Calculus I (MAT 145) Dr. Day Wednes day , April 3, 2013

1. Calculus I (MAT 145)Dr. Day Wednesday, April 3, 2013 • Optimization: Using Extreme Values to Solve Contextual Problems (4.7) MAT 145

2. Derivative Patterns You Must Know • The Derivative of a Constant Function • The Derivative of a Power Function • The Derivative of a Function Multiplied by a Constant • The Derivative of a Sum or Difference of Functions • The Derivative of a Polynomial Function • The Derivative of an Exponential Function • The Derivative of a Logarithmic Function • The Derivative of a Product of Functions • The Derivative of a Quotient of Functions • The Derivatives of Trig Functions • Derivatives of Composite Functions (Chain Rule) • Implicit Differentiation • Logarithmic Differentiation MAT 145

3. Position, Velocity, Acceleration An object is moving in a positive direction when v(t) > 0. An object is moving in a negative direction when v(t) < 0. An object speeds up when v(t) and a(t) share same sign. An object slows down when v(t) and a(t) have opposite signs. An object changes directions when v(t) = 0 and v(t) changes sign. The average velocity over a time interval is found by comparing net change in position to length of time interval (SLOPE!). The instantaneous velocity at a specific point in time is found by calculating v(t) for the specified point in time. The net change in position over a time interval is found by calculating the difference in the positions at the start and end of the interval. The total distance traveled over a time interval is found by first determining the times when the object changes direction, then calculating the displacement for each time interval when no direction change occurs, and then summing these displacements. MAT 145

4. Absolute and Relative Extremes • Extreme Value Theorem: For f(x) continuous on a closed interval, there must be extreme values. If f is continuous on a closed interval [a,b] then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some number c and d in [a,b]. • Fermat’s Theorem: If f has a local max or min at x = c and if f ’(c) exists then f ’(c) = 0. • Absolute (Global) Extreme: An output of a function such that it is either the greatest (maximum) or the least (minimum) of all possible outputs. A global extreme CAN be at an endpoint! • Relative (Local) Extreme: An output of a function such that it is either the greatest (maximum) or the least (minimum) in some small open neighborhood along the x-axis. A local extreme CANNOT be at an endpoint! • Critical Point: An interior point (not an endpoint) on f(x) such that f’(x= 0 or f ’(x) is undefined. Any critical number MUST have a function output! MAT 145

5. Absolute and Relative Extremes Ways to Find Extrema • Local Extremes: Examine behavior at critical points (interior). • Absolute Extremes: Examine behavior at critical points and at endpoints. Example Determine critical numbers, absolute extrema, and relative extrema for the unrestricted function (all possible domain values) and then for the restricted domain [−1,2]. MAT 145

6. Absolute and Relative Maximums and Minimums • Problems 1, 2, and 3: PIP page 44 (top). • Must every continuous function have critical points on a closed interval? Explain. • Can an increasing function have a local max? Explain. • Use the graph of f’(x) on PIP page 44 (bottom) to describe a strategy for identifying the global and local extrema of f, given f’(x). MAT 145

7. First Derivative Test, Concavity, Second Derivative Test (4.3) Determining Increasing or Decreasing Nature of a Function If f’(x) > 0, then f is _?_. If f’(x) < 0, then f is _?_. Using the First Derivative to Determine Whether an Extreme Value Exists: The First Derivative Test (and first derivative sign charts) If f’ changes from positive to negative at x=c, then f has a _?_ _?_ at c. If f’ changes from negative to positive at x=c, then f has a _?_ _?_ at c. If f’ does not change sign at x=c, then f has neither a local max or min at c. Concavity of f If f’’(x) > 0 for all x in some interval I, then the graph is concave up on I. If f’’(x) < 0 for all x in some interval I, then the graph is concave down on I. Second derivative Test If f’(c) = 0 and f’’(c) > 0, then f has a local min at c. If f’(c) = 0 and f’’(c) < 0, then f has a local max at c. MAT 145

8. Pulling it all together • For f(x) on PIP page 49, #4, use calculus to determine and justify: • All x-axis intervals for which f is increasing • All x-axis intervals for which f is decreasing • The location and value of every local & absolute extreme • All x-axis intervals for which f is concave up • All x-axis intervals for which f is concave down • The location of every point of inflection. MAT 145

9. Info about f from f ’ Here’s a graph of g’(x). Determine all intervals over which g is increasing and over which g is decreasing. Identify and justify where all local extremes occur. MAT 145

10. Info about f from f ’’ Here’s a graph of h”(x). Determine all intervals over which h is concave up and over which h is concave down. Identify and justify where all points of inflection occur. MAT 145

11. Optimization • Use PIP problems to practice solving optimization problems: pp 63-64 MAT 145

12. Assignments WebAssign • Ch 4: Derivative Applications • three assignments on WA for completion; due starting tonight Also • Gateway Quiz #5 today! • Test #4: Friday, April 5 • Through all we’ve done in Chapter 4—and previous chapters. Please review previous TEST results and see me with questions, corrections, and concerns! MAT 145