Miss Battaglia AP Calculus AB/BC

1. 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

2. 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.

3. 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]

4. Intervals on Which f is Increasing or Decreasing Find the open intervals on which is increasing or decreasing.

5. 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.

6. Applying the First Derivative Test Find the relative extrema of the function in the interval (0,2π)

7. Applying the First Derivative Test Find the relative extrema of

8. Applying the First Derivative Test Find the relative extrema of

9. Classwork/Homework • Read 3.3 Page 179 #1, 8, 12, 21, 27, 29, 35, 43, 45, 63, 67, 79, 99-103