INCREASING AND DECREASING FUNCTIONS AND THE FIRST DERIVATIVE TEST

1. INCREASING AND DECREASING FUNCTIONS AND THE FIRST DERIVATIVE TEST Section 3.3

2. When you are done with your homework, you should be able to… • Determine intervals on which a function is increasing or decreasing • Apply the first derivative test to find relative extrema of a function

3. Ptolemy lived in 150AD. He devised the 1st “accurate” description of the solar system. What branch of mathematics did he create? • Calculus • Number Theory • Trigonometry • Statistics

4. Definitions of Increasing and Decreasing Functions • A function f is increasing on an interval if for any two numbers and in the interval, implies . • A function is decreasing on an interval if for any two numbers and in the interval, implies . How does this relate to the derivative?!

5. Theorem: Test for Increasing and Decreasing Functions • Let f be a function that is continuous on the closed interval and differentiable on the open interval • If for all , then f is increasing on . • If for all , then f is decreasing on . • If for all , then f is constant on .

6. Guidelines for Finding Intervals on which a Function is Increasing or Decreasing Let f be continuous on the interval To find the open intervals on which f is increasing or decreasing, use the following steps. • Locate the critical numbers of in and use these numbers to determine the test intervals. • Determine the sign of at one test value in each of the intervals. • Use the previous theorem to determine whether f is increasing or decreasing on each interval. *These guidelines are also valid if the interval • is replaced by an interval of the form

7. The weight (in pounds) of a newborn infant during its 1st three months of life can be modeled by the equation below, where t is measured in months. Determine when the infant was gaining weight and losing weight. • The infant lost weight for approximately the first month and then gained for the 2nd and 3rd. • The infant lost weight for approximately .56 month and then gained thereafter. • The infant gained weight for approximately the first month and then lost weight during the 2nd and 3rd months. • The infant lost weight for approximately .56 month and then gained weight during the 2nd and 3rd months.

8. Fibonacci lived in 1200. Why was he famous? • He introduced the Hindu-Arabic system of numeration (aka “base 10”) • He developed the study of numbers in the form of 1, 1, 2, 3, 5, 8, 13, 21, . . . • He found all integer solutions to • All of the above.

9. Theorem: The First Derivative Test Let c be a critical number of a function f that is continuous on an open interval I containing c. If f is differentiable on the interval, except possibly at c, then can be classified as follows: •  If changes from negative to positive at c, then f has a relative minimum at • If changes from positive to negative at c, then f has a relative maximum at • If is positive on both sides of c or negative on both sides of c, then is neither a relative minimum or relative maximum.

11. Find the relative extrema of the function and the intervals on which it is increasing or decreasing. • Relative min @ (5, 0), relative max @ (5, -75/16), increasing on , decreasing on . • Relative max @ (5, 0), relative min @ (5, -75/16), increasing on , decreasing on . • None of the above