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3.3 Increasing and Decreasing and the First Derivative Test Objective: Determine intervalues in which a function is increasing or decreasing and apply the First Derivative Test. Miss Battaglia AP Calculus AB/BC. Increasing and Decreasing Functions.

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## Miss Battaglia AP Calculus AB/BC

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**3.3 Increasing and Decreasing and the First Derivative**TestObjective: Determine intervalues in which a function is increasing or decreasing and apply the First Derivative Test. Miss Battaglia AP Calculus AB/BC**Increasing and Decreasing Functions**A function f is increasing on an interval for any two numbers x1and x2 in the interval, x1<x2 implies f(x1)<f(x2) A function f is decreasingon an interval for any two numbers x1 and x2 in the interval, x1<x2 implies f(x1)>f(x2) Increasing! Pierre the Mountain Climbing Ant is climbing the hill from left to right. Decreasing! Pierre is walking downhill.**Test for Increasing and Decreasing Functions**Let f be a function that is continuous on the closed interval [a,b] and differentiable on the open interval (a,b). • If f’(x)>0 for all x in (a,b), then f is increasing on [a,b] • If f’(x)<0 for all x in (a,b), then f is decreasing on [a,b] • If f’(x)=0 for all x in (a,b), then f is contant on [a,b]**Intervals on Which f is Increasing or Decreasing**Find the open intervals on which is increasing or decreasing.**The First Derivative Test**• Find the first derivative. • Set the derivative equal to zero and solve for x. • Put the critical numbers you found on a number line (dividing it into regions). • Pick a value from each region, plug it into the first derivative and note whether your result is positive or negative. • Indicate where the function is increasing or decreasing.**Applying the First Derivative Test**Find the relative extrema of the function in the interval (0,2π)**Applying the First Derivative Test**Find the relative extrema of**Applying the First Derivative Test**Find the relative extrema of**Classwork/Homework**• Read 3.3 Page 179 #1, 8, 12, 21, 27, 29, 35, 43, 45, 63, 67, 79, 99-103

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