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Math 1431
LAB session 6
Quiz 11
Example 1:
The function f is graphed below on the interval [-4, 4].Give the number of values c between -4 and 4 which
satisfy the conclusion of the mean value theorem for f.
Example 2:
A) 29
B) 25
Determine if Rolles Theorem applies to the function f (x) = x5
interval that satisfy the theorem.
C)- 29
D) 27
x on [-1, 0]. If so, find all numbers c on the
A) 29
Example 3:
B) 25
f (x) = x5
C)- 29
D) 27
x
p
Determine if the function f (x) = 5 x x satisfies the Mean Value Theorem on [4, 9]. If so, find all numbers c
on the interval that satisfy the theorem.
f (x) = x25x+3
f (x) = 7x
7 sin x
f (x) = 3 sin2 (x)
f (x) = 4x2 (3
f (x) = x4
x)2
x2
f (x) = 13 x3
3 2
x
2
f (x) = 4x4
2x2 + 10
10x + 17
f (x) =
f (x) = x2 + 8x + 19
A) 29 B) 25 C)- 29
f (x) = x5
Example 4:
p
f (x) = 5 x
x
x2
5 6x
4+x
2
D)
87
>
< 3x if 0  x  1
f (x) = x 4 if 1 < x  5
>
:
6 x if 5 < x  7
Find the intervals on which the following function increases.
f (x) =
5x
x2 +3
5
x2
f (x) = 5 x
Example 5:
f (x) =
5x
x2 +3
Find the intervals on which f (x) = 4x2 (3
A) 29
B) 25
f (x) = x5
p
f (x) = 5 x
x)2
C)- 29
decreases.
D) 27
x
x2
5x
x2 +3
Example 6:
f (x) =
Find the intervals on which
f (x) = 7x 7 sin x increases for 0≤x≤2π.
f (x) = 4x2 (3 x)2
5
f (x) = 7x
Example 7:
7 sin x
f (x) = 3 sin2 (x)
Find the intervals on which
f (x) = 4x2 (3
A) 29
B) 25
f (x) = x5
C)- 29
D) 27
x
5x
x2 +3
A) 29
f (x) = 7x
7 sin x
f (x) = 4x2 (3
f (x) = x4
x)2
3 2
x
2
f (x) =
D) 27
x
x2
2
A)
5x 9
B) 25
x2 +3
5
f (x)7 =
f (x) = 7x
sinx
x
10x + 17
2
f (x) = 3fsin
(x)(x)
=
f (x) = 4x4 2x2 + 10
Question
#:
f (x) = 4x2 (3
8
> 3x if 0  x  1
B)not applicable<
f (x) = x 4 if 1 < x  5
>
:
6 x if 5 < x  7
D)-1; 0; 1
f (x) = x4
x2
C)- 29
x
p
5 x
x2
5x
x2 +3
on [-2, 2]. If so, find all numbers c on
f (x) = 7x
7 sin x
f (x) = 3 sin2 (x)
A) f (7) local minimum; f (3.8) local maximum
f (x) = 4x2 (3
B) f (3) local minimum; f (5) local maximum
f (x) = x4
Question #:
A)( 2, 5)
C)(0, 3)
x)2
x2
C) f ( 2.5) local minimum; f (5.8) local maximum
1
Find the intervals on which the following function decreases f (x) = 3 x3
D) f (0) local minimum
D) 27
x)2
f (x) =
5 6x to the function
Determine if Rolles Theorem applies
f (x) =
the
interval
that 1,
satisfy
C)(0,
3) D)(
3) the
[ (3,theorem.
1) 4 + x
C)2; 4
C)- 29
p
f (x) = 5 x
x2
f (x) = 13 x3
B) 25
f (x) = x5
f (x) = 3 sin2 (x)
A)5; 9
x)2
x
p
f (x) = 5 x
f (x) =
decreases for 0≤x≤π.
