Lesson 58 – Quotient Rule 10/15/2015 IB Math SL1 - Santowski 1

Lesson Objectives 0 1. Develop the quotient rule us 0 2. Use the quotient rule to evaluate derivatives 0 2. Apply the quotient rule to an analysis of functions 0 3. Apply the quotient rule to real world problems 10/15/2015IB Math SL1 - Santowski2

Fast Five 0 1. If H = f/g, solve for f 0 2. Evaluate and interpret 0 3. Solve for x if 0 4. Solve for x if 0 5. Simplify 10/15/2015IB Math SL1 - Santowski3

(A) Derivatives of Rational Functions – The Quotient Rule 0 Since the derivative of a product does not equal the product of the derivatives, what about a quotient? 0 Would the derivative of a quotient equal the quotient of the derivatives? 0 Since quotients are in one sense nothing more than products of a function and a reciprocal  we would guess that the derivative of a quotient is not equal to the quotient of the derivatives 10/15/2015IB Math SL1 - Santowski4

(A) Derivatives of Rational Functions – The Quotient Rule 0 If a rational function can be written in the form,rewrite f(x) as ?????? And determine the expression for d/dx of R(x) 10/15/2015IB Math SL1 - Santowski5

(B) Quotient Rule - Derivation 0 First, set up a division and then rearrange the division to produce a multiplication so that we can apply the product rule developed earlier 10/15/2015IB Math SL1 - Santowski6

(C) Examples Using the Quotient Law 0 Differentiate each of the following rational functions. Simplify the derivative. 0 ex 1. 0 ex 2. 0 ex 3. 10/15/2015IB Math SL1 - Santowski7

(C) Examples Using the Quotient Law 0 Determine the equation of the line normal to at x = 0 10/15/2015IB Math SL1 - Santowski8

(D) Function Analysis of Rational Functions 0 Make a 2 column worksheet. In the first column, briefly state what needs to be done and why and in the second column, do it! 0 For what intervals is the function concave down? 10/15/2015IB Math SL1 - Santowski9

(D) Function Analysis of Rational Functions - HINTS 0 Determine the domain of f(x) 0 Determine the vertical asymptotes and the behaviour along the VA using limits 0 Determine the horizontal asymptotes using limits 0 Determine the x- and y-intercepts 10/15/2015IB Math SL1 - Santowski10

(D) Function Analysis of Rational Functions 0 Again, make a 2 column worksheet. In the first column, briefly state what needs to be done and why and in the second column, simply do what you said needs to be done! 0 Find the intervals of increase/decrease and max/min for the given function. Then sketch the function based on your intervals. 10/15/2015IB Math SL1 - Santowski11

(D) Function Analysis of Rational Functions - HINTS 0 Determine the domain of f(x) 0 Determine the vertical asymptotes and the behaviour along the VA using limits 0 Determine the horizontal asymptotes using limits 0 Determine the x- and y-intercepts 10/15/2015IB Math SL1 - Santowski12

(D) Function Analysis of Rational Functions 0 Again, make a 2 column worksheet. In the first column, briefly state what needs to be done and why and in the second column, simply do what you said needs to be done! 0 Find the intervals of increase/decrease and extrema for the given function. Sketch f(x) as well. 10/15/2015IB Math SL1 - Santowski13

(E) Applications - Economics 0 Suppose that the total cost in hundreds of dollars of producing x barrels of oil is given by the function C(x) = 4x x Determine the following. 0 (a) the cost of producing 5000 barrels of oil 0 (b) the cost of producing 5001 barrels of oil 0 (c) the cost of producing the 5001st barrel of oil 0 (d) C `(5000) = the marginal cost at a production level of 5000 barrels of oil. Interpret. 0 (e) The production level that minimizes the average cost (where AC(x) = C(x)/x)) 10/15/2015IB Math SL1 - Santowski14

(E) Applications 0 The number of used cars that Qassem sells per week, t weeks after dropping out of Calculus class, is 0 (a) At what rate is the number of sales changing after the 1 st week? After the 5 th week? 0 (b) Does the number of sales per week decrease at any time during the first ten week period? 10/15/2015IB Math SL1 - Santowski15

(G) Internet Links 0 Calculus I (Math 2413) - Derivatives - Product and Quotient Rule Calculus I (Math 2413) - Derivatives - Product and Quotient Rule 0 Visual Calculus Visual Calculus solving derivatives step-by-step from Calc101 solving derivatives step-by-step from Calc101 10/15/2015IB Math SL1 - Santowski16

(F) Homework 0 Text, S4.2, p234 0 (1) Algebra: Q1-28 odds 0 (2) Word Problems: Q (3) Word problems: Q50,51,53,54,55 from pg (4) “A” Level, Worksheet (p72), Q2,3,5,7,8 & (p81), Q2,3,4 10/15/2015IB Math SL1 - Santowski17