Math Section TTH 5:30-7:00pm SR 116. James West 620 PGH
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1 Math 1431 Section TTH 5:30-7:00pm SR 116 James West 620 PGH Office Hours: 2:30 4:30pm TTH in the CASA Tutoring Center or by appointment
2 Class webpage:
3 4. Find the extrema for: ( x) f = 2 x 2 x 9.
4 Popper10 1. Find the values of x that give relative extrema for the function whose derivative is f' ( ) = ( + ) 2 ( ) A) Local maximum: x = 1 Local minimum: x = 1 B) Local maxima: x = 1, 3 Local minimum: x = 1 C) Local minimum: x = 2 D) Local maximum: x = 1 Local minimum: x = 2 E) None of these x x 1 x 2.
5 2. Let f (x) be a polynomial function with x = 3 as a critical number. If f (3)> 0, then which of the following statements is true? A) (3, f (3)) is a relative minimum B) (3, f (3)) is a zero C) (3, f (3)) is a relative maximum D) f (x) is always increasing E) none of these
6 What is the difference between the first and second derivative test? What are we finding with these tests?
7 Absolute Extreme Values Let c be a point in the domain of f; c may be an interior point or an endpoint. We say that f has an absolute minimum at c if f ( x) f ( c) for all x in the domain of f, f has an absolute maximum at c if f ( x) f ( c) for all x in the domain of f. Finding the absolute minimum and maximum values of a continuous function defined on a closed bounded interval [ a,b ]: 1. Find the critical points for f in the interval ( a,b ). 2. Evaluate the function at each of these critical points and at the endpoints. 3. The smallest of these computed values is the absolute minimum value, and the largest is the absolute maximum value of f.
8 Ex 1: Find the locations of all absolute minima and maxima for 3 2 x x 3x 9x 4 on the interval [ 6, 3]. ( ) f = + +
9 Ex 2: Find the critical numbers and classify all extreme values of f π π x = tan x x x < 3 2 ( )
10 3. Let f (x) = (x +2) 4 4 The point ( 2, 4) is:
11 Ex 3: Find the critical numbers and classify all extreme values of f x) = 2 x + 5x, x [ 20, 2 ( 1 ]
12 4. Find the coordinates of the absolute max and/or absolute min of f on the interval [ 1, 2]. ( x) f = 20 ( 2 x + 1) A) Max: (0, 20) Min: None B) Max: ( 1, 10) Min: (2, 4) C) Max: None Min: None D) Max: (0, 20) Min: (2, 4)
13 Quick factoring review: 6a 2 + 7a 5 = 6a 2 + 7a 5 = 6ab 1 + 7a 2 b 2 =
14 5. Find the absolute maximum and absolute minimum of f on the interval (0, 3). f ( x) = 3 2 x + x + 3x + 1 x + 1 A) Maximum: (1, 2) C) Maximum: None Minimum: (3, 2) Minimum: None B) Maximum: (1, 2) D) Maximum: None Minimum: None Minimum: (3, 2) E) None of these
15 Discuss the extrema for f (x) on the domain [ 4, 5.2] 1) at x = 4 2) at x = 3 f (x) ) at x = ) at x = ) at x = 1 What changes if the domain were (, )? 5 6) at x = 4 7) at x = 5.2
16 POPPER10 6. The graph of f (x) is shown. Give the number of critical values of f. a. 2 b. 3 c. 4 d. 5 Assume the domain of f is all real numbers. f (x) e. none of these
17 7. The graph of f (x) is shown. Give the number of local minima of f. a. 1 b. 2 c. 3 d. 4 Assume the domain of f is all real numbers. f (x) e. none of these
18 8. The graph of f (x) is shown. Give the number of local maxima of f. a. 1 b. 2 c. 3 d. 4 Assume the domain of f is all real numbers. f (x) e. none of these
19 9. The graph of f (x) is shown. Give the number of intervals of increase of f. a. 1 b. 2 c. 3 d. 4 Assume the domain of f is all real numbers. f (x) e. none of these
20 10. The graph of f (x) is shown. Give the number of intervals of decrease of f. a. 1 b. 2 c. 3 d. 5 Assume the domain of f is all real numbers. f (x) e. none of these
21 11. The graph of f (x) is shown. Classify the critical value at 3 or state that the value is not a critical value. a. local maximum b. local minimum c. neither d. not a critical value Assume the domain of f is all real numbers. f (x) e. none of these
22 12. The graph of f (x) is shown. Classify the critical value at 2 or state that the value is not a critical value. a. local maximum b. local minimum c. neither d. not a critical value Assume the domain of f is all real numbers. f (x) e. none of these
23 13. The graph of f (x) is shown. Classify the critical value at 1 or state that the value is not a critical value. a. local maximum b. local minimum c. neither d. not a critical value Assume the domain of f is all real numbers. f (x) e. none of these
24 14. The graph of f (x) is shown. Classify the critical value at 0 or state that the value is not a critical value. a. local maximum b. local minimum c. neither d. not a critical value Assume the domain of f is all real numbers. f (x) e. none of these
25 15. The graph of f (x) is shown. Classify the critical value at 1 or state that the value is not a critical value. a. local maximum b. local minimum c. neither d. not a critical value Assume the domain of f is all real numbers. f (x)
26 Find critical numbers and classify extrema. f( x) = x x < 3 2 x + x 3 x 2 5x 4 2< x< 5
27 Concavity and Points of Inflection Suppose f > 0 on an interval. f > 0 f is increasing f is increasing f is concave up Suppose f > 0 on an interval. What are the possible shapes for the graph of f over this interval?
28 Suppose f < 0 on an interval. f < 0 f is decreasing f is decreasing f is concave down Suppose f < 0 on an interval. What are the possible shapes for the graph of f over this interval?
29 Points of inflection occur where the concavity changes. Identify the inflection points and the intervals of concavity of the function graphed below. A B
30 Identify the inflection points and the intervals of concavity of the function graphed below.
31 Determine the intervals of concavity and inflection points for y = x 4 and y = x 3
32 If (c, f (c)) is a point of inflection of f, what conclusion can be made about f (c)? True or False: If f (c) = 0, then (c, f (c)) is a point of inflection of f.
33 Determine the intervals of concavity and inflection points for f (x) = x 3 3x 2 + 2x 1
34 Given f (x), find all extrema and points of inflection and tell where the graph is increasing and decreasing, concave up and concave down. 3 2 f x = 2x 5x 4x + 2 ( )
35 f (x) A F G B C D H I E Using the graph of f (x), complete the chart by indicating whether the functions are positive, negative, or zero at the indicated points. D and H are POI s. A B C D E F G H I f (x) f (x) f (x)
= c, we say that f ( c ) is a local
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