Assignment Date Period - Uplift Education...Kuta Software - Infinite Calculus Name_____ Assignment...
Transcript of Assignment Date Period - Uplift Education...Kuta Software - Infinite Calculus Name_____ Assignment...
©u u210R143j hKEuHt4as nSwo1f2tVwlagr7eE ELyLVCE.B S EAjl4lk FruiogGhUt5sf srGeisieZrhvPeZdp.9 r 6MtandseI twki0trhY 8I8nbfRiInDiotweS ACSaUlpcnuHlzuHsB.K Worksheet by Kuta Software LLC
Kuta Software - Infinite Calculus Name___________________________________
Period____Date________________Assignment
For each problem, find all points of relative minima and maxima.
1)
y =
x3 − 5
x2 + 7
x − 5
x
y
−8 −6 −4 −2 2 4 6 8
−8
−6
−4
−2
2
4
6
8
Relative minimum: (
7
3,
−86
27 )Relative maximum: (1, −2)
For each problem, find all points of relative minima and maxima. You may use the provided graph to
sketch the function.
2)
y =
x3 − 6
x2 + 9
x + 1
x
y
−8 −6 −4 −2 2 4 6 8
−8
−6
−4
−2
2
4
6
8
Relative minimum: (3, 1)Relative maximum: (1, 5)
-1-
©R u2f081i37 uKgu8t2a1 eSoo7f4tmwGa1r5ei 4LkLnCx.Y y jAhlnlg arXiUgnhutrsi HrjeFsZeFrivgepdm.v h iMJavdNep zw6iUtNho qIKnUfZi9noivtleh mCCaolDcTuQl9uns8.l Worksheet by Kuta Software LLC
For each problem, find all points of relative minima and maxima.
3)
y =
−
x3 − 3
x2 − 1
Relative minimum: (−2, −5)Relative maximum: (0, −1)
4)
y =
x4 − 2
x2 + 3
Relative minima: (−1, 2), (1, 2)Relative maximum: (0, 3)
5)
y =
x4 −
x2
Relative minima: (
−
2
2,
−1
4), (
2
2,
−1
4)
Relative maximum: (0, 0)
6)
y =
−
2
x2 − 4
Relative minimum: (0,
1
2 ) No relative maxima.
7)
y =
(
2
x − 8)
2
3
Relative minimum: (4, 0) No relative maxima.
8)
y =
−
1
5
(
x − 4)
5
3 −
2
(
x − 4)
2
3
Relative minimum: (0,
−
123
2
5)
Relative maximum: (4, 0)
Critical thinking questions:
9) Give an example function
f (
x) where
f
''(0) = 0 and there is no relative minimum or maximum at
x = 0.
Many answers. Ex:
f (
x) = 0,
x,
x3, etc
10) Give an example function
f (
x) where
f
''(0) = 0 and there is a relative maximum at
x = 0.
Many answers. Ex:
f (
x) =
−
x4
-2-
Create your own worksheets like this one with Infinite Calculus. Free trial available at KutaSoftware.com
©y F2G0B1g3K jKAuDtpay SSeoGfGtaw5aMrpe4 3LzLLCV.0 W JA4lXl8 DrviIg3httUsW grleusfe3rSvVeydT.r e uM2ahd8e4 iwVi6tjh6 AIWnOfCiUnUihtNee MC3aKlGcQuXlgufsN.W Worksheet by Kuta Software LLC
Kuta Software - Infinite Calculus Name___________________________________
Period____Date________________Absolute Extrema
For each problem, find all points of absolute minima and maxima on the given closed interval.
1) y = −x3
− 6x2
− 9x + 3; [−3, −1]
x
y
−8 −6 −4 −2 2 4 6 8
−8
−6
−4
−2
2
4
6
8
Absolute minimum: (−3, 3)Absolute maximum: (−1, 7)
2) y =8
x2 + 4
; [0, 5]
x
y
−8 −6 −4 −2 2 4 6 8
−8
−6
−4
−2
2
4
6
8
Absolute minimum: (5, 8
29)Absolute maximum: (0, 2)
3) y = x3
+ 6x2
+ 9x + 3; [−4, 0]
Absolute minima: (−4, −1), (−1, −1)Absolute maxima: (0, 3), (−3, 3)
4) y = x4
− 3x2
+ 4; [−1, 1]
Absolute minima: (−1, 2), (1, 2)Absolute maximum: (0, 4)
5) y =x
2
3x − 6; [3, 6]
Absolute minimum: (4, 8
3)Absolute maxima: (3, 3), (6, 3)
6) y = (x + 2)2
3; [−4, −2]
Absolute minimum: (−2, 0)Absolute maximum: (−4,
3
4)
-1-
©t W2r0j1G3l GK2u3t5ae WS1ozfzthwTa0rTej cL9LQCL.m Z fAZlSlD IrLiXgehPtwsh srbeDsPeJrUvReEd5.J 0 DMvaCdfeB MwqiMtvh9 LIqnkfti6nqiEt2eN uCWaYlLc8uYlCu4sK.Q Worksheet by Kuta Software LLC
For each problem, find all points of absolute minima and maxima on the given interval.
