MAT 125 – Applied Calculus 3.6 – Implicit Differentiation and Related Rates

Today’s Class  We will be learning the following concepts today:  Differentiating Implicitly  Related Rates Dr. Erickson Implicit Differentiation and Related Rates 2

 Up to now, we have dealt with functions expressed in the form y = f (x), that is, functions in which the dependent variable y is expressed explicitly in terms of the independent variable x.  However, not all functions are expressed in this form. Some equations are expressed implicitly as a function of x. For example,  We can find the derivative using implicit differentiation Implicit Differentiation and Related Rates 3 Introduction Dr. Erickson

Implicit Differentiation  We need to assume that dy/dx exists. Dr. Erickson Implicit Differentiation and Related Rates 4

Find the derivative using implicit differentiation Implicit Differentiation and Related Rates 5 Example 1 Dr. Erickson

Find the derivative using implicit differentiation Implicit Differentiation and Related Rates 6 Example 2 Dr. Erickson

Example 3  Find the second derivative of each function. Dr. Erickson Implicit Differentiation and Related Rates 7

Example 4  Find an equation of the tangent line to the graph of the function f defined by the equation at the indicated point. Dr. Erickson Implicit Differentiation and Related Rates 8

Implicit differentiation is a useful technique for solving a class of problems knows as related-rates problems. In related rate problems, the idea is to compute the rate of change of one quantity in terms of the rate of change of another quantity (which may be more easily measured) Implicit Differentiation and Related Rates 9 Related Rates Dr. Erickson

Solving Related-Rates Dr. Erickson Implicit Differentiation and Related Rates 10

a)If A is the area of a circle with radius r and the circle expands as time passes, find dA/dt in terms of dr/dt. b)Suppose oil spills from a ruptured tanker and spreads in a circular pattern. If the radius of the oil spill increases at a constant rate of 1 m/s, how fast is the area of the spill increasing when the radius is 30 m? Implicit Differentiation and Related Rates 11 Example 5 Dr. Erickson

Example 6 Dr. Erickson Implicit Differentiation and Related Rates 12

Example 7 Effect of Price on Supply Suppose the wholesale price of a certain brand of medium-sized eggs p (in dollars per carton) is related to the weekly supply x (in thousands of cartons) by the equation If 25,000 cartons of eggs are available at the beginning of a certain week and the weekly supply is falling at a rate of 1000 cartons/week, at what rate is the wholesale price changing? Dr. Erickson Implicit Differentiation and Related Rates 13

Example 8 Dr. Erickson Implicit Differentiation and Related Rates 14

Example 9 Dr. Erickson Implicit Differentiation and Related Rates 15

Example 10 Dr. Erickson Implicit Differentiation and Related Rates 16

Next Class  We will discuss the following concepts:  Determining the Intervals where a Function is Increasing or Decreasing  Relative Extrema & Finding Relative Extrema  Determining Intervals of Concavity  Inflection Points  The Second Derivative Test  Comparing the First and Second Derivative  Please read through Section 4.1 – Applications of the First Derivative and Section 4.2 – Applications of the Second Derivative in your text book before next class. Dr. Erickson Implicit Differentiation and Related Rates 17