INTRODUCTORY MATHEMATICAL ANALYSIS
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1 INTRODUCTORY MATHEMATICAL ANALYSIS For Business, Economics, and the Lie and Social Sciences Chapter 11 Dierentiation 011 Pearson Education, Inc.
2 Chapter 11: Dierentiation Chapter Objectives To compute derivatives by using the limit deinition. To develop basic dierentiation rules. To interpret the derivative as an instantaneous rate o change. To apply the product and quotient rules. To apply the chain rule. 011 Pearson Education, Inc.
3 Chapter 11: Dierentiation Chapter Outline 11.1) The Derivative 11.) Rules or Dierentiation 11.3) The Derivative as a Rate o Change 11.4) The Product Rule and the Quotient Rule 11.5) The Chain Rule and the Power Rule 011 Pearson Education, Inc.
4 Chapter 11: Dierentiation 11.1 The Derivative Tangent line at a point: (tangent is Latin or touching) (a) (a) a a The slope o a curve at P is the slope o the tangent line at P. The slope o the tangent line at (a, (a)), Za h is m tan lim 011 Pearson Education, Inc. ( z) ( a) ( a h) ( a) z a lim z a h 0 h
5 secant line The slope o the tangent line at (, ()) is m tan lim ( z) ( ) ( h) ( ) z lim z h 0 z h 011 Pearson Education, Inc.
6 11.1 The Derivative E Find the slope o the tangent line to the curve y () at the point (1, 1). Solution: ( h) ( ) m tan lim h 0 lim h 0 h ( 1 h) ( 1) ( 1 h) ( 1) h lim h 0 h The derivative o a unction is the unction denoted and deined by ' ( ) lim ( z) ( ) ( h) ( ) z lim z h 0 h 011 Pearson Education, Inc.
7 11.1 The Derivative E I () 3, ind an equation o the tangent line to the graph o at (1, 7). Solution: ' ( ) lim h 0 ( ) 3 ( 3)? ( h) ( ) ( h) ( h) h lim h 0 h slope Find the equation or the tangent line 011 Pearson Education, Inc.
8 011 Pearson Education, Inc.
9 Derivative o a unction () (written as () and read as prime ) ' ( ) lim h 0 ( h) ( ) h I (a) can be ound, then is said to be dierentiable at a. y d ( ) d y' dy ( ) y'( a) dy d a d Dy 011 Pearson Education, Inc.
10 E A Function with a Vertical Tangent Line at 0 d d ( ) Find. Solution: d d ( ) h lim 0 h h 1? E a. For (), Will it be continuous or all? 1 p b. For (p), Will it be continuous or all? Will the derivative eist or all? 011 Pearson Education, Inc.
11 I is dierentiable at a, then is continuous at a. The converse is not necessarily true a unction that is continuous at a is not always dierentiable at a. A unction can ail to be dierentiable at a point a i either ( a h) ( a) lim h 0 h does not eist, or is ininite. 011 Pearson Education, Inc.
12 In the ormer case, we sometimes have a cusp on the graph, and in the latter case, we get a point o vertical tangency. 011 Pearson Education, Inc.
13 Chapter 11: Dierentiation 11. Rules or Dierentiation Rules or Dierentiation: RULE 1 Derivative o a Constant: RULE Derivative o n : RULE 3 Constant Factor Rule: RULE 4 Sum or Dierence Rule d d 011 Pearson Education, Inc. d d d d d d ( c) 0 ( n) n 1 n ( c( ) ) c '( ) ( ( ) ± g( ) ) '( ) ± g' ( )
14 11. Rules or Dierentiation E Derivatives o Constant Functions a. d d ( 5 ) 0 ( ) 10 b. I g, then g' ( ). 011 Pearson Education, Inc.
15 E Rewriting Functions in the Form n Dierentiate the ollowing unctions: Solution: a. y dy d 1 1 ( 1/ ) 1 b. y y' 6 0 c. h ( ) h' ( ) Pearson Education, Inc.
16 Eample 5 Dierentiating Sums and Dierences o Functions Dierentiate the ollowing unctions: a. b. 5 ( ) 3 F F' ( ) ( ) ( 4 3 ) '( ) c. ( ) 3 '( ) 011 Pearson Education, Inc.
