Section 11.3 d i E The Derivative as a Rate of Change a l l u d Dr. Abdulla Eid b A College of Science . r D MATHS 104: Mathematics for Business II Dr. Abdulla Eid (University of Bahrain) Rate of Change 1 / 11
Rate of change Recall: If a line has a slope m = 3, it means that for every one step to the right, we move along the line 3 steps up. In this case, as x increases, d i y increases. E If a line has a slope m = − 2, it means that for every 1 step to the a l l right, we move along the line 2 steps down. In this case, as x u d increases, y decreases. b A For general function y = f ( x ) , for every step to the right, how many steps to go up/down? How do we measure that change in y ? . r If x changes by 1, an estimate of the change in y is dy D dx . Definition The derivative of y = f ( x ) can be interpreted as rate of change of y in term of x . Dr. Abdulla Eid (University of Bahrain) Rate of Change 2 / 11
Definition Let y = f ( x ) be a function, then The rate of change of f ( x ) is d i f ′ ( x ) E a l The relative rate of change of f ( x ) is l u d b f ′ ( x ) A f ( x ) . r D The percentage rate of change of f ( x ) is f ′ ( x ) f ( x ) · 100% Dr. Abdulla Eid (University of Bahrain) Rate of Change 3 / 11
Example (Old Final Exam Question) It is projected that x months from now, the 3 2 + 5000. At what population of a certain town will be P ( x ) = 2 x + 4 x percentage rate of change will the population be changing with respect to time 9 months from now? d i E Solution: a l l u Percentage rate = P ′ ( x ) d P ( x ) · 100% b A 1 2 + 6 x 2 . = · 100% r D 3 2 + 5000 2 x + 4 x Now we substitute x = 9 to get Percentage rate = 0.1288% Dr. Abdulla Eid (University of Bahrain) Rate of Change 4 / 11
Exercise (Old Exam Question) Consider the cost function c ( q ) = 1.3 q 2 + 0.2 q − 8. Determine the percentage rate of change of c with respect to q when q = 10. d i E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Rate of Change 5 / 11
Marginal Cost Recall: The total–cost of any manufacturer is calculated based on the quantity that is being produced. usually, we write this as c = f ( q ) d i E Definition a l The rate of change of c with respect to q is called marginal cost , l u d marginal cost = dc b A dq . r D Definition The average cost per unit for a total cost function is given by c = c q Note: c = qc . Dr. Abdulla Eid (University of Bahrain) Rate of Change 6 / 11
Example (Old Exam Question) Find the marginal cost function if the average cost function is c = 2 q + 10000 q 2 d i Solution: Recall that E marginal cost = dc a dq l l u We need first to find the cost function which is given by d b � 2 q + 10000 � A c ( q ) = qc = q q 2 . r = 2 q 2 + 10000 D q = 2 q 2 + 10000 q − 1 hence, marginal cost = dc dq = 4 q − 10000 q − 2 Dr. Abdulla Eid (University of Bahrain) Rate of Change 7 / 11
Exercise Find the marginal cost function if the average cost function is c = 0.002 q 2 − 0.5 q + 60 + 7000 q d Find the marginal cost for q = 15 and q = 25 i E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Rate of Change 8 / 11
Example Find the marginal revenue function if the revenue function is r = 2 q ( 30 − 0.1 q ) d Find the marginal revenue at q = 10, and q = 20. i E Solution: Recall that a marginal revenue = dr l l u dq d b We need first to rewrite the revenue function which is given by A r ( q ) = 2 q ( 30 − 0.1 q ) . r D = 60 q − 0.2 q 2 hence, marginal revenue = dr dq = 60 − 0.4 q Dr. Abdulla Eid (University of Bahrain) Rate of Change 9 / 11
Continue... To find the marginal revenue at q = 10 and q = 20, we substitute in the d derivative to get i E a l l marginal revenue = dr u = 60 − 0.4 ( 10 ) = d dq q = 10 b A marginal revenue = dr = 60 − 0.4 ( 20 ) = dq q = 20 . r D Dr. Abdulla Eid (University of Bahrain) Rate of Change 10 / 11
Exercise Find the marginal revenue function if the revenue function is r ( q ) = 240 q + 40 q 2 − 2 q 3 Find the marginal cost for q = 15 and q = 25 d i E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Rate of Change 11 / 11
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