MATH 12002 - CALCULUS I § 3.3: Graphing Example Professor Donald L. White Department of Mathematical Sciences Kent State University D.L. White (Kent State University) 1 / 14
Example Example Let 2 x 2 / 3 − 3 f ( x ) = 3 5 x 5 / 3 . Determine intervals where f is increasing, intervals where f is decreasing, the location of all local maxima and minima, intervals where f is concave up, intervals where f is concave down, and the location of all inflection points. D.L. White (Kent State University) 2 / 14
Example We need to determine the signs of f ′ and f ′′ for f ( x ) = 3 2 x 2 / 3 − 3 5 x 5 / 3 . First, x − 1 / 3 − x 2 / 3 f ′ ( x ) = 1 x 1 / 3 − x 2 / 3 = 1 − ( x 1 / 3 )( x 2 / 3 ) = x 1 / 3 1 − x = x 1 / 3 . Hence f ′ ( x ) = 0 when x = 1 and f ′ ( x ) is undefined when x = 0. D.L. White (Kent State University) 3 / 14
Example Using f ′ ( x ) = x − 1 / 3 − x 2 / 3 , we have 3 x − 4 / 3 − 2 − 1 3 x − 1 / 3 f ′′ ( x ) = � 1 − 1 2 � = x 4 / 3 + x 1 / 3 3 − 1 � 1 + 2 x � = x 4 / 3 3 � 2( 1 � − 1 2 + x ) = 3 x 4 / 3 − 2( x + 1 2 ) = 3 x 4 / 3 . Hence f ′′ ( x ) = 0 when x = − 1 2 and f ′′ ( x ) is undefined when x = 0. D.L. White (Kent State University) 4 / 14
Example √ x x 1 / 3 has a nonlinear factor x 1 / 3 = Notice that f ′ ( x ) = 1 − x 3 √ √ x ) 4 . − 2( x + 1 2 ) has a nonlinear factor x 4 / 3 = x 4 = ( 3 3 and f ′′ ( x ) = 3 x 4 / 3 How do we determine the signs of these factors? If a is a real number and n is an odd integer, a and a n have the same sign. Applying this to a = x 1 / 3 and n = 3, we see that x 1 / 3 and ( x 1 / 3 ) 3 = x have the same sign. More generally, we have If a is a real number and n is an odd integer, √ a all have the same sign. then a , a n , and a 1 / n = n Now x 4 / 3 = ( x 1 / 3 ) 4 , so x 4 / 3 is always positive (or 0). D.L. White (Kent State University) 5 / 14
Example − 2( x + 1 2 ) 2 x 2 / 3 − 3 f ( x ) = 3 5 x 5 / 3 , f ′ ( x ) = 1 − x x 1 / 3 , f ′′ ( x ) = 3 x 4 / 3 − 1 0 1 2 0 1 − x + + + − x 1 / 3 0 + + − − ✲ f ′ ( x ) X 0 + − − − − 2 − − − − x + 1 0 − + + + 2 0 3 x 4 / 3 + + + + ✛ f ′′ ( x ) 0 X + − − − ✲ D D I D Inc-Dec MIN MAX ✛ U D D D Concave INF ☞ ✎ ☞ ✍ Shape D.L. White (Kent State University) 6 / 14
Example − 2( x + 1 2 x 2 / 3 − 3 2 ) f ( x ) = 3 5 x 5 / 3 , f ′ ( x ) = 1 − x x 1 / 3 , f ′′ ( x ) = 3 x 4 / 3 − 1 0 1 2 D D I D Inc-Dec MIN MAX U D D D Concave INF ✟ ☛ ✟ ✡ Shape f is increasing on (0 , 1), f is decreasing on ( −∞ , 0) ∪ (1 , ∞ ); f has a local minimum at x = 0, f has a local maximum at x = 1. f is concave up on ( −∞ , − 1 2 ), f is concave down on ( − 1 2 , 0) ∪ (0 , ∞ ); f has an inflection point at x = − 1 2 . D.L. White (Kent State University) 7 / 14
Example In order to sketch the graph of f , we will need to plot the points whose x coordinates are in the sign chart. 5 x 5 / 3 at these points: 2 x 2 / 3 − 3 We need to evaluate f ( x ) = 3 2 ) 2 / 3 − 3 f ( − 1 3 2 ( − 1 5 ( − 1 2 ) 5 / 3 2 ) = ( − 1 2 ) 2 / 3 � 3 2 − 3 5 ( − 1 � = 2 ) � � � 3 1 2 + 3 � = ( − 2) 2 / 3 10 � � � 15 1 10 + 3 � = 4 1 / 3 10 1 4 · 18 9 = 10 = 4 ≈ 1 . 13 √ √ 3 5 3 5 (0) 5 / 3 = 0 2 (0) 2 / 3 − 3 3 f (0) = 2 (1) 2 / 3 − 3 5 (1) 5 / 3 = 3 3 2 − 3 5 = 15 10 − 6 10 = 9 f (1) = 10 = 0 . 9 . Hence the points ( − 1 9 4 ) ≈ ( − 0 . 5 , 1 . 13), (0 , 0), and (1 , 9 10 ) = (1 , 0 . 9) √ 2 , 5 3 are on the graph. D.L. White (Kent State University) 8 / 14
Example It is also useful to have the x -intercepts; that is, x values where f ( x ) = 0. We can factor f ( x ) as follows: 2 x 2 / 3 − 3 3 5 x 5 / 3 f ( x ) = 3 x 2 / 3 � 1 2 − 1 5 x 3 / 3 � = 3 x 2 / 3 � 1 2 − 1 � = 5 x . Therefore, f ( x ) = 0 when x 2 / 3 = 0, and so x = 0 (also the y -intercept), or when 1 2 − 1 5 x = 0, and so x = 5 2 = 2 . 5. Hence the x -intercepts are x = 0 and x = 2 . 5. D.L. White (Kent State University) 9 / 14
Example − 1 0 1 2 D D I D Inc-Dec MIN MAX U D D D Concave INF ✟ ☛ ✟ ✡ Shape ✻ 2 q (1 , 0 . 9), MAX q ( − 1 1 2 , ≈ 1 . 13), INF ✛ ✲ (2 . 5 , 0), INT q q − 1 1 2 3 (0 , 0), MIN ❄ D.L. White (Kent State University) 10 / 14
Example D.L. White (Kent State University) 11 / 14
Example D.L. White (Kent State University) 12 / 14
Example D.L. White (Kent State University) 13 / 14
Example D.L. White (Kent State University) 14 / 14
Recommend
More recommend
