Curve Sketching (IV) (x 1) 2
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- Ross Harrell
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1 . Consider the following polynomial function:f() = 9 ( ) 5 ( + ) 3 Sketch the grapf of the function f().. Consider the following polynomial function:f() = ( 3) 6 + Sketch the grapf of the function f(). 3. Consider the following polynomial function:f() = 5 ( ) 3 ( + 3) Sketch the grapf of the function f().. Consider the following polynomial function:f() = 6 ( ) 5 ( + ) 3 Sketch the grapf of the function f(). 5. Consider the following polynomial function:f() = 96 ( 3) 5 ( + 3) 3 Sketch the grapf of the function f(). 6. Consider the following polynomial function:f() = 6 Sketch the grapf of the function f(). 7. Consider the following polynomial function:f() = 6 Sketch the grapf of the function f(). ( ) 5 ( + 3) ( ) ( + 3) 3 8. Consider the following polynomial function:f() = 9 ( 3) Sketch the grapf of the function f(). 9. Consider the following polynomial function:f() = 8 ( )3 5 ( + ) 3 Sketch the grapf of the function f(). 0. Consider the following polynomial function:f() = 3 ( + ) 3 Sketch the grapf of the function f(). c 009 La Citadelle of
2 Solutions:. f() = 9 ( ) 5 ( + ) 3 Domain The domain is D f = R. Symmetry f( ) = 9 ( ) 5 ( + ) 3 f( ) f() f( ) f() Therefore the function f() is neither even nor odd function. Zeros The zero(s) of the function f() is(are): = = Sign Chart for f() y-intercept y int = f(0) = 9 (0 ) 5 (0 + ) 3 = 0.68 (, ) (, ) (, ) f() Asymptotes The function f() does not have any kind of asymptotes. Critical Numbers f () = d d 9 ( 5 ) ( + ) 3 = d d 9 ( ) ( + ) 3 5 = [3 + 7] 5 5 ( + ) Critical numbers are the solutions of the equation f () = 0 5 [3 + 7] = 0 5 ( + ) 3 = 7 3 =.308 The critical number(s) is(are): = =.308 = p3= or Sign Chart for First Derivative f () (, ) (,.308).308 (.308, ) (, ) f() f () + DNE Increasing and Decreasing Intervals The function f() is increasing over (, ) (,.308) (, ) The function f() is decreasing over (.308, ) c 009 La Citadelle of
3 Maimum and Minimum Points The function f() has a maimum point at (.308, 0.75) The function f() has a minimum point at (, 0.000) Concavity Intervals The second derivative of the function f() is given by: f () = d [3 + 7] = d 5 5 ( + ) 5 5 ( + ) 7 ( ) The second derivative f () is zero when f () = 0 or: 5 The second derivative f () is zero at: = = The second derivative f () does not eist at: = ( + ) 7 ( ) = 0 5 Sign Chart for the Second Derivative f () (,.307).307 (.307, ) (, 0.308) ( 0.308, ) f() f () 0 + DNE 0 + Inflection Points The infllection point(s) is(are): (-.307,-0.598) (-,0.000) (-0.308,0.6) Graph y (.308, 0.75) ( 0.308, 0.6) (, 0.000) (, 0.000) (.307, 0.598) c 009 La Citadelle 3 of
4 . f() = ( 3) 6 + Domain The domain is D f = (, ). Symmetry f( ) = ( 3) 6 + f( ) f() f( ) f() Therefore the function f() is neither even nor odd function. Zeros The zero(s) of the function f() is(are): = 3 Sign Chart for f() y-intercept y int = f(0) = 6 (0 3) 0 + = (, ) (, 3) 3 (3, ) f() DNE DNE Asymptotes The function f() has a vertical asymptote at =. Critical Numbers f () = d ( 3) = d d 6 + d 6 ( 3) ( + ) = 3 [3 + 7] 3 ( + ) 3 Critical numbers are the solutions of the equation f () = 0 3 [3 + 7] = 0 3 ( + ) 3 3 = 7 3 =.333 The critical number(s) is(are): = = 3 Sign Chart for First Derivative f () Increasing and Decreasing Intervals The function f() is increasing over (3, ) The function f() is decreasing over (, 3) Maimum and Minimum Points (, ) (, 3) 3 (3, ) f() DNE DNE 0 f () DNE DNE 0 + or c 009 La Citadelle of
