Class 1--Math Review and Use of Calculus

                                                                     MGT 5149

 

I. NOTES ON CALCULUS

Must be able to differentiate a function:

 

given the function {f(x)}  y=x³ -3x² + 5x +10

 

the derivative of the function is:

 

     {f'(x) or dy/dx}        = 3x²  -  6x  +  5  

 

the 2nd derivative is:

 

{f''(x)}     =  6x - 6

 

The 1st derivative gives the rate of change of the original function  (the slope), the 2nd derivative gives the rate of change of the marginal function.

 

USES:  given a u-shaped function, what is the maximum pt., or min. pt.

 

If the first derivative is zero (slope of 0), then the curve is flat at that point (a max or min).

 

             

And if the second deriv. is negative that indicates it is a max., whereas if the second deriv. is positive, it is a minimum.

 

So solving a cost minimization problem may simply involve taking the derivative of the cost function, then setting equal to 0 and solving for "x"  which in this case would be quantity.

 

Example: TR = 2000Q -2Q²       TC = 5000 +100Q +50Q² + 2Q³

 

Given a TR and TC function. One solution would involve taking the first derivative of each function, to get MC and MR. Then set them equal since profit maximization will occur where MC = MR.  Solve for Q. 

 

That is: MR = d(TR)/dQ = 2000 - 4Q

 

And:     MC = d(TC)/dQ = 0 +100 + 100Q + 6Q²

 

Finally:  2000 -4Q = 100 + 100Q + 6Q²      6Q²+ 104Q -1900 = 0

 

Use the quadratic formula to get two roots:  11.13  ,  - 28.46  (π = $6947.90)

 

Alternative: Profit will be TR-TC.  So subtract to get a profit function, then take the deriv. and set equal to 0.  (get two roots to the quadratic equation:                ).

 

 

 

II. Other Math Topics:

A. Rules of exponents

1) X^a(X^b) = X^(a+b)

2) (X^a)^b = X^(a*b)

3) (XY)^a = X^aY^a

4) (X/Y)^a =X^a/Y^a

 

5) X^a/X^b = X^(a-b)

 

 

B. Rules of Logarithms

1) LogXY = LogX + LogY

2) Log(X/Y) = LogX - LogY

3) Log X^a = aLogX

4) Log1 = 0