17 E Dierentiating Sums and Dierences o Functions d. y dy d E: Find the tangent line to the curve ( ) 1 when 011 Pearson Education, Inc.
18 E Finding an Equation o a Tangent Line 1. Find an equation o the tangent line to the curve 4 when 1. y 3. Find all points on the curve where the slope is 4. y Pearson Education, Inc.
19 Chapter 11: Dierentiation 11.3 The Derivative as a Rate o Change Average velocity is given by Velocity at time t is given by v ave v s t t lim t 0 ( t t) ( t) t t ( t) ( t) dierentiation 011 Pearson Education, Inc.
20 Secant line Slope o secant line average velocity Tangent line: Slope o tangent line instantaneous velocity 011 Pearson Education, Inc.
21 E Finding Average Velocity and Velocity Suppose the position unction o an object moving along a number line is given by s (t) t 1, where t is in seconds and s is in meters. a. Find the average velocity over the interval [5, 5.5]. b. Find the velocity when t 5 sec. 011 Pearson Education, Inc.
22 11.3 The Derivative as a Rate o Change E Finding Average Velocity and Velocity Solution: a. When t 5, v ave s t ( t t) ( t) ( 5 0.5) ( 5) ( 5.5) ( 5) 0.5 t 0.5 b. Velocity at time t is given by m/s ds v 4t dt When t 5, the velocity is ds dt t 5 4 ( 5) 0m/s 011 Pearson Education, Inc.
23 y dy d dy y d 011 Pearson Education, Inc.
24 11.3 The Derivative as a Rate o Change I y (), then y ( ) ( ) average rate o change o with respect to over the interval rom to y dy d lim y instantaneous rate o y with respect to 0 change o E. Find the rate o change o y 4 with respect to, and evaluate it when 3 and when Pearson Education, Inc.
25 11.3 The Derivative as a Rate o Change E Rate o Change o Volume A spherical balloon is being illed with air. Find the rate o change o the volume o air in the balloon with respect to its radius. Evaluate this rate o change when the radius is 3 t. 4 v π 3 ( 3) r 011 Pearson Education, Inc.
26 11.3 The Derivative as a Rate o Change Applications o Rate o Change to Economics Total-cost unction is c (q). Marginal cost is the cost o one additional unit o any item produced or bought in quantity. It is deined as: dc dq change o cost change o quantity 011 Pearson Education, Inc.
27 Total-revenue unction is r (q). Marginal revenue is deined as the rate o change o the total revenue with respect to the total number o units sold. dr dq revenue received rom selling one additional unit o output Relative and Percentage Rates o Change The relative rate o change o () is. ' ( ) ( ) The percentage rate o change o () is ' ( ) ( ) ( 100% ) 011 Pearson Education, Inc.
28 I a manuacturer s average-cost equation is c q 1. Find the cost unction. 0.0q q. Find the total cost when 50 units are produced. 3. Find the marginal-cost unction. 4. What is the marginal cost when 50 units are produced? 5. Find the total cost or producing 51 units. 011 Pearson Education, Inc.
29 11.3 The Derivative as a Rate o Change E Relative and Percentage Rates o Change y ( ) Determine the relative and percentage rates o change o when Pearson Education, Inc.
30 11.4 The Product Rule and the Quotient Rule The Product Rule d d ( ( ) g( ) ) '( ) g( ) ( ) g' ( ) 011 Pearson Education, Inc.
31 011 Pearson Education, Inc The Product and Quotient Rule E 1 Applying the Product Rule Find F (). ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )( ) ' d d d d F F ( ) ( )( ) ( ) ( )( ) H G Find G () and H ()
32 011 Pearson Education, Inc. 1) 1)(3 1)( ( : Given y Find y. Dierentiating the product o three actors ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) h g h g h g h g g g h h g g d d h h g d d
33 011 Pearson Education, Inc. E. I, ind F (). The Quotient Rule ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ' ' g g g g d d ( ) F ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ' d d d d F
34 11.4 The Product and Quotient Rule E Dierentiate the ollowing unctions. a. ( ) '( ) 5 3 c. ( ) '( ) b. ( ) ( 3 ) '( ) d. ( ) '( ) Pearson Education, Inc.
