ISMT11ET_C04_E

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1 Chapter Applications of Derivatives y ± ) ( )( ± Ê the graph is concave up on Š _ß Š ß _, concave $ " " $ don on Š ß Ê points of inflection at Š ß $ % $ % 9 "Î$ Î$ 9 "Î$ Î$. y a 7 b Ê y a 7b () a b, y ± )( ± "! " Ê the graph is rising on ( _ß ) ("ß_), falling on ( ß" ) Ê 7 a local maimum is 7 at, a local 7 &Î$ "Î$ "Î$ &Î$ &Î$ minimum is 7 at ; y a b a b, y )( Ê! the graph is concave up on (!ß _), concave don on ( _ß!) Ê a point of inflection at (!ß!). y sin Ê y cos, y [ ± ± ] Ê the graph is rising on ˆ ß, falling Î$ Î$ Î$ Î$ on ˆ ˆ ß ß Ê local maima are at at, local minima are at at ; y sin, y [ ± ± ± ] Ê the Î$ Î! Î Î$ graph is concave up on ˆ ˆ, concave don on ˆ ß! ß ß ˆ!ß Ê points of inflection at ˆ ß, (!ß!), ˆ ß 6. y tan Ê y sec, y ( ± ± ) Ê the graph is rising on ˆ ß Î Î$ Î$ Î ˆ, falling on ˆ a local maimum is at, a local minimum is ß ß Ê at ; y (sec )(sec )(tan ) asec b(tan ), y ( ± ) Ê the graph is concave up on ˆ 0 ß, Î! Î concave don on ˆ ß0 Ê a point of inflection at (0 ß! ) 7. If 0, sin kk sin if 0, sin kk sin ( ) sin. From the sketch the graph is rising on ˆ, ˆ ˆ ß!ß ß, falling on ˆ ß, ˆ ˆ ß! ß ; local minima are at 0 at! ; local maima are at 0 at ; concave up on ( ß) ( ß ), concavedon on ( ß 0) (!ß ) Ê points of inflection are ( ß! ) ( ß! ) 8. y cos Ê y sin, y [ ± ± ± ] Ê rising on $ Î% Î% & Î% $ Î ˆ ˆ, falling on ˆ ˆ ß ß ß ß Ê local maima are at, at at, local minima are at at ; y cos, y [ ± ± ] Ê concave up on ˆ ß ˆ ß, concave don on Î Î $ Î ˆ ß Ê points of inflection at Š ß Š ß

2 Section. Concavity Curve Sketching 9. When y, then y ( ) y. The curve rises on (ß_) falls on ( _ß ). At there is a minimum. Since y 0, the curve is concave up for all. 0. When y ', then y ( ") y. The curve rises on ( _ß) falls on ( ß_ ). At there is a maimum. Since y 0, the curve is concave don for all. $. When y, then y ( )( ) y 6. The curve rises on ( _ß ) ("ß _) falls on ( ß). At there is a local maimum at a local minimum. The curve is concave don on ( _ß 0) concave up on (!ß _). There is a point of inflection at 0.. When y (6 ), then y (6 ) (') ( )( ") y ( ) ( ") ( ). The curve rises on ( _ß" ) ( $ß_ ) falls on ("ß $ ). The curve is concave don on ( _ß ) concave up on (ß _). At there is a point of inflection. $. When y 6, then y 6 6( ) y ( ). The curve rises on (!ß ) falls on ( _ß 0) (ß _). At 0 there is a local minimum at a local maimum. The curve is concave up on ( _ß ") concave don on ("ß _). At there is a point of inflection.

3 6 Chapter Applications of Derivatives $. When y ( ), then y ( ) y 6( ). The curve never falls there are no local etrema. The curve is concave don on ( _ß ) concave up on (ß _). At there is a point of inflection. % $. When y, then y ( )( ) " " y Š Š. The curve rises on ( "ß!) ("ß _) falls on ( _ß ) (!ß "). At there are local minima at 0 a local maimum. The curve is concave up on Š _ß " Š ß_ concave don on Š ß. At there are points of inflection. " " " " % $ 6. When y 6, then y Š Š y ( )( ). The curve rises on Š _ß Š!ß, falls on Š ß! Š ß _. At there are local maima at 0 a local minimum. The curve is concave up on ( "ß ") concave don on ( _ß ) ("ß _). At there are points of inflection. $ % $ 7. When y, then y ( $) y ( ). The curve rises on a_ß$ b falls on a$ß _ b. At there is a local maimum, but there is no local minimum. The graph is concave up on a!ß b concave don on a_ß! b aß _ b. There are inflection points at 0.

4 Section. Concavity Curve Sketching 7 % $ $ 8. When y, then y 6 ( ) y ( ). The curve rises on ˆ falls on ˆ ß ß. There is a local minimum at, but no local maimum. The curve is concave up on ( _ß ) (!ß _), concave don on ( ß0). At 0 there are points of inflection. & % % $ $ 9. When y, then y 0 ( ) $ y ( ). The curve rises on ( _ß! ) (%ß_), falls on (!ß%). There is a local maimum at 0, a local minimum at. The curve is concave don on ( _ß ) concave up on ( ß_ ). At there is a point of inflection. % % $ $ ", y $ ( ). The curve is rising 0. When y ˆ, then y ˆ () ˆ ˆ " ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ on ( _ß ) (0 ß_ ), falling on (ß0). There is a local maimum at a local minimum at 0. The curve is concave don on ( _ß %) concave up on (%ß _). At there is a point of inflection.. When y sin, then y "cos y sin. The curve rises on (!ß ). At 0 there is a local absolute minimum at there is a local absolute maimum. The curve is concave don on (!ß ) concave up on ( ß ). At there is a point of inflection.. When y sin, then y "cos y sin. The curve rises on (!ß ). At 0 there is a local absolute minimum at there is a local absolute maimum. The curve is concave up on (!ß ) concave don on ( ß ). At there is a point of inflection.

