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Day 1a examples

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Day 1a examples

  1. 1. Increasing and Decreasing FunctionsRecall again :)If f > 0 on an interval, the f is increasing on the intervalIf f < 0 on an interval, the f is decreasing on the intervalEx. Given f(x) = x3 + 3x2 - 24x + 18. Determine theintervals on which f(x) is increasing or decreasing.Step 1: Find f (x)Step 2: Factor the derivativeStep 3: Find the critical numbersRecall 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 intervalsStep 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. 2. Recall, graphically critical points 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. 3. Ex. Find the critical numbers for the given functions: ex a) x-2 b) x3/5(4 - x)
  4. 4. Given that c is 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 bglobal maximum = highest point over entire domain of fglobal minmum = lowest point over entire domain of f*can only have one of each!
  5. 5. First Derivative TestIf c 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 leftof c and positive to the right of c- f has a local maximum at x = c if f is positive to the leftof c and negative to the right of c Local Maximum Local Minimum
  6. 6. Ex. Use the first derivative test to find the local maximumand minimum values of f if f(x) = x4 + 2x3 + x2 - 8

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