35 E. 011 Pearson Education, Inc.
36 I total national income C consumption Marginal propensity to consume dc di The rate o change o consumption with respect to national income 011 Pearson Education, Inc.
37 Saving Income Consumption (S I C) ds di di dc di di dc 1 marginal di propensity to save Marginal propensity to save 1 marginal propensity to consume 011 Pearson Education, Inc.
38 MPS(Marginal Propensity to Save) indicates what household sector does with etra income ( bonus, ta return, raised income ). I MPS is 0.5, then 5% o etra income goes to saving. By U.S. data, U.S. national saving is -3.9% at 3/13/ Pearson Education, Inc.
39 Finding Marginal Propensities to Consume and to Save I the consumption unction is given by C 6 3I 4 I 3 determine the marginal propensity to consume and the marginal propensity to save when I Pearson Education, Inc.
40 Marginal Revenue E. Here is the demand unction or a certain product, where p denotes the price per unit or q units. Find the marginalrevenue unction. p q 750 q Pearson Education, Inc.
41 EX. Suppose that the savings unction o a country is S I 8 where the national income (I ) and the national savings (S) are measured in billions o dollars. Find the country s marginal propensity to consume and its marginal propensity to save when the national income is $150 billion. I I 011 Pearson Education, Inc.
42 11.5 The Chain Rule Suppose you are asked to dierentiate the unction 1 (Composite unction) Let, let 1 In Leibniz notation, i and and both are dierentiable unctions, then is the product o the derivatives o and g. 011 Pearson Education, Inc.
43 Chain Rule: dy d dy du du d Power Rule: d ( u ) d nu n n 1 du d 011 Pearson Education, Inc.
44 E Using the Chain Rule ? b. I y u 3u and u 4, ind dy/d. or dy d dy d dy du du d 011 Pearson Education, Inc. d du ( )( ) ( 4u 3 4 4) d d ( ) ( u 3u 4) [ ( ) ]( ) ( ) ( 3 3)( ) 8 6 [ 3 ]( ) 8 6
45 E Using the Chain Rule I and 4, ind E Using the Power Rule I y ( 1) 100, ind y. 011 Pearson Education, Inc.
46 E: 1 a. I y ind dy/d. b. I y ( ) 4 ( 3 4), ind y Pearson Education, Inc.
47 Revenue Unit Price Quantity Marginal Revenue: the rate o change o the total dollar value received with respect to the total number o units sold, 011 Pearson Education, Inc.
48 E: A monopolist who employs m workers inds that the produce 1 units per day. The total revenue r in dollars is given by a) What is the price per unit (to the nearest cent) when there are 1 workers? b) Determine the marginal revenue when there are 1 workers. 011 Pearson Education, Inc.
49 During a lu outbreak at a school o 763 children, the number o inected children, D, is epressed as a unction o the number o susceptible children, H, by H D 00 ln H What is the maimum possible number o inected children? Justiy your answer. 011 Pearson Education, Inc.
50 The total-cost unction or a manuacturer is given by 5q C q where c is in dollars. Find the marginal cost when 15 units are produced. 011 Pearson Education, Inc.
51 A manuacturer has ound that when m employees are working, the number o units o product produced per day is q 10 m The demand equation or the product is 8qp where p is the selling price when the demand or the product is q units per day. a. Determine the manuacturer s marginal-revenue product when m40 b. Find the relative rate o change o revenue with respect to the number o employees when m40 c. Suppose it would cost the manuacturer $400 more per day to hire an additional employee. Would you advise the manuacturer to hire the 41 st employee? Why or why not? 011 Pearson Education, Inc ,
52 I p rate o q 13 q 4 is a demand equation, ind the change o pricep with respect to quantity q. 011 Pearson Education, Inc.
53 I p 0.15q 450 is a demand equation, ind the marginal- revenue unction. 011 Pearson Education, Inc.
54 I c 0.04q 1.5 ind the marginal cost when q q is an average- cost unction, ind the 011 Pearson Education, Inc.
Differentiation. The main problem of differential calculus deals with finding the slope of the tangent line at a point on a curve.
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