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Calculus and Analytic Geometry I Cloud County Community College Fall, 2012 Instructor: Timothy L. Warkentin Chapter 04: Applications of Derivatives • • • • • • • • 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 Extreme Values of Functions The Mean Value Theorem Monotonic Functions and the First Derivative Test Concavity and Curve Sketching Indeterminate Forms and L’Hôpital’s Rule Applied Optimization Newton’s Method Antiderivatives Chapter 04 Overview • The rate at which things happen is of fundamental importance to every field of study in which measurement is a priority. Derivatives (functions whose range elements are rates/slopes) are used to describe how quantities are changing in time. 04.01: Extreme Values of Functions 1 • Extreme range values and their relationship to open/closed intervals. Example 1 • The Extreme Value Theorem. • For functions y = f [x] extreme range values are found at domain values where f ′ = 0, where f ′ is undefined, or at the endpoints of the domain. Domain values where f ′ = 0 or where f ′ is undefined are called critical points. • The search for extreme range values begins by identifying domain values associated with horizontal tangent lines. Critical domain values are either local maximums, local minimums or inflection points. Examples 2 – 4 04.02: The Mean Value Theorem 1 • Rolle’s Theorem (a special case of the MVT). Example 1 • The Mean Value Theorem (MVT) for Derivatives unites the ideas of Average Rate of Change and Instantaneous Rate of Change (a consequence of Rolle’s Theorem). • MVT: There is at least one point in (a,b) where the slope of the curve is equal to the slope of the secant line on [a,b]. • If y = f [x] is continuous on [a,b] and differentiable on (a,b) then there is at least one point c in (a,b) at which f [b] f [a] f ' [c ] . ba Examples 2 & 3 • MVT Corollary 1: Functions with zero derivatives are constant. • MVT Corollary 2: Functions with the same derivative differ by a constant. Example 4 04.03: Monotonic Functions and the First Derivative Test 1 • Definition of an Increasing/Decreasing function (section 01.01). • A function that is increasing or decreasing on an interval is said to monotonic on that interval. • Critical points subdivide the domain into nonoverlapping intervals on which the derivative is either positive or negative. Example 1 • MVT Corollary 3: If the first derivative is Positive/Negative the function is Increasing/Decreasing. • The First Derivative Test (local min/max text). Examples 2 & 3 04.04: Concavity & Curve Sketching 1 • The graph of a differentiable function is concave up/down on an interval if the first derivative is increasing/decreasing (the second derivative is positive/negative). • The Second Derivative Test for Concavity. Examples 1 & 2 • Inflection Point: A point where the tangent line exists and the concavity changes. Examples 3 – 6 • The Second Derivative Test for Local Extrema. Example 7 • Graphing functions. Examples 8 – 10 04.04: Concavity & Curve Sketching 2 • Understanding y f [x] . 1. Graph the function with a graphing utility. 2. Identity any transformative elements. 3. Find the domain and range. 4. Identify any symmetries. 5. Identify any discontinuities. 6. Find any asymptotes or holes. 7. Find any x and y intercepts. 8. Find the first and second derivatives. 9. Find any extreme points and identify local/global maximums/minimums. 10. Find the intervals where the function is increasing and decreasing. 11. Find any inflection points and find the intervals on which the curve is concave up and concave down. 12. Re-graph the function with any asymptotes and significant points plotted and labeled. 04.05: Indeterminate Forms and L’Hôpital’s Rule 1 • Many difficult limit problems can be solved by application(s) of L’Hôpital’s Rule. If the limit attempted yields the indeterminate forms 0/0 or ∞/∞ then the following can be applied. Examples 1 & 3 f [ x] f '[ x] lim lim x a g[ x ] x a g '[ x] • This rule can be applied recursively until an acceptable form is found. Example 2 • Some other indeterminate forms that can be transformed into the required 0/0 or ∞/∞ are ± ∞/ ± ∞, ∞*0, ∞ - ∞, 1∞, and ∞0. Examples 4 – 8 04.06: Applied Optimization 1 • Optimization means finding the best possible solution to a particular problem. Considering that problems often have an infinite number of solutions, the ability to find the single best solution for many problems illustrates the power of Calculus. • Where an extreme value occurs is not the same as the extreme value. • Solving Applied Optimization Problems (textbook procedure). Examples 1 – 5 04.07: Newton’s Method 1 • This section is not covered. 04.08: Antiderivatives 1 • Definition of the Antiderivative: A function F is an antiderivative of f if the derivative of F is f. Example 1 • MVT Corollary 2: Functions with the same derivative differ by a constant (usually written as C). • The value of C may be determined if an ordered pair solution of F is known. Examples 2 & 5 • The set of all antiderivatives of f [x] (an infinite set) is called the indefinite integral of f [x] and is denoted by the single symbol: f [ x] dx . Examples 3, 4, & 6