Solve using calculus

W

writer2019

New member
Joined
Apr 1, 2019
Messages
19
Emma is starting a new business. he knows that most business when they start run at loss. Emma predict that in the first month she will lose $2200, in the second he will loose $1800 and in the third she will lose $1400 and this pattern will continue with each passing month (like arithmetic sequence where a=-2200 and d=400). Given the cumulative (running) total profit after n months is given by S(n)=n/2 (2a+(n-1)d) use calculus to determine:

a) The last month Emma will incur a net loss
b) The amount of loss during that month (given tn=a+(n-1)d)
c) The maximum cumulative loss (minimum cumulative profit) that Emma will experience


My working for this is quite stupid in my opinion. I try to work out these question by using the arithmetic sequence formula when we are asked to use calculus. I am not sure where to start to use calculus to solve this
Here is my working out:

a) t1+(n-1)d
-2200 + (7-1)*400 = 200
Thus during the seventh month it will be the start of the positive.

-2200 + (6-1)*400 = -200
In the sixth month it will be the last month of a loss.

b) the loss on the last month will be $-200.

The maximum cumulative loss by Emma will be

n/2 (2a+(n-1)d)
6/2 (2(-2200)+(6-1)*400 = $-7200 loss

Is my working out for the answer correct and if they are how can I use calculus to solve the questions?
 
writer2019 said:
Emma is starting a new business. he knows that most business when they start run at loss. Emma predict that in the first month she will lose $2200, in the second he will loose $1800 and in the third she will lose $1400 and this pattern will continue with each passing month (like arithmetic sequence where a=-2200 and d=400). Given the cumulative (running) total profit after n months is given by S(n)=n/2 (2a+(n-1)d) use calculus to determine:

a) The last month Emma will incur a net loss
b) The amount of loss during that month (given tn=a+(n-1)d)
c) The maximum cumulative loss (minimum cumulative profit) that Emma will experience


My working for this is quite stupid in my opinion. I try to work out these question by using the arithmetic sequence formula when we are asked to use calculus. I am not sure where to start to use calculus to solve this
Here is my working out:

a) t1+(n-1)d
-2200 + (7-1)*400 = 200
Thus during the seventh month it will be the start of the positive.

-2200 + (6-1)*400 = -200
In the sixth month it will be the last month of a loss.

b) the loss on the last month will be $-200.

The maximum cumulative loss by Emma will be

n/2 (2a+(n-1)d)
6/2 (2(-2200)+(6-1)*400 = $-7200 loss

Is my working out for the answer correct and if they are how can I use calculus to solve the questions?
Click to expand...
Not sure how calculus can be used. But I would start by showing how to find the answer in a) instead of just confirming that 6 works.
 
Regarding the use of calculus - could it be that you added some of your prior knowledge into the text of the question? Perhaps you needed to find the "profit at month n equation" by doing something with S(n)=n/2 (2a+(n-1)d)? This might result in a slightly different equation to the one that you used.

I agree with lev888 about how you found 6 months, there's a direct way instead trying different numbers.
 
Cubist said:
Regarding the use of calculus - could it be that you added some of your prior knowledge into the text of the question? Perhaps you needed to find the "profit at month n equation" by doing something with S(n)=n/2 (2a+(n-1)d)? This might result in a slightly different equation to the one that you used.
Click to expand...

Actually this still wouldn't involve calculus. The rate of change of the running profit isn't the same as the profit for one month. One month profit would be "S(n) - S(n-1)" which indeed comes out as "a+d*(n-1)"

So I'm stumped! It might be worth you double checking the exact wording of the question though.
 
I agree. It's clear that you didn't show us the exact problem as given to you; at least you made many typos, but I think you changed more than that. So please show us the original (even if it's in a different language!). We need to see your thoughts separate from the original problem.

But if it actually says to use calculus, all I can think is that since calculus is just an extension of algebra, and series are often taught along with calculus, they just want you to use algebra and the given formulas.
 
writer2019 said:
Emma is starting a new business. he knows that most business when they start run at loss. Emma predict that in the first month she will lose $2200, in the second he will loose $1800 and in the third she will lose $1400 and this pattern will continue with each passing month (like arithmetic sequence where a=-2200 and d=400). Given the cumulative (running) total profit after n months is given by S(n)=n/2 (2a+(n-1)d) use calculus to determine:

a) The last month Emma will incur a net loss
b) The amount of loss during that month (given tn=a+(n-1)d)
c) The maximum cumulative loss (minimum cumulative profit) that Emma will experience


My working for this is quite stupid in my opinion. I try to work out these question by using the arithmetic sequence formula when we are asked to use calculus. I am not sure where to start to use calculus to solve this
Here is my working out:

a) t1+(n-1)d
-2200 + (7-1)*400 = 200
Thus during the seventh month it will be the start of the positive.

-2200 + (6-1)*400 = -200
In the sixth month it will be the last month of a loss.

b) the loss on the last month will be $-200.

The maximum cumulative loss by Emma will be

n/2 (2a+(n-1)d)
6/2 (2(-2200)+(6-1)*400 = $-7200 loss

Is my working out for the answer correct and if they are how can I use calculus to solve the questions?
Click to expand...
You wrote b) the loss on the last month will be $-200. That is not correct. You should say that the loss on the last month will be $200. A loss of $-200 is actually a gain of $200. I just thought that I should point this out.
 
Hi the wording of this question is correct. I copied it straight as I found it. I have attached a copy for your reference. The working out is mine, so I apologise for that. Please see the question and guide me in the right path to solving the question through calculus. Like I said i figured the easiest way to solve this was through sequences but you are correct there must be a straightforward method to solve this but I am unable to get anything.


2019-11-13_192333.jpg
 
Well, no, you didn't type it exactly as it was given; you changed the name, some pronouns, and so on. But I was sure the aside about a=-2200 was your addition, as it can easily be deduced from what was given before. And it does say to use calculus, which none of us can make sense of. It might be helpful if you told us the context -- what calculus would you be expected to use?

Anyway, apart from the fact that you don't show how you got n=7, your answer looks fine.

The straightforward method is not to avoid sequences (that's what you're told to do!), but to use algebra. Write the inequality a + (n-1)d > 0, with the appropriate constants, and solve for n. You'll find that the smallest value of n is 7.
 
I can now see a way of using calculus in this. I'm not a fan of this question at all. It seems very contrived. Anyway...


In part a) the marks awarded are much higher. Therefore this is the difficult part, and this is probably where they want you to use the calculus. You could answer this by starting with...

In an arithmetic sequence the variable n is an integer. In order to use calculus, I will substitute nr a real number in place of n for the purpose of finding the turning point of S()

... <perform the calculation. Maybe also prove that it is a minimum?>

The turning point was nr=<xxx>.

<This might not be integer, if it isn't then include something like the following>

This means that the last month where Niles will incur a net loss will be one of:- [math]\mathit{n_1}=\left\lfloor{\mathit{n_r}}\right\rfloor[/math]; or [math]\mathit{n_2}=\left\lceil{\mathit{n_r}}\right\rceil[/math]
I substitute these values back into the arithmetic sequence formula to determine which one is the correct, integer, answer.
 
...or I guess you could say that S() is symmetric around the turning point, therefore you can round nr to the nearest integer
 
Cubist said:
I can now see a way of using calculus in this. I'm not a fan of this question at all. It seems very contrived. Anyway...
Click to expand...

For the benefit of others who may find this thread, never use calculus for this type of problem unless explicitly asked to do so. Prefer the method that Dr Peterson points out above. And be careful whether you are asked for the first month of profit or the last month of loss :)
 
You must log in or register to reply here.
Top