Clicker Question 1 A population of mice starts at 25 and grows at a continuous rate of 7% per month. In about how many months will there be 200 mice? A.

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Presentation transcript:

Clicker Question 1 A population of mice starts at 25 and grows at a continuous rate of 7% per month. In about how many months will there be 200 mice? A. 7 months B. 20 months C. 25 months D. 30 months E. 54 months

Clicker Question 2 The temperature TMP of a soda placed in a 40 degree refrigerator is given by TMP = 30e -0.2 t + 40 where t is in minutes. At what rate is the soda’s temperature changing at time t = 5 minutes? A degrees/minB degrees/min C degrees/minD – 5.1 degrees/min E degrees/min

Clicker Question 3 Suppose I have 50 grams of a radioactive substance whose half-life is 100 years. How much will remain 20 years from now? A. About 41.1 grams B. About 43.5 grams C. About 32.7 grams D. About 25.0 grams E. About 21.1 grams

Using the Derivative (3/27/09) The derivatives (1 st and 2 nd ) of a function can be used to investigate important properties of that function. Direction and shape: Increasing, decreasing, concave up, concave down. Important points on the function: Local and global maxima and minima (“extrema”), critical points, inflection points.

Local Maxima and Minima The function f has a local maximum at p if f (p) is greater than or equal to all the values of f for points near p. The function f has a local minimum at p if f (p) is less than or equal to all the values of f for points near p. In either case, such a point is often called a local extreme.

Critical Points A point p in the domain of f is called a critical point (or critical number) of f if either f '(p) = 0 or f '(p) does not exist. Often the point (p, f (p) ) on the graph is called the critical point (as opposed to just p itself). In this case, f (p) is called a critical value.

Clicker Question 4 What are the critical numbers of f (x ) = (1/3) x 3 – x 2 – 8x + 5 ? A. 5 B. 0 and -2 C. 0, -2, and 4 D. - 4 and 2 E. -2 and 4

Critical Points and Local Extrema A critical point may indicate a local extreme place of f, but not necessarily (example: f (x) = x 3 at x = 0). We can say that if f has a local extreme at p and if f is differentiable at p, then f '(p) = 0 for sure! To see if a critical point is an extreme, we need to check using values of f or of f ' near p, or check f ' ' at p.

Global Extremes A function f has a global (or absolute) maximum at (p, f (p)) if f (p) is greater than or equal to all values of f. A function f has a global (or absolute) minimum at (p, f (p)) if f (p) is less than or equal to all values of f. To find the global max or min of f on a closed interval (i.e., an interval including its endpoints): Find and compare the values of f at all critical points and at the endpoints !

Assignment We will have a regular lab on Monday on critical and inflection points of functions. In preparation for that, please read today’s slides and Section 4.1! For Wednesday, in Section 4.1 please do Exercises 1, 3, 5, 7, 11, 15, 21, 26, 29, 31, 37, 41, 43, 47, 49, 53.