5
3 2
x
2
10x + 17
B)( 1, 2) [ (5, 1)
D)( 1, 3) [ (3, 1)
A)5; 9
B)not applicable
C)2; 4
D)-1; 0; 1
5
5
f (x) = 7x
7 sin x
f (x) = 3 sin2 (x)
Quiz 12
f (x) = 4x2 (3
f (x) = x4
Example 8:
f (x) = 13 x3
Find the critical numbers of f (x) = 4x4
x)2
x2
3 2
x
2
10x + 17
2x2 + 10 and classify all local extreme values.
D) 27
5
10x + 17
0
Example 9:
Find the critical numbers of the following function and classify all local extreme values.
f (x) =
5 6x
4+x
5
f (x) = x5
x
Example 10:
f (x) =
p
f (x) = 5 x
Find the critical numbers of
f (x) =
x2
f (x) = x2 + 8x + 19
5 6x
4+x
and classify all extreme values given -5≤x≤5
5x
x2 +3
f (x) = 7x
7 sin x
f (x) = 3 sin2 (x)
f (x) = 4x2 (3
f (x) = x4
x)2
x2
f (x) = 13 x3
3 2
x
2
f (x) = 4x4
2x2 + 10
10x + 17
5
f (x) =
5 6x
4+x
2
(x) =
+ 8x numbers
+ 19
Example 11:fFind
thexcritical
of f and classify the extreme values given
8
>
< 3x if 0  x  1
f (x) = x 4 if 1 < x  5
>
:
6 x if 5 < x  7
5
f (x) = x4
x2
f (x) = 13 x3
3 2
x
2
f (x) = 4x4
2x2 + 10
10x + 17
For the following questions you are given the graph of the derivative of f(x)
Question #:
A) 29
List all critical points for function f(x)
B) 25
C)- 29
f (x) = x5
D) 27
f (x) =
x
5 6x
4+x
2
f (x) = x + 8xp+ 19
f (x) = 5 x
f (x) =
x
8
>
< 3x if 0  x  1
f (x) = x 4 if 1 < x  5
7 sin x
>
:
6 x if 5 < x  7
5x
x2 +3
f (x) = 7x
f (x) = 3 sin2 (x)
A)-1.5; 4
B)-4; 3; 5
f (x) = 4x2 (3
C)3; 5
D)-4; 3
x)2
A)f (7) local minimum; f (3.8) local maximum
4
2
f (5) local fmaximum
(x)2 = xB)
2 x
A)
C)- 29 D) 27
9
5
C)f ( 2.5) local
minimum;
f (5.8) local maximum
1 3
ff(x)
(x)==3xx5
3 2
x2 x
B)f (3) local minimum;
Df (0) local minimum
10x + 17
p
4
2
ff(x)
(x)==4x
5 x 2xx + 10
Question #:
f (x) =
5x
x2 +3
f (x) = 7x
Classify all critical points of function f(x)
5
5 6x
4+x
f (x) =
7 sin x
8
>
< 3x if 0  x  1
f (x) = 3 sin2 (x)
f (x) = x 4 if 1 < x  5
>
:
6 x if 5 < x  7
f (x) = 4x2 (3 x)2
A) f (7) local
minimum; f (3.8) local maximum
f (x) = x4 x2
B) f (3) local
minimum; f (5) local maximum
f (x) = 13 x3 32 x2 10x + 17
C)
f ( =2.5)
f (5.8) local maximum
f (x)
4x4 local
2x2minimum;
+ 10
D) f (0) local minimum
f (x) =
Question #:
5 6x
4+x
8
>
< 3x
f (x) = x 4
>
:
6 5x
if 0  x  1
if 1 < x  5
if 5 < x  7
A) f (7) local minimum; f (3.8) local maximum
B) f (3) local minimum; f (5) local maximum
Find the intervals where function f(x) decreases
C) f ( 2.5) local minimum; f (5.8) local maximum
D) f (0) local minimum
A)( 1, 3) [ (0, 3)
C)(0, 3)
B)( 2, 2)
D)( 1, 3) [ (3, 1)
5