7) y = x3
− 3x2
− 3; (0, 3)
x
y
−8 −6 −4 −2 2 4 6 8
−8
−6
−4
−2
2
4
6
8
Absolute minimum: (2, −7)No absolute maxima.
8) y = (5x + 25)1
3; [−2, 2]
x
y
−8 −6 −4 −2 2 4 6 8
−8
−6
−4
−2
2
4
6
8
Absolute minimum: (−2, 3
15)Absolute maximum: (2,
3
35)
9) y = x3
− 3x2
+ 6; [0, ∞)
Absolute minimum: (2, 2)No absolute maxima.
10) y = x4 − 2x
2 − 3; (0, ∞)
Absolute minimum: (1, −4)No absolute maxima.
11) y =4
x2 + 2
; (−5, −2]
No absolute minima.
Absolute maximum: (−2, 2
3 )
12) y = −1
6(x + 1)
7
3 +
14
3(x + 1)
1
3; (−5, 0)
Absolute minimum: (−3, −43
2)No absolute maxima.
-2-
Create your own worksheets like this one with Infinite Calculus. Free trial available at KutaSoftware.com
Calculus Maximus WS 5.1: Extrema on an Interval
Page 1 of 8
Name_________________________________________ Date________________________ Period______
Worksheet 5.1—Extrema on an Interval
Show all work. No calculator unless otherwise stated.
1. Let f be the functioned defined on [ ]1,2− by ( ) 2/33 2f x x x= − .
(a) What is its maximum value? (b) What is its minimum value?
2. Sketch the graph of a function f that is continuous on [ ]1,5 and has an absolute minimum at 1x = , an
absolute maximum at 5x = , a local maximum at 2x = , and a local minimum at 4x = .
3. Sketch the graph of f by hand and use your sketch to find the absolute and local extrema of f.
(a) ( )1 , 0 2
2 4, 2 3
x xf x
x x
− ≤ <#= $
− ≤ ≤%(b) ( )
2
2
, -1 0
2 , 0 1
x xf x
x x
! ≤ <#= $
− ≤ ≤#&
Calculus Maximus WS 5.1: Extrema on an Interval
Page 2 of 8
4. Find the critical values of the function
(a) ( ) 4 3 23 4 6x t t t t= + − (b) ( )
2
1
1
zf z
z z
+=
+ +(c) ( ) 2/3 5/3
5g t t t= +
(d) ( ) ( )1g t t t= − (e) ( ) 4 tang θ θ θ= − (f) ( ) lnf x x x=
(g) ( ) 3 2G x x x= − (h) ( ) 2xf x xe= (i) ( ) 2 3g x x= +
Calculus Maximus WS 5.1: Extrema on an Interval
Page 3 of 8
5. Find the absolute extrema of f on the given interval.
(a) ( ) 3 22 3 12 1f x x x x= − − + , [ ]2,3− (b) ( ) ( )
321f x x= − , [ ]1,2− (c) ( )
2
2
4
4
xf x
x
−=
+, [ ]4,4−
(d) ( ) ( )38f t t t= − , [ ]0,8 (e) ( ) sin cosf x x x= + , 0,
3
π" #$ %& '
(f) ( ) 2cosf x x x= − , [ ],π π−
(g) ( )ln x
f xx
= , [ ]1,3 (h) ( ) xf x xe−= , [ ]0,2 (i) ( ) 2x xf x e e− −= − , [ ]0,1
(Hint: factor out 2xe− from ( )f x! )
Calculus Maximus WS 5.1: Extrema on an Interval
Page 4 of 8
6. Show that 5 is a critical value of the function ( ) ( )3
2 5g x x= + − , but g does not have a local extreme
value at 5.
