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Lecture

MAT136H1 Lecture Notes - Rate Function, Antiderivative, Marginal Cost


Department
Mathematics
Course Code
MAT136H1
Professor
all

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8.4 Challenging Integral Applications
Economics & Biology Applications
Question #1 (Easy): Marginal Cost Function Using the Net Change Theorem
Strategy
Net Change Theorem is practical approach that is equivalent to the Fundamental Theorem of Calculus
Part 2. Thus, given the marginal rate function, first take the integral to find the anti-derivative, then plug
in the net change values as the beginning and end points to determine the total cost over the interval.
Sample Question
Marginal cost function of publishing copies of books is given by     ,
where is in thousands, as well as the cost is in thousands of dollars. The fixed start-up cost is .
1) What is the cost of publishing the first  copies of the book?
2) What about the next  copies?
Solution
1) First, since the marginal cost function is provided, the total cost function needs to be
determined. This is simply the anti-derivative of . To find the cost from to , noting
that is in thousands:         
  
 
   
  

Note that start-up cost of , ie   which represents the constant factor which can
in no way be determined by taking the anti-derivative if it were not given by the question
needed to be added, because producing 0 number of books does not require any start-up cost.
Therefore, the cost of publishing the first  copies of the book is .
2) Next, publishing the next  books is:      

   
  
 

 
 
 
Note that   gets eliminated due to the net change theorem.
Therefore, the cost for publishing the next  copies of the book is .
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