# Business Models with Exp. and Nat. Log and Diff. rules LAB I

Business Models with Exp. and Nat. Log and Diff. rules LAB I 150 150 Affordable Capstone Projects Written from Scratch

MTH 241 Business Models with Exp. and Nat. Log and Diff. rules LAB I
Name:__________________
Recitation Time: 8 9 10 11 12 1 2 3 4 5
1) A small company has the marketing information that 35 units will sell daily at a price of \$34.75 per unit, and that sales will rise to 36 units per day at a price of \$33.06 per unit.
Use this information to create a linear demand function, then create the associated revenue function and find the price that will yield the maximum revenue.
2) A small company has decided to model its demand function with the exponential function xexP05.200)(−= , where x is the number of items produced and sold daily and P(x) is the price
per unit in \$.
a) Is this model consistent with the marketing information from question1?
b) Does P(x) have the right shape to be a demand function?
c) Develop the associated revenue function.
d) Find the price that will yield the maximum revenue for this model.
e) Graph both functions and compare them to the graphs of the functions you worked with in problem 1.
f) Which models do you think are better?
Why?
3) For a different item in a different company, the cost function associated with producing
x items is: )50ln(5)(costxx⋅=
a) What is the marginal cost at a production level of 10 items?
b) How would you interpret that result?
c) What is the average cost function?
d) What is the derivative of the average cost function?
e) What is the value of that derivative function you just found in 2d at a production level of 50 items?
f) How would you interpret that result?
Recall:
If )()(xhxgy⋅= then )(‘)()()(‘xhxgxhxgdxdy⋅+⋅= [product rule]
If )()(xhxgy= then 2))(()(‘)()(‘)(xhxhxgxgxhdxdy⋅−⋅= [quotient rule]
If ))((xgfy= then )(‘))((‘xgxgfdxdy⋅= [chain rule]
Ex. 4) For F and G functions and given that:
1)5(−=F,3)5(‘−=F4)5(=G2)5(‘=G
a) If )()()(xGxFxP⋅=, then evaluate )5(‘P.
)5(‘P=_____________________
b) If )()()(xGxFxQ=, then evaluate )5(‘Q.
)5(‘Q=_____________________
Ex. 5) For R and Q functions and given that:
5)1(=−Q, 4)1(−=−R, 7)5(=R;
3)1(‘−=−Q, 2)1(‘=−R, 1)5(‘−=R
If ))(()(xQRxf=, then evaluate )1(‘−f.
)1(‘−f=_________________
Ex 6. Differentiate )()(xefxh= )(‘xh=________________
Ex. 7 Differentiate ))(ln()(xfxg= )(‘xg=________________
Ex. 8 Differentiate )()ln()(xfxxk= )(‘xk=________________
Ex 9. Differentiate )()(xfexmx= )(‘xm=________________
Ex 10. Differentiate xexfxn)()(= )(‘xn=________________
Ex 11. Differentiate )()ln()(xfxxp= )(‘xp=________________
Ex 12. Differentiate )()(xfexq= )(‘xq=________________

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