7. Prove that the function ( ) 101 511f x x x x= + + + has neither a local maximum nor a local minimum by
analyzing the derivative function.
MULTIPLE CHOICE
8. Find all the critical values, 0x , of the function ( ) 5 sin5g x x x= + in ( )0,∞ , where 0, 1, 2, n = K .
(A) 0
3 1
5
nx π
+= (B)
05
nx π= (C)
0
2 1
5
nx π
+= (D)
0
4 1
5
nx π
+= (E)
0
1
5
nx π
+=
Calculus Maximus WS 5.1: Extrema on an Interval
Page 5 of 8
9. Find all the critical values of the function ( ) 2cosf x x x= + on the interval ( ),π π− . Hint: think of
where y x= has a critical value, then look at two different cases for ( )f x! .
(A) 5 5
, , 0, , 6 6 6 6
π π π π− − (B) 0x = (C) , 0,
6 6
π π− (D)
2,
3 3
π π−
(E) 5 5
, 0, 6 6
π π− (F)
2,
3 3
π π−
10. Determine the absolute maximum value of ( )2
5 2
14
xf x
x
+=
+ on the interval [ ]2,4− .
(A) 1
18 (B)
13
30 (C)
8
7 (D)
1
2 (E) None
11. Find all the critical values of f when ( ) ( )24/5
5f x x x= − .
(A) 5
0, 7
(B) 10
, 57
(C) 5
, 57
(D) 5
0, , 57
(E) 10
0, 7
(F) 10
0, , 57
Calculus Maximus WS 5.1: Extrema on an Interval
Page 6 of 8
12. Let f be the function defined by ( ) 21 2f x x x= − + on [ ]1,1− .
(i) Find the derivative of f
(A) ( )2
2
1 2
1
xf x
x
−" =
−
(B) ( )2
2
1
2
xf x
x
−" = (C) ( )
2
2
2
1
xf x
x
−" =
−
(D) ( ) 21f x x! = −
(E) ( )2
2
1
xf x
x
! =−
(F) ( ) 22 1f x x x! = −
(ii) Find all the critical points of f in ( )1,1− .
(A) 1
4 (B)
1
4± (C)
1
2± (D)
1
2
± (E) 1
2 (F)
1
2
(iii) Determine the absolute maximum value of f on [ ]1,1− .
(A) 7
2 (B)
5
2 (C) 1 (D)
3
2 (E) 2 (F) 3
Calculus Maximus WS 5.1: Extrema on an Interval
Page 7 of 8
13. Let f be the function defined by ( ) 2sin cosf x x x= − on [ ]0,2π .
(i) Find the derivative of f .
(A) ( ) ( )sin 2cos 1f x x x! = + (B) ( ) ( )cos 2sin 1f x x x! = + (C) ( ) ( )sin 2cos 1f x x x! = − +
(D) ( ) ( )cos 1 2sinf x x x! = − (E) ( ) ( )cos 1 2sinf x x x! = − + (F) ( ) ( )sin 2cos 1f x x x! = −
(ii) Find al the critical values of f in ( )0,2π .
(A) 5 3
, , , 6 2 6 2
π π π π (B)
5, ,
3 3
π ππ (C)
7 3 11, , ,
2 6 2 6
π π π π (D)
2 4 3, , ,
2 3 3 2
π π π π
(E) 2 4
, , 3 3
π ππ (F)
11, ,
6 6
π ππ
(iii) Determine the absolute maximum value of f on [ ]0,2π .
(A) 1− (B) 5
4− (C)
5
4 (D) 1 (E)
3
4− (F)
3
4
Calculus Maximus WS 5.1: Extrema on an Interval
Page 8 of 8
14. Let f be the function defined by ( )2
12
16
f x xx
! "= −$ %
& ', 0x ≠ . Determine the absolute maximum value
of f on ( ], 1−∞ − . Hint: find the domain of f, the critical values of f, then look at the sign of f ! for all
values in the specified interval.
(A) 17
8− (B) No max value (C)
3
2− (D)
3
2 (E)
17
8
15. An advertisement is run to stimulate the sale of cars. After t days, 1 48t≤ ≤ , the number of cars sold is
given by ( ) 2 34000 45N t t t= + − . On what day does the maximum rate of growth (that’s ( )N t! ) of
sales occur?
(A) day 17 (B) day 13 (C) day 15 (D) day 16 (E) day 14