5 The function f() has a minimum point at (3, 0.000) Concavity Intervals The second derivative of the function f() is given by: f () = d 3 d 3 [3 + 7] = ( + ) 3 6 ( + ) 5 ( ) The second derivative f () is zero when f () = 0 The second derivative f () is zero at: The second derivative f () does not eist at: = or: 6 ( + ) 5 ( ) = 0 Sign Chart for the Second Derivative f () Inflection Points The infllection point(s) is(are): Graph (, ) (, ) f() DNE DNE f () DNE DNE + y (3, 0.000) c 009 La Citadelle 5 of
6 3. f() = 5 ( ) 3 ( + 3) Domain The domain is D f = R. Symmetry f( ) = 5 ( ) 3 ( + 3) f( ) f() f( ) f() Therefore the function f() is neither even nor odd function. Zeros The zero(s) of the function f() is(are): = = 3 Sign Chart for f() y-intercept y int = f(0) = 5 (0 ) 3 (0 + 3) =.73 (, 3) 3 ( 3, ) (, ) f() Asymptotes The function f() does not have any kind of asymptotes. Critical Numbers f () = d d 5 ( ) 3 ( + 3) = d d 5 ( )( + 3) 3 = 3 + 3[7 + ] 5 Critical numbers are the solutions of the equation f () = 0 or 3 + 3[7 + ] = = 7 = 0.3 The critical number(s) is(are): = 3 = 0.3 p3= Sign Chart for First Derivative f () (, 3) 3 ( 3, 0.3) 0.3 ( 0.3, ) f() f () Increasing and Decreasing Intervals The function f() is increasing over (, 3) ( 0.3, ) The function f() is decreasing over ( 3, 0.3) Maimum and Minimum Points The function f() has a maimum point at ( 3, 0.000) c 009 La Citadelle 6 of
7 The function f() has a minimum point at ( 0.3,.738) Concavity Intervals The second derivative of the function f() is given by: f () = d d [7 + ] = 5 (8 + 6) 3 ( + 3) The second derivative f () is zero when f () = 0 The second derivative f () is zero at: =.86 or: 5 (8 + 6) = 0 3 ( + 3) The second derivative f () does not eist at: = 3 Sign Chart for the Second Derivative f () (, 3) 3 ( 3,.86).86 (.86, ) f() f () DNE 0 + Inflection Points The infllection point(s) is(are): (-.86,-0.57) Graph y ( 3, 0.000) (.86, 0.57) ( 0.3,.738) c 009 La Citadelle 7 of
8 . f() = 6 ( ) 5 ( + ) 3 Domain The domain is D f = R. Symmetry f( ) = 6 ( 5 ) ( + ) 3 f( ) f() f( ) f() Therefore the function f() is neither even nor odd function. Zeros The zero(s) of the function f() is(are): = = Sign Chart for f() y-intercept y int = f(0) = 6 (0 ) 5 (0 + ) 3 = (, ) (, ) (, ) f() Asymptotes The function f() does not have any kind of asymptotes. Critical Numbers f () = d d 6 ( 5 ) ( + ) 3 = d d 6 ( ) ( + ) 3 5 = [3 + ] 80 5 ( + ) Critical numbers are the solutions of the equation f () = 0 80 [3 + ] = 0 5 ( + ) 3 = 3 =.077 The critical number(s) is(are): = =.077 = p3= or Sign Chart for First Derivative f () (, ) (,.077).077 (.077, ) (, ) f() f () + DNE Increasing and Decreasing Intervals The function f() is increasing over (, ) (,.077) (, ) The function f() is decreasing over (.077, ) Maimum and Minimum Points c 009 La Citadelle 8 of
