4.3 - How Derivatives Affect the Shape of a Graph
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1 4.3 - How Derivatives Affect the Shape of a Graph 1. Increasing and Decreasing Functions Definition: A function f is (strictly) increasing on an interval I if for every 1, in I with 1, f 1 f. A function f is (strictly) decreasing on an interval I if for every 1, in I with 1, f f 1. Eample: The graph of f is given below. y f is increasing on 0,0.8,.5,4. f is decreasing on 0.8,.5, 4,. -10 f 1
2 Theorem: Suppose that f is differentiable on an interval I. (i) If f 0 for all in I, then f is increasing on I. (ii) If f 0 for all in I, then f is decreasing on I. Proof: Let 1 and be in I and 1. Then by the Mean Value Theorem, we know there eists a value c in 1, such that f c f f 1 1. If f c 0, then f f that implies f f 1 0 because 1. Hence, f f 1, that is, f is increasing. In a similar way, we can show (ii).
3 Eample: The graph of f is given below. Determine graphically the interval on which f is increasing. y f is increasing on 0,1.6, 3.1,4.7 because f f Eample: Find the intervals where f is increasing and decreasing. Verify your answers by graphing both f and f. 3
4 Eample: Find the intervals where f is increasing and decreasing. Verify answers with graphs of f and f. Step I: Compute f : f Step II: Find values of at which f 0: 4 and 1. Step III: Check sign changes of f over intervals:, 4, 4,1, 1, f 5 36 f 0 4 f 36, interval, 4 4,1 1, sign of f So, f is increasing on, 4, 1, and is decreasing on 4, red f, green f y 4
5 . First Derivative Test For Local Etrema (maima or minima): Theorem: Suppose that f is continuous on a, b and c in a, b is a critical number. (i) If f changes sign from positive to negative at c, then f c is a local maimum of f. The number c is called a local maimum point. (ii) If f changes sign from negative to positive at c, then f c is a local minimum of f The number c is called a local minimum point. (iii) If f does not change sign at c, then f c is not a local etremum. Eample: Let f e 3. Find all critical numbers and use the 1st Derivative Test to classify each as the location of a local maimum, local minimum of neither. 5
6 Eample: Let f e 3. Find all critical numbers and use the 1st Derivative Test to classify each as the location of a local maimum, local minimum of neither. Step I: Find the domain of f : D f,. Step II: Compute f and find all critical numbers: f e 3 3 e 3 e 3 3. (1) Critical number of type (i): f 0 0 or 3. () Critical number of type (ii): None. Step III: Check sign change of f over intervals:,0, 0, 3, 3, f 1 1 e f e f 1 1 e 3 1 0, interval,0 0, 3 3, sign of f 3 is a local maimum point and 0 is a local minimum point. 6
7 Eample: Let f 4 sketch the graph of f. Step I: horizontal and vertical asymptotes:. Find all asymptotes and etrema, and Horizontal asymptote: lim 4 1, y 1 Vertical asymptote: 4 0,, Step II: Compute f and find all critical numbers: f (1) Critical number of type (i): f 0, 8 0, 0. () Critical number of type (ii): f is not defined at but they are not in D f, so none. Step III: Check sign change of f over,0, 0, : f 0 for in, 0 and f 0 for in 0,. 0 is a local maimum point. 7
8 y y f, -.- y f 8
9 Eample: Let f 5/3 3 /3. Find all critical numbers and use the 1st Derivative Test to classify each as the location of a local maimum, local minimum of neither. Step I: Find the domain of f : D f,. Step II: Compute f and find all critical numbers: f 5 3 /3 1/3 1/3 5 3 (1) Critical number of type (i): f () Critical number of type (ii): f is not defined when 0. Step III: Check sign change of f over intervals:,0, 0, 6 5, 6 5, f f f , interval,0 0, 6 5 sign of f - 6 5, 9
10 So, 0 is a local maimum point and 6 5 point. is a local minimum y y f 10
11 Eample: Sketch a graph of a function with the given properties: (i) f 0 1, f 5 (ii) f 0 0, f 0 f 0, for 0 and ; f 0 for 0 Eample: Sketch a graph of a function with the given properties: (i) f 1. 0, f 3 0, f 0 does not eist (ii) f 3 0, f 0 does not eist f 0, for 0 and 3; f 0 for
12 3. Concave up and concave down For a function f that is differentiable on an interval I, the graph of f is If f is concave up on a, b, then the secant line passing through points 1, f 1 and, f for any 1 and in a, b are above the curve y f between 1, f 1 and, f. If f is concave down on a, b, then the secant line passing through points 1, f 1 and, f for any 1 and in a, b are below the curve y f between 1, f 1 and, f. y y a concave up a concave down 1
13 Eample: The graph of f is given below. Determine graphically the interval on which f is (1)concave up; () concave up and decreasing. (1) f is concave up on 1. 8, () f is concave up and decreasing on 1. 8,.5 y f How can we determine algebraically where f is concave up and where f is concave down? 13
