University:
Glendale Community College
Course:
MAT187 | Precalculus
Academic year:

2021
Views:

47

Pages:

32
Author:

Kasey Weeks

Section 3.1 Increasing and Decreasing functions and Relative Maxima and Minima (Minimum Homework: 1 – 11 odds 13, 17, 21, 25) A point is a relative (local) maximum point provided it is higher than any point close to it. The relative (local) maximum value is the y-coordinate of the relative maximum point. A point is a relative (local) minimum point provided it is lower than any point close to it. The relative (local) minimum value is the y-coordinate of the relative minimum point. It would be proper to say the following with regards to the graph in the left panel. • (−2,58) is a relative maximum point • The relative maximum value is 𝑦 = 58 which occurs when 𝑥 = −2 • (3, −67) is a relative minimum point. • The relative minimum value is 𝑦 = −67 which occurs when 𝑥 = 3. We are going to need to be able to describe regions in graphs in this section. We need to be able to describe three regions in this graph using only round parenthesis and only x-values. The regions we need to describe are: • Left of relative maximum point • Right of relative minimum point • Between the relative maximum and minimum points. Left of relative maximum point Region starts at the far-left end of the x-axis (𝑥 = −∞) Region ends at the relative maximum point (𝑥 = −2) Right of relative minimum point Region starts at relative minimum point (𝑥 = 3) Region ends at far-right end of x-axis (𝑥 = ∞) Region between the relative maximum and relative minimum points. Region starts at relative maximum point (𝑥 = −2) Region ends at relative minimum point (𝑥 = 3) This is how the regions will need to be described in chapter 3. (only x-values and only round parenthesis) • Left of relative maximum point (−∞, −2) • Right of relative minimum point (3, ∞) • Between the relative maximum and minimum points. (−2,3) Using a Graph to Determine Where a Function is Increasing, Decreasing A graph of a function is increasing over an open interval provided the y-coordinates of the points in the interval get larger, or equivalently the graph gets higher as it moves from left to right over the interval. A graph of a function is decreasing over an open interval provided the 𝑦 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 of the points in the interval get smaller, or equivalently the graph gets lower as it moves from left to right over the interval. This function has a relative maximum point of (−2,16). It has a relative minimum point of (2, −16) The graph of the function is increasing • left of relative maximum point (−∞, −2) • right of relative minimum point (2,∞) The graph of the function is decreasing • between the relative maximum and minimum points. (−2,2) For example: Use the graph of f(x) to determine: a) interval(s) where the graph is increasing. b) interval(s) where the graph is decreasing. c) the coordinates of local maximum point, if any d) the relative maximum value e) the coordinates of the relative minimum point if any f) the relative minimum value a) Increasing between the relative minimum point and the relative maximum point. 