9 The function f() has a maimum point at (.077, 0.56) The function f() has a minimum point at (, 0.000) Concavity Intervals The second derivative of the function f() is given by: f () = d [3 + ] = d 80 5 ( + ) 00 5 ( + ) 7 ( ) The second derivative f () is zero when f () = 0 or: 00 The second derivative f () is zero at: =.09 5 = 0.55 The second derivative f () does not eist at: = ( + ) 7 ( ) = 0 5 Sign Chart for the Second Derivative f () (,.09).09 (.09, ) (, 0.55) 0.55 (0.55, ) f() f () 0 + DNE 0 + Inflection Points The infllection point(s) is(are): (-.09,-0.7) (-,0.000) (0.55,0.30) Graph y (.077, 0.56) (0.55, 0.30) (, 0.000) (, 0.000) (.09, 0.7) c 009 La Citadelle 9 of
10 5. f() = 96 ( 3) 5 ( + 3) 3 Domain The domain is D f = R. Symmetry f( ) = 96 ( 5 3) ( + 3) 3 f( ) f() f( ) f() Therefore the function f() is neither even nor odd function. Zeros The zero(s) of the function f() is(are): = 3 = 3 Sign Chart for f() y-intercept y int = f(0) = 96 (0 3) 5 (0 + 3) 3 = 0. (, 3) 3 ( 3, 3) 3 (3, ) f() Asymptotes The function f() does not have any kind of asymptotes. Critical Numbers f () = d d 96 ( 5 3) ( + 3) 3 = d d 96 ( 3) ( + 3) 3 ( 3) 3 5 = [3 + 5] ( + 3) Critical numbers are the solutions of the equation f () = ( 3) 3 [3 + 5] = 0 5 ( + 3) 3 = 5 3 =.7 The critical number(s) is(are): = 3 =.7 = 3 p3= or Sign Chart for First Derivative f () (, 3) 3 ( 3,.7).7 (.7, 3) 3 (3, ) f() f () + DNE Increasing and Decreasing Intervals The function f() is increasing over (, 3) ( 3,.7) (3, ) The function f() is decreasing over (.7, 3) Maimum and Minimum Points c 009 La Citadelle 0 of
11 The function f() has a maimum point at (.7, 0.9) The function f() has a minimum point at (3, 0.000) Concavity Intervals The second derivative of the function f() is given by: f () = d ( 3) 3 ( 3) [3 + 5] = d ( + 3) ( + 3) 7 ( ) The second derivative f () is zero when f () = 0 or: 300 ( 3) ( + 3) 7 ( ) = 0 The second derivative f () is zero at: = =.5 = The second derivative f () does not eist at: = 3 Sign Chart for the Second Derivative f () (, 3.8) 3.8 ( 3.8, 3) 3 ( 3,.5).5 (.5, 3) 3 (3, ) f() f () 0 + DNE Inflection Points The infllection point(s) is(are): (-3.8,-0.56) (-3,0.000) (-.5,0.33) Graph y (.7, 0.9) (.5, 0.33) ( 3, 0.000) (3, 0.000) ( 3.8, 0.56) c 009 La Citadelle of
12 6. f() = 6 ( ) 5 ( + 3) Domain The domain is D f = R\{ 3}. Symmetry f( ) = 6 f( ) f() ( ) 5 ( + 3) f( ) f() Therefore the function f() is neither even nor odd function. Zeros The zero(s) of the function f() is(are): = Sign Chart for f() y-intercept y int = f(0) = (0 ) 6 = (0 + 3) (, 3) 3 ( 3, ) (, ) f() + DN E Asymptotes The function f() has a vertical asymptote at = 3. Critical Numbers f () = d ( ) d 6 = d 5 ( + 3) d 6 ( ) ( + 3) 5 = [8 + 3] 80 5 ( + 3) 7 Critical numbers are the solutions of the equation f () = 0 80 [8 + 3] = 0 5 ( + 3) 7 3 = 3 = The critical number(s) is(are): = 3 = = p3= or Sign Chart for First Derivative f () (, ) (, 3) 3 ( 3, ) (, ) f().563 DN E 0 f () 0 + DNE 0 + Increasing and Decreasing Intervals The function f() is increasing over (, 3) (, ) The function f() is decreasing over (, ) ( 3, ) c 009 La Citadelle of