14 Theorem: Suppose that f is differentiable on an interval a,b. Then the graph of f is (a) concave up on a, b, if f is increasing on a,b ; and (b) concave down on a, b, iff is decreasing on a,b. Or, suppose that f eists on a, b. The graph of f is (a) concave up on a, b, if f 0 for all in a,b ; (b) concave down on a, b, iff 0 for all in a,b. 4. Inflection points: Definition: Suppose that f is continuous on the interval a, b. Let c be in a, b. Then the point c, f c is called an inflection point of f if the graph of f changes concavity at the point c, f c. 14
15 Note that: changes concavity at c, f c f changes from increasing to decreasing at c, f c f changes from positive to negativeor from negative to positive at c, f c. Eample: Let the graph of f be given at right. Find (1) the coordinate of each inflection point of f; () where the graph of f is concave up. y f (1) f 0 when, 0, 1 and. f does not change sign at 0. So, the coordinates of inflection points of f are, 1 and. () f 0 for 0, 0 1, and f 0 for, 1. So, the graph of f is concave up on, 0 0, 1,. 15
16 Eample: Let the graph of f be given at right. Find (1) the coordinate of each inflection point of f; () where the graph of f is concave up. y f f 0 when 0. 8,. 5, 4. f 0 f is increasing for ,. 5 4; f 0 f is decreasing for , 4 5. (1) So, 0. 8,. 5, 4 are the coordinates of inflection points. () The graph of f is concave up on 0, ,4. 16
17 Eample: Let f Find (1) all inflection points of f; () where the graph of f is concave up and is concave down. Verify your answers by graphing both f and f. (1i.) Compute f : f , f (1ii.) Solve f 0 : (1iii.) Check signs of f over intervals:, 3, 3, f f , interval, 3 sign of f 3, Since f changes sign at the point where 3, 3, 79 is an inflection point of f. The graph of f is concave up on 3, and is concave down on, 3. Verify the results with the graph of f. 17
18 y red f, green f, blue f 18
19 5. Second Derivative Test: Theroem: Suppose that f is continuos on the interval a, b and f c 0, for some c in a, b. (a) If f c 0, then f c is a local maimum and (b) if f c 0, then f c is a local minimum. Eample: The graph of f is given below. Suppose that we know f 1 0, f 0 and f 4 0. Determine if f 1, f and f 4 are local maimum, local minimum or neither. y f 1 0 and f 4 0, so f 1 and f 4 are local maimum values. f 0, f is a local minimum value. f 19
20 Eample: Let f 1 1/3. Find (1) the intervals of increase and decrease; () all local etrema; (3) the intervals of concavity; (4) all inflection points; and (5) sketch the graph of f based on the information in a.-d. The domain of f :, Compute f and f : f / Find critical numbers of f : type (i): f 0 f /
21 , 3 1 3, , , type (ii): f is not defined : 1. Determine the sign change of f over, , , 1, 1, , 1.544, f , f f , f interval, ,1 1, , f 1
22 Find where f 0: Find where f is not defined: 1 Determine the sign change of f over intervals:,1 and 1, f f interval,1 1, f State the results: (1) f is increasing on, , , and is decreasing on , 1, 1, () By the first derivative test, f is a local maimum and f is a local minimum. (3) f is concave up on 1, and is concave down on, 1. (4) f changes concavity at 1 and 1 is in the domain of f so it is an inflection point of f.
23 y 1 (5) Sketch the graph of f based on the information in a.-d
24 Eample: Let f e cos. Find (1) the intervals of increase and decrease; () all local etrema; (3) the intervals of concavity; (4) all inflection points; and (5) sketch the graph of f based on the information in a.-d. The domain of f : D f, Compute f and f : f e cos e sin e cos sin f e cos sin e sin cos e sin Find critical numbers of f : type (i): f 0 e cos sin 0, cos sin 0, sin cos, tan 1 4 n, n 0, 1,,... 4
25 type (ii): f is not defined: None. Determine the sign change of f over , 5, 4 5 4, 4, 4, 3, 3, 4 7, 7, , f e cos e 0, f e cos 0 e 0 f 0 1, f e cos e 0, f e cos e 0 interval 9 4, , 4 4, 3 4 4, sign of f Find where f 0:e sin 0, n, n 0,1,,... Find where f is not defined: None. Determine the sign change of f e sin over intervals:...,,,0, 0,,,,... f 3 e3 / sin 3 0, f 1 e / sin 1 0 5
26 f 1 e / sin 1 0, f 3 e / sin 3 0 interval,,0 0,, sign of f State the results: (1) f is increasing on , 5 4, 4, 3 4, 5, 9 4 4,... () f c is a local maimum for c 5 4, 3, 9, f c is a local minimum for c... 4, 5,... 4 (3) f is concave up on...,, 0,,... and is concave down on...,0,,... (4) Inflection points of f are: n (5) Sketch the graph of f based on the information in (1)-(4) 6
27 y 30 y
28 Eample: Sketch a graph of a function with the given properties: (i) f 0 (ii) f 0, for all ; f 0 1 (iii) f 0 for 0, f 0 for 0, f 0 0 Eample: Sketch a graph of a function with the given properties: (i) f 0 0, f 1 1, f 1 1 (ii) f 0, for 1 and 0 1, f 0 for 1 0 and 1; (iii) f 0 for 0 and 0 8
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