines! (d) Use your lines from (c) to estimate the instantaneous rate of change at 𝑡 = 3 and 𝑡 = 6. You should explain what you are doing! (e) According to the graph, at what 𝑡 value(s) is the graph changing the fastest?
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a) There is a discontinuity at t = 10, because the limit as t approaches from left and right are not
equal.
lim− 𝑓(𝑥) = 14 ≠ lim+ 𝑓(𝑥) = 21
𝑡→10

𝑡→10

b) Practically it means the revenue before t =10 (October 2018) is much less than the revenue after
and at t =10 (October 2018).

lim 𝑓(𝑥) = 14

𝑡→10−

This means the revenue before t =10(October 2018) is 14000 dollars.

l...


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