13 Maimum and Minimum Points The function f() has a minimum point at (,.563) (, 0.000) Concavity Intervals The second derivative of the function f() is given by: f () = d [8 + 3] = d 80 5 ( + 3) ( + 3) ( ) The second derivative f () is zero when f () = 0 The second derivative f () is zero at: The second derivative f () does not eist at: = 3 or: 00 ( + 3) ( ) = 0 5 Sign Chart for the Second Derivative f () Inflection Points The infllection point(s) is(are): Graph (, 3) 3 ( 3, ) f() DNE f () + DNE + y (,.563) (, 0.000) c 009 La Citadelle 3 of
14 7. f() = 6 ( ) ( + 3) 3 Domain The domain is D f = ( 3, ). Symmetry f( ) = 6 f( ) f() ( ) ( + 3) 3 f( ) f() Therefore the function f() is neither even nor odd function. Zeros The zero(s) of the function f() is(are): = Sign Chart for f() y-intercept y int = f(0) = (0 ) 6 = 0.07 (0 + 3) 3 (, 3) 3 ( 3, ) (, ) f() DNE DNE Asymptotes The function f() has a vertical asymptote at = 3. Critical Numbers f () = d ( ) d 6 = d ( + 3) 3 d 6 ( ) ( + 3) 3 = [5 + 7] 6 ( + 3) 7 Critical numbers are the solutions of the equation f () = 0 6 [5 + 7] = 0 ( + 3) 7 3 = 7 = The critical number(s) is(are): = 3 = Sign Chart for First Derivative f () Increasing and Decreasing Intervals The function f() is increasing over (, ) The function f() is decreasing over ( 3, ) Maimum and Minimum Points (, 3) 3 ( 3, ) (, ) f() DNE DNE 0 f () DNE DNE 0 + or c 009 La Citadelle of
15 The function f() has a minimum point at (, 0.000) Concavity Intervals The second derivative of the function f() is given by: f () = d [5 + 7] = d 6 ( + 3) 7 56 ( + 3) ( ) The second derivative f () is zero when f () = 0 The second derivative f () is zero at: The second derivative f () does not eist at: = 3 or: 56 ( + 3) ( ) = 0 Sign Chart for the Second Derivative f () Inflection Points The infllection point(s) is(are): Graph (, 3) 3 ( 3, ) f() DNE DNE f () DNE DNE + y (, 0.000) c 009 La Citadelle 5 of
16 8. f() = 9 ( 3) Domain The domain is D f = [0, ). Symmetry f( ) = 9 ( 3) f( ) f() f( ) f() Therefore the function f() is neither even nor odd function. Zeros The zero(s) of the function f() is(are): = 3 = 0 Sign Chart for f() y-intercept y int = f(0) = 9 (0 3) 0 = (, 0) 0 (0, 3) 3 (3, ) f() DN E Asymptotes The function f() does not have any kind of asymptotes. Critical Numbers f () = d d 9 ( 3) = d d 9 ( 3) = 36 3 [9 + 3] 3 Critical numbers are the solutions of the equation f () = [9 + 3] = = 3 9 = The critical number(s) is(are): = 0 = = 3 Sign Chart for First Derivative f () Increasing and Decreasing Intervals (, 0) 0 (0, 0.333) (0.333, 3) 3 (3, ) f() DN E f () DNE DNE The function f() is increasing over (0, 0.333) (3, ) The function f() is decreasing over (0.333, 3) Maimum and Minimum Points The function f() has a maimum point at (0.333, 0.600) or c 009 La Citadelle 6 of
17 The function f() has a minimum point at (3, 0.000) Concavity Intervals The second derivative of the function f() is given by: f () = d d 36 3 [9 + 3] = 3 7 ( ) The second derivative f () is zero when f () = 0 The second derivative f () is zero at: 5 =.77 The second derivative f () does not eist at: = 0 or: 7 ( ) = 0 Sign Chart for the Second Derivative f () Inflection Points The infllection point(s) is(are): (.77,0.385) Graph (, 0) 0 (0,.77).77 (.77, ) f() DN E f () DNE DNE 0 + y (0.333, 0.600) (.77, 0.385) (3, 0.000) c 009 La Citadelle 7 of
