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Calculus and Analytic Geometry I

Calculus and Analytic Geometry I. Cloud County Community College Fall, 2012 Instructor: Timothy L. Warkentin. Chapter 04: Applications of Derivatives. 4.1 Extreme Values of Functions 4.2 The Mean Value Theorem 4.3 Monotonic Functions and the First Derivative Test

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Calculus and Analytic Geometry I

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  1. Calculus and Analytic Geometry I Cloud County Community College Fall, 2012 Instructor: Timothy L. Warkentin

  2. Chapter 04: Applications of Derivatives • 4.1 Extreme Values of Functions • 4.2 The Mean Value Theorem • 4.3 Monotonic Functions and the First Derivative Test • 4.4 Concavity and Curve Sketching • 4.5 Indeterminate Forms and L’Hôpital’s Rule • 4.6 Applied Optimization • 4.7 Newton’s Method • 4.8 Antiderivatives

  3. 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.

  4. 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

  5. 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 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

  6. 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 non-overlapping 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

  7. 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

  8. 04.04: Concavity & Curve Sketching 2 • Understanding . • Graph the function with a graphing utility. • Identity any transformative elements. • Find the domain and range. • Identify any symmetries. • Identify any discontinuities. • Find any asymptotes or holes. • Find any x and y intercepts. • Find the first and second derivatives. • Find any extreme points and identify local/global maximums/minimums. • Find the intervals where the function is increasing and decreasing. • Find any inflection points and find the intervals on which the curve is concave up and concave down. • Re-graph the function with any asymptotes and significant points plotted and labeled.

  9. 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 • 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

  10. 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

  11. 04.07: Newton’s Method 1 • This section is not covered.

  12. 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: . Examples 3, 4, & 6

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