Day 1a examples

Day 1a examples

• 1.
  Increasing and DecreasingFunctions Recall again :) If f ' > 0 on an interval, the f is increasing on the interval If f ' < 0 on an interval, the f is decreasing on the interval Ex. Given f(x) = x3 + 3x2 - 24x + 18. Determine the intervals on which f(x) is increasing or decreasing. Step 1: Find f '(x) Step 2: Factor the derivative Step 3: Find the critical numbers Recall Critical point = any value in the domain where f '(c) = 0 or f '(x) does not exist. Step 4: Plot the critical numbers on a number line to divide the function into intervals Step 5: Test each interval to determine if f '(x) is positive or negative and determine if f is increasing or decreasing on the intervals
• 2.
  Recall, graphically criticalpoints occur where: - a tangent is horizontal f '(x) = 0 - the graph has a sharp corner or cusp - a tangent is vertical f '(x) is undefined cusp corner
• 3.
  Ex. Find thecritical numbers for the given functions: ex a) x-2 b) x3/5(4 - x)
• 4.
  Given that cis a critical number of a function f: f has a local maximum at x = c if for all x near c, f(c) ≥ f(x) f has a local minimum at x = c if for all x near c, f(c) ≤ f(x) a c d b global maximum = highest point over entire domain of f global minmum = lowest point over entire domain of f *can only have one of each!
• 5.
  First Derivative Test Ifc is a critical number and if f ' changes sign at c then - f has a local minimum at x = c if f ' is negative to the left of c and positive to the right of c - f has a local maximum at x = c if f ' is positive to the left of c and negative to the right of c Local Maximum Local Minimum
• 6.
  Ex. Use thefirst derivative test to find the local maximum and minimum values of f if f(x) = x4 + 2x3 + x2 - 8