18 9. f() = 8 ( )3 5 ( + ) 3 Domain The domain is D f = R. Symmetry f( ) = 8 ( )3 5 ( + ) 3 f( ) f() f( ) f() Therefore the function f() is neither even nor odd function. Zeros The zero(s) of the function f() is(are): = = Sign Chart for f() y-intercept y int = f(0) = 8 (0 )3 5 (0 + ) 3 = 0.5 (, ) (, ) (, ) f() Asymptotes The function f() does not have any kind of asymptotes. Critical Numbers f () = d d 8 ( 5 )3 ( + ) 3 = d d 8 ( )3 ( + ) 3 ( ) 5 = [8 + ] 0 5 ( + ) Critical numbers are the solutions of the equation f () = 0 0 ( ) [8 + ] = 0 5 ( + ) 3 = 8 = The critical number(s) is(are): = = = p3= or Sign Chart for First Derivative f () (, ) (, 0.667) ( 0.667, ) (, ) f() f () DNE Increasing and Decreasing Intervals The function f() is increasing over ( 0.667, ) (, ) The function f() is decreasing over (, ) (, 0.667) Maimum and Minimum Points c 009 La Citadelle 8 of
19 The function f() has a minimum point at ( 0.667, 0.99) Concavity Intervals The second derivative of the function f() is given by: f () = d ( ) [8 + ] = d 0 5 ( + ) 00 5 ( + ) 7 ( ) The second derivative f () is zero when f () = 0 or: 00 ( + ) 7 ( ) = 0 The second derivative f () is zero at: =.9 5 = 0.0 = The second derivative f () does not eist at: = Sign Chart for the Second Derivative f () (,.9).9 (.9, ) (, 0.0) 0.0 ( 0.0, ) (, ) f() f () + 0 DNE Inflection Points The infllection point(s) is(are): (-.9,0.353) (-,0.000) (-0.0,-0.90) (,0.000) Graph y (.9, 0.353) (, 0.000) (, 0.000) ( 0.0, 0.90) ( 0.667, 0.99) c 009 La Citadelle 9 of
20 0. f() = 3 ( + ) 3 Domain The domain is D f = [, ). Symmetry f( ) = 3 ( + ) 3 f( ) f() f( ) f() Therefore the function f() is neither even nor odd function. Zeros The zero(s) of the function f() is(are): = 0 = Sign Chart for f() y-intercept y int = f(0) = 03 (0 + ) 3 = (, ) (, 0) 0 (0, ) f() DN E Asymptotes The function f() does not have any kind of asymptotes. Critical Numbers f () = d d 3 ( + ) 3 = d d 3 ( + ) 3 = + [5 + ] Critical numbers are the solutions of the equation f () = 0 + [5 + ] = 0 3 = 5 = The critical number(s) is(are): = = = 0 Sign Chart for First Derivative f () (, ) (, 0.800) ( 0.800, 0) 0 (0, ) f() DN E f () DNE DNE Increasing and Decreasing Intervals The function f() is increasing over ( 0.800, 0) (0, ) The function f() is decreasing over (, 0.800) Maimum and Minimum Points or c 009 La Citadelle 0 of
21 The function f() has a minimum point at ( 0.800, 0.53) Concavity Intervals The second derivative of the function f() is given by: f () = d d + [5 + ] = 6 The second derivative f () is zero when f () = 0 ( + ) 5 ( ) The second derivative f () is zero at: 5 = = or: 6 ( + ) 5 ( ) = 0 The second derivative f () does not eist at: = Sign Chart for the Second Derivative f () (, ) (, 0.559) ( 0.559, 0) 0 (0, ) f() DN E f () DNE DNE Inflection Points The infllection point(s) is(are): (-0.559,-0.095) (0,0.000) Graph y ( 0.559, (0, 0.000) 0.095) ( 0.800, 0.53) c 009 La Citadelle of
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