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔 (−8,2) b) Decreasing in two intervals. To left of relative minimum point: (−∞, −8) To right of relative maximum point: (2, ∞) Decreasing: (−∞, −8) ∪ (2, ∞) . c) Relative maximum point (2,52) d) Relative maximum value 𝑦 = 52 which occurs when 𝑥 = 2. e) Relative minimum point (−8, −448) f) Relative minimum value: 𝑦 = −448 which occurs when 𝑥 = −8. We need to use Calculus to find the intervals where a graph is increasing and decreasing and to find the maximum and minimum points. Here are the steps to find the intervals where a graph is increasing and decreasing. 1) Find the derivative of the given function. 2) Find the critical numbers for the derivative. • Any value of x that makes the derivative equal to zero. • Any value of x that makes the derivative undefined. 3) Plot the critical numbers on a number line that also includes −∞ 𝑎𝑛𝑑 ∞. 4) Create interval(s) using only round parenthesis. 5) Pick a number inside the interval and plug it into the derivative. 6) Determine whether the graph is increasing or decreasing in that interval. • Graph is increasing when the result is positive. • Graph is decreasing when the result is negative. 7) Place directional arrows in each interval to signify whether the graph in increasing or decreasing. • Determine if each critical point is a relative maximum point, a relative minimum point, or neither. 8) Find the y-coordinates of any relative maximum or relative minimum points. 9) Write your answer. Example: 𝑓 (𝑥 ) = 𝑥 3 − 24𝑥 2 + 18 Find the following: a) f’(x) b) the critical numbers c) interval(s) where the graph is increasing. d) interval(s) where the graph is decreasing. e) the coordinates of relative maximum point, if any f) the relative maximum value g) the coordinates of the relative minimum point if any h) the relative minimum value See next few pages for work / answers: 1) Find the derivative of the given function. 𝑓 ′ (𝑥 ) = 3𝑥 2 − 48𝑥 2) Find the critical numbers for the derivative. • Any value of x that makes the derivative equal to zero. • Any value of x that makes the derivative undefined. 3𝑥 2 − 48𝑥 = 0 3𝑥 (𝑥 − 16) = 0 3𝑥 = 0 𝑥 − 16 = 0 𝑥=0 𝑥 = 16 Critical numbers 𝑥 = 0, 16 3) Plot the critical numbers on a number line that also includes −∞ 𝑎𝑛𝑑 ∞. | −∞ | 0 | 16 | ∞ 4) Create interval(s) using only round parenthesis. (−∞, 0) | −∞ (0,16) | 0 | 16 (16, ∞) | ∞ 5) Pick a number inside the interval and plug it into the derivative. (−∞, 0) 𝑐ℎ𝑜𝑜𝑠𝑒 𝑥 = −1 𝑓 ′ (−1) = 3(−1)2 − 48(−1) = 51 (0,16) 𝑐ℎ𝑜𝑜𝑠𝑒 𝑥 = 1 𝑓 ′ (1) = 3(1)2 − 48(1) = −45 (16, ∞) 𝑐ℎ𝑜𝑜𝑠𝑒 𝑥 = 17 𝑓 ′ (17) = 3(17)2 − 48(17) = 51 6) Determine whether the graph is increasing or decreasing in that interval. • Graph is increasing when the result is positive. • Graph is decreasing when the result is negative. (−∞, 0) positive - increasing (0,16) negative - decreasing (16, ∞) positive - increasing 7) Place directional arrows in each interval to signify whether the graph in increasing or decreasing. • Determine if each critical point is a relative maximum point, a relative minimum point, or neither. | −∞ | 0 | 16 | ∞ Relative maximum point at 𝑥 = 0 Relative minimum point at 𝑥 = 16 8) Find the y-coordinates of any relative maximum or relative minimum points. y-coordinate of relative maximum: 𝑦 = 𝑓 (0) = (0)3 − 24(0)2 + 18 = 18 relative maximum point (0,18) y-coordinate of relative minimum: 𝑦 = 𝑓 (16) = (16)3 − 24(16)2 + 18 = −2030 relative