tailieunhanh - Calculus and its applications: 2.1

"Calculus and its applications: " - Using first derivatives to find maximum and minimum values and sketch graphs have objective: find relative extrema of a continuous function using the first-derivative test, sketch graphs of continuous functions. | 2012 Pearson Education, Inc. All rights reserved Slide Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs OBJECTIVE Find relative extrema of a continuous function using the First-Derivative Test. Sketch graphs of continuous functions. 2012 Pearson Education, Inc. All rights reserved Slide DEFINITIONS: A function f is increasing over I if, for every a and b in I, if a f (b). (If the input a is less than the input b, then the output for a is greater than the output for b.) Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs 2012 Pearson Education, Inc. All rights reserved Slide THEOREM 1 If f (x) > 0 for all x in an interval I, then f is increasing over I. If f (x) Slide DEFINITION: A critical value of a function f is any number c in the domain of f for which the tangent line at (c, f (c)) is horizontal or for which the derivative does not exist. That is, c is a critical value if f (c) exists and f (c) = 0 or f (c) does not exist. Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs p. 200 The definition of critical value has a period at the end of the first c and a capital “in the domain ” . The grammar here makes no sense. 2012 Pearson Education, Inc. All rights reserved Slide DEFINITIONS: Let I be the domain of f : f (c) is a relative minimum if there exists within I an open interval I1 containing c such that f (c) ≤ f (x) for all x in I1; and F (c) is a relative maximum if there exists within I an open interval I2 containing c such that f | 2012 Pearson Education, Inc. All rights reserved Slide Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs OBJECTIVE Find relative extrema of a continuous function using the First-Derivative Test. Sketch graphs of continuous functions. 2012 Pearson Education, Inc. All rights reserved Slide DEFINITIONS: A function f is increasing over I if, for every a and b in I, if a f (b). (If the input a is less than the input b, then the output for a is greater than the output for b.) Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs 2012 Pearson Education, Inc. All rights reserved Slide THEOREM 1 If f (x) > 0 for all x in an interval I, then f is increasing over I. If f (x) < 0 for all x in an interval I, then