5 8 Chapter Applications of Derivatives "Î& " %Î& *Î&. When y, then y y. The curve rises on ( _ß _) there are no etrema. The curve is concave up on ( _ß!) concave don on (!ß _). At 0 there is a point of inflection. Î& $Î& 6 )Î&. When y, then y y. The curve is rising on (0 ß_ ) falling on ( _ß! ). At 0 there is a local absolute minimum. There is no local or absolute maimum. The curve is concave don on ( _ß!) (!ß _). There are no points of inflection, but a cusp eists at 0. Î$ "Î$. When y, then y %Î$ y. The curve is rising on ( _ß!) ("ß _), falling on (!ß "). There is a local maimum at 0 a local minimum at. The curve is concave up on ( _ß!) (!ß _). There are no points of inflection, but a cusp eists at 0. Î$ 6. When y ˆ Î$ &Î$, then "Î$ Î$ "Î$ y ( ) %Î$ 0 "Î$ %Î$ y ( ). The curve is rising on (!ß ") falling on ( _ß!) ("ß _). There is a local minimum at 0 a local maimum at. The curve is concave up on ˆ " _ß concave don on ˆ " ß! (0 ß_ ). There is a point " of inflection at a cusp at When y 8 a8 b "Î, then "Î "Î "Î ( )( ) ÊŠ Š " y a8 b () ˆ a8 b ( ) a8 b a8 b $ " " y ˆ a8 b ( ) a8 ba8 b ( ) a b Éa8 b $. The curve is rising on ( ß ), falling on Š ß Š ß. There are local minima, local maima at. The curve is concave up on Š ß! concave don on Š!ß. There is a point of inflection at 0.

6 Section. Concavity Curve Sketching 9 $Î 8. When y a b, then y ˆ a b ( ) ÊŠ Š "Î " y ( ) a b ( ) ˆ a b ( ) 6( ")( ) ÊŠ Š. The curve is rising on "Î "Î Š ß! falling on Š!ß. There is a local maimum at 0, local minima at. The curve is concave don on ( "ß ") concave up on Š ß" Š "ß. There are points of inflection at. ( ) a b( ") () ( )( ) ( ) ( )( ) a b ( ) ( ) % ( ) $ 9. When y, then y y. The curve is rising on ( _ß ") ( $ß _), falling on ("ß ) (ß $ ). There is a local maimum at a local minimum at. The curve is concave don on ( _ß ) concave up on (ß _). There are no points of inflection because is not in the domain. $ a b (6) a b a b a b y " " a $ ba b a 6 b (6) ˆ % a % b 6( )( ") a b $ 0. When y, then y. The curve is rising on ( _ß _) so there are no local etrema. The curve is concave up on ( _ß" ) (!ß"), concave don on ( "ß! ) ("ß _). There are points of inflection at ", 0,., kk. When y k k, then, kk, kk, kk " y y. The, k k ", kk " curve rises on ( "ß!) ("ß _) falls on ( _ß ) (0ß). There is a local maimum at 0 local minima at. The curve is concave up on ( _ß) ("ß _), concave don on ( "ß "). There are no points of inflection because y is not differentiable at (so there is no tangent line at those points).

7 0 Chapter Applications of Derivatives, Ÿ 0. When y k k, then, 0 Ú 0 " 0 y Û y. 0 a b / " / Ü 0 The curve is rising on (0 ß_ ) falling on ( _ß0). There is a absolute minimum of 0 at 0. The curve is concave don on ( _ß 0) (0 ß _). / / e / /. When y e, then y e e ˆ y / / / / e ˆ ˆ e e e Š ˆ. The curve is rising on ( ß_ ) ( _ß0) falling on (0ß). The curve is concave don on ( _ß0) concave up on (0 ß_ ). There is a local minimum of e at, but there are no inflection points. e e e a be. y, Ê y Ê y ± ± Ê the graph is rising on 0 ( ß_ ), falling on ( _ß0) (0ß); a local minimum is e at ae e e bae e bab ˆ e ; y Ê y ± Ê the graph is concave up on 0 (0 ß_ ), concave don on ( _ß0), but has no inflection points.. y lna b, Ê y Ê y ( ± ) Ê the graph is rising on 0 Š ß0, falling on Š 0ß ; a local minimum is ln at ˆ ababab ˆ a b a b 0; y Ê y ( ) Ê the graph is concave don on Š ß. 6. y aln b, Ê y ln ˆ ab aln b ln a ln b Ê y ( ± ± Ê the 0 e graph is rising on a0ße b a ß_ b, falling on ae, b; a local maimum is e at e a local minimum is 0 at ; y ln ˆ ˆ a lnb a ln b Ê y ( ± Ê the graph is concave up on 0 e ae, _ b, concave don on a0, e bê point of inflection at ae, e b.