minimum point (16, −2030) 9) Write your answer. a) f’(x) 𝑓 ′ (𝑥 ) = 3𝑥 2 − 48𝑥 b) the critical numbers 𝑥 = 0,16 c) interval(s) where the graph is increasing. (−∞, 0) ∪ (16, ∞) d) interval(s) where the graph is decreasing. (0,16) e) the coordinates of relative maximum point if any (0, 18) f) the relative maximum value 𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑚𝑎𝑥𝑖𝑚𝑢𝑚 𝑣𝑎𝑙𝑢𝑒 𝑦 = 18 𝑤ℎ𝑖𝑐ℎ 𝑜𝑐𝑐𝑢𝑟𝑠 𝑤ℎ𝑒𝑛 𝑥 = 0 g) the coordinates of the relative minimum point if any (16, −2030) h) the relative minimum value 𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑚𝑖𝑛𝑖𝑚𝑢𝑚 𝑣𝑎𝑙𝑢𝑒 𝑦 = −2030 𝑤ℎ𝑖𝑐ℎ 𝑜𝑐𝑐𝑢𝑟𝑠 𝑤ℎ𝑒𝑛 𝑥 = 16 (Minimum Homework: 1 – 11 odds 13, 17, 23, 27) 1-12: Find: a) interval(s) where the graph is increasing. b) interval(s) where the graph is decreasing. c) the coordinates of relative maximum point, if any d) the relative maximum value e) the coordinates of the relative minimum point if any f) the relative minimum value 1) 2) a) interval(s) where the graph is increasing. (−∞, −2) ∪ (2, ∞) b) interval(s) where the graph is decreasing. (−2,2) c) the coordinates of relative maximum point if any (−2,4) d) the relative maximum value 𝑦 = 4 𝑤ℎ𝑒𝑛 𝑥 = −2 e) the coordinates of the relative minimum point if any (2, −4) f) the relative minimum value 𝑦 = −4 𝑤ℎ𝑒𝑛 𝑥 = 2 3) 4) a) interval(s) where the graph is increasing. (−2,2) b) interval(s) where the graph is decreasing. (−∞, −2) ∪ (2, ∞) c) the coordinates of relative maximum point if any (−2,7) d) the relative maximum value 𝑦 = 7 𝑤ℎ𝑒𝑛 𝑥 = 2 e) the coordinates of the relative minimum point if any (−2, −1) f) the relative minimum value 𝑦 = −1 𝑤ℎ𝑒𝑛 𝑥 = −2 5) 6) a) interval(s) where the graph is increasing. (−1,0) ∪ (1, ∞) b) interval(s) where the graph is decreasing. (−∞, −1) ∪ (0,1) c) the coordinates of relative maximum point if any (0,0) d) the relative maximum value 𝑦 = 0 𝑤ℎ𝑒𝑛 𝑥 = 0 e) the coordinates of the relative minimum point if any (−1, −1)𝑎𝑛𝑑 (1, −1) f) the relative minimum value 𝑦 = −1 𝑤ℎ𝑒𝑛 𝑥 = −1,1 7) 8) a) interval(s) where the graph is increasing. (−1, ∞) b) interval(s) where the graph is decreasing. (−∞, −1) c) the coordinates of relative maximum point, if any none d) the relative maximum value none e) the coordinates of the relative minimum point if any (−1,6) f) the relative minimum value 𝑦 = 6 𝑤ℎ𝑒𝑛 𝑥 = −1 9) 10) a) interval(s) where the graph is increasing. (−∞, −1) b) interval(s) where the graph is decreasing. (−1, ∞) c) the coordinates of relative maximum point if any (−1,3) d) the relative maximum value 𝑦 = 3 𝑤ℎ𝑒𝑛 𝑥 = −1 e) the coordinates of the relative minimum point if any none f) the relative minimum value none 11) 12) a) interval(s) where the graph is increasing. (−3,1) ∪ (1, ∞) b) interval(s) where the graph is decreasing.(−∞, −7) ∪ (−7, −3) c) the coordinates of relative maximum point, if any none d) the relative maximum value none e) the coordinates of the relative minimum point if any (−3, 4.32) f) the relative minimum value 𝑦 = 4.32 𝑤ℎ𝑒𝑛 𝑥 = −3 (Minimum Homework: 1 – 11 odds 13, 17, 21, 25) #13 – 26: For each function find the following: a) f’(x) b) the critical numbers c) interval(s) where the graph is increasing. d) interval(s) where the graph is decreasing. e) the coordinates of relative maximum point if any f) the relative maximum value g) the coordinates of the relative