Complete the following argument that sinx#x for any x in the interval (0,1): • For any x in (0, 1), the slope of the secant line through the points (0,0) and (x , sinx) is sinx-sin-'0 sinx since sin-0=0. x-0 • Therefore, by [ Select] for any such x, there is some number c in the interval (0, x) so that d sin dx sin [ Select ] d sin • Now, since 1 >1 for all x in (0, 1), we know that sin-x (x)= dx and since [ Select ] 1. VI-x- • So sinx [ Select] x, for any x in (0 , 1), and we are done.
Complete the following argument that sinx#x for any x in the interval (0,1): • For any x in (0, 1), the slope of the secant line through the points (0,0) and (x , sinx) is sinx-sin-'0 sinx since sin-0=0. x-0 • Therefore, by [ Select] for any such x, there is some number c in the interval (0, x) so that d sin dx sin [ Select ] d sin • Now, since 1 >1 for all x in (0, 1), we know that sin-x (x)= dx and since [ Select ] 1. VI-x- • So sinx [ Select] x, for any x in (0 , 1), and we are done.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Complete the following argument that sinx#x for any x in the interval (0,1):
sinx-sin-0
-1
sinx
X.
since sin¬0=0.
-1
• For any x in (0,1), the slope of the secant line through the points (0,0) and (x, sin'x) is
%D
X-0
d sin-
• Therefore, by [Select]
[ Select
for any
such x, there is some number c in the interval (0,x) so that
dx
sinx
-1
[ Select ]
sin¬!
d sin
• Now, since
1
(x)=
V1-x²
and since
>1 for all x in (0, 1), we know that
[ Select ]
V1.
dx
VI-x2
V1-x²
• So sinx
[ Select ]
for any x in (0,1), and we are done.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F254dd9ef-d4f1-406d-9d55-ac92a7c50632%2F2065d45f-5ef1-487f-9d23-1bbda5aed9bf%2Fqvd8zvw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Complete the following argument that sinx#x for any x in the interval (0,1):
sinx-sin-0
-1
sinx
X.
since sin¬0=0.
-1
• For any x in (0,1), the slope of the secant line through the points (0,0) and (x, sin'x) is
%D
X-0
d sin-
• Therefore, by [Select]
[ Select
for any
such x, there is some number c in the interval (0,x) so that
dx
sinx
-1
[ Select ]
sin¬!
d sin
• Now, since
1
(x)=
V1-x²
and since
>1 for all x in (0, 1), we know that
[ Select ]
V1.
dx
VI-x2
V1-x²
• So sinx
[ Select ]
for any x in (0,1), and we are done.
![V [ Select ]
the Intermediate Value Theorem
the Mean Value Theorem
Rolle's Theorem
the Extreme Value Theorem
some (other) Theorem
Complete the following argument that sinx#x for any x in the interval (0,1):
sin-x-sin-
sin-x
• For any x in (0 ,1), the slope of the secant line through th points (0 ,0) and (x , sin¬'x) is
since sin-0=0.
х-0
d sin-!
• Therefore, by [Select ]
for any such x, there is some number c in the interval (0, x) so that
dx
sin-!
[ Select ]
d sin-!
1
1
->1 for all x in (0,1) , we know that
sin-';
• Now, since
-(x)=
dx
[ Select ]
and since
1.
is greater than
is less than
• So sin-x
[v [ Select ]
x, for any x in (0,1), and we are done.
is equal to
is greater than
is less than
is equal to
V [ Select ]
c.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F254dd9ef-d4f1-406d-9d55-ac92a7c50632%2F2065d45f-5ef1-487f-9d23-1bbda5aed9bf%2F1e9nz3e_processed.jpeg&w=3840&q=75)
Transcribed Image Text:V [ Select ]
the Intermediate Value Theorem
the Mean Value Theorem
Rolle's Theorem
the Extreme Value Theorem
some (other) Theorem
Complete the following argument that sinx#x for any x in the interval (0,1):
sin-x-sin-
sin-x
• For any x in (0 ,1), the slope of the secant line through th points (0 ,0) and (x , sin¬'x) is
since sin-0=0.
х-0
d sin-!
• Therefore, by [Select ]
for any such x, there is some number c in the interval (0, x) so that
dx
sin-!
[ Select ]
d sin-!
1
1
->1 for all x in (0,1) , we know that
sin-';
• Now, since
-(x)=
dx
[ Select ]
and since
1.
is greater than
is less than
• So sin-x
[v [ Select ]
x, for any x in (0,1), and we are done.
is equal to
is greater than
is less than
is equal to
V [ Select ]
c.
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