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Math 142 Lecture Notes for Section 5.1 Section 5.1 - 1 The First Derivative From previous lectures we can recall the following facts about the first derivative: Consequently, we can use these facts to analyze the behavior of a function simply by understanding what the derivative is telling us. Definition 5.1.1: A partition number of f 0 (x) is a value of x such that Definition 5.1.2: A critical value of f 0 (x) is a value of x such that Strategy for using partition numbers and critical values of a function f (x): • Step 1: • Step 2: • Step 3: • Step 4: Math 142 Lecture Notes for Section 5.1 2 Example 5.1.3: Find the critical values and partitions numbers for the following functions and then determine where the function is increasing or decreasing. (a) f (x) = 3x2 − 5x + 3 (b) f (x) = x+4 x−2 (c) y = x ln x3 − 3x (d) f (x) = (x − 3)2 − ln(x − 3) 2 Math 142 Lecture Notes for Section 5.1 3 Definition 5.1.4: Relative (Local) Extrema: • For a function f , a point f (c) is called a relative (local) maximum if there exists an interval (m, n) containing c such that • For a function f , a point f (c) is called a relative (local) minimum if there exists an interval (m, n) containing c such that • if f (c) is either a relative (local) maximum or minimum, then we can call f (c) a relative (local) extremum of f . First Derivative Test: to (1) If f 0 (x) changes from and at x = c there is a at x = c, then we have that f (x) is going from . to to (2) If f 0 (x) changes from and at x = c there is a at x = c, then we have that f (x) is going from . to (3) If the sign of f 0 (x) is the same on both sides of x = c, then at x = c there is . All of this material adds a fifth step to the process discussed previously. This step is: Step 5: Math 142 Lecture Notes for Section 5.1 4 Example 5.1.5: Determine the intervals where the following functions are increasing and decreasing and find the local extrema. (a) f (x) = 3x4 + 8x3 − 30x2 − 72x . 12 (b) y = 4 − 25 x2 Math 142 Lecture Notes for Section 5.1 (c) g(x) = (x − 3)ex Suggested Homework: 1-41(odd), 45-53(odd), 61, 63, 73, 75, 87, 89. 5