minimum point if any h) the relative minimum value 13) 𝑓(𝑥 ) = 𝑥 2 − 6𝑥 + 3 14) 𝑓(𝑥 ) = 2𝑥 2 − 8𝑥 + 1 a) f’(x) 𝑓 ′ (𝑥 ) = 4𝑥 − 8 b) the critical numbers 𝑥 = 2 c) interval(s) where the graph is increasing. (2, ∞) d) interval(s) where the graph is decreasing. (−∞, 2) e) the coordinates of relative maximum point if any none f) the relative maximum value none g) the coordinates of the relative minimum point if any (2, −7) h) the relative minimum value 𝑦 = −7 𝑤ℎ𝑒𝑛 𝑥 = 2 #13 – 26: For each function find the following: a) f’(x) b) the critical numbers c) interval(s) where the graph is increasing. d) interval(s) where the graph is decreasing. e) the coordinates of relative maximum point if any f) the relative maximum value g) the coordinates of the relative minimum point if any h) the relative minimum value 15) 𝑓(𝑥 ) = 𝑥 2 − 3 Skipping for time 16) 𝑓(𝑥 ) = 𝑥 2 + 5 #13 – 26: For each function find the following: a) f’(x) b) the critical numbers c) interval(s) where the graph is increasing. d) interval(s) where the graph is decreasing. e) the coordinates of relative maximum point if any f) the relative maximum value g) the coordinates of the relative minimum point if any h) the relative minimum value 17) 𝑓(𝑥 ) = 𝑥 3 − 12𝑥 + 4 18) 𝑓(𝑥 ) = 𝑥 3 − 48𝑥 + 18 a) f’(x) 𝑓 ′ (𝑥 ) = 3𝑥 2 − 48 b) the critical numbers 𝑥 = 4, −4 c) interval(s) where the graph is increasing. (−∞, −4) ∪ (4, ∞) d) interval(s) where the graph is decreasing. (−4,4) e) the coordinates of relative maximum point if any (−4,146) f) the relative maximum value 𝑦 = 146 𝑤ℎ𝑒𝑛 𝑥 = −4 g) the coordinates of the relative minimum point if any (4, −110) h) the relative minimum value 𝑦 = −110 𝑤ℎ𝑒𝑛 𝑥 = 4 #13 – 26: For each function find the following: a) f’(x) b) the critical numbers c) interval(s) where the graph is increasing. d) interval(s) where the graph is decreasing. e) the coordinates of relative maximum point if any f) the relative maximum value g) the coordinates of the relative minimum point if any h) the relative minimum value 19) 𝑓(𝑥 ) = −𝑥 3 − 3𝑥 2 + 45𝑥 − 5 9 21) 𝑓(𝑥 ) = 𝑥+2 𝑥−5 20) 𝑓(𝑥 ) = −𝑥 3 − 6𝑥 2 + 24𝑥 − 22) 𝑓(𝑥 ) = 𝑥+3 𝑥−7 −10 a) f’(x) 𝑓 ′ (𝑥 ) = (𝑥−7)2 b) the critical numbers 𝑥 = 7 c) interval(s) where the graph is increasing. never d) interval(s) where the graph is decreasing. (−∞, 7) ∪ (7, ∞) e) the coordinates of relative maximum point if any none f) the relative maximum value none g) the coordinates of the relative minimum point if any none h) the relative minimum value none #13 – 26: For each function find the following: a) f’(x) b) the critical numbers c) interval(s) where the graph is increasing. d) interval(s) where the graph is decreasing. e) the coordinates of relative maximum point if any f) the relative maximum value g) the coordinates of the relative minimum point if any h) the relative minimum value 23) 𝑓(𝑥 ) = 𝑥−4 𝑥+1 25) 𝑓(𝑥 ) = 𝑥𝑒 3𝑥 24) 𝑓(𝑥 ) = 𝑥−6 𝑥+3 26) 𝑓(𝑥 ) = 𝑥𝑒 2𝑥 a) f’(x) 𝑓 ′ (𝑥 ) = 𝑒 2𝑥 (2𝑥 + 1) b) the critical numbers 𝑥 = −1/2 1 c) interval(s) where the graph is increasing. (− , ∞) 2 1 d) interval(s) where the graph is decreasing. (−∞, − ) 2 e) the coordinates of relative maximum point if any none f) the relative maximum value none 1 −1 g) the coordinates of the relative minimum point if any (− , ) 2 2𝑒 h) the relative minimum value 𝑦 = − 1 2𝑒 𝑤ℎ𝑒𝑛 𝑥 = − 1 2