d. What do your results in (b) and (c) thus tell you about f'(x)? e. By emulating the steps taken above, use the limit definition of the derivative to argue convincingly that [cos(x)] = =sin(x).
d. What do your results in (b) and (c) thus tell you about f'(x)? e. By emulating the steps taken above, use the limit definition of the derivative to argue convincingly that [cos(x)] = =sin(x).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Plz complete it
![es
3. In this exercise, we explore how the limit definition of the derivative more
formally shows that (sin(x)] = cos(x). Letting f(x) = sin(x), note that the
limit definition of the derivative tells us that
a.
f'(x) = lim
h→0
b.
Recall the trigonometric identity for the sine of a sum of angles a and 3:
sin(a + 3) = sin(a) cos(3) + cos(a) sin(3). Use this identity and some
algebra to show that
f'(x) = lim
sin(a+h)- sin(x)
h
h→0.
sin(a) (cos(h)-1) + cos(x) sin(h)
Next, note that as h changes, a remains constant. Explain why it therefore
makes sense to say that
f'(x) = sin(x) - lim
h 0
lim
A-0
h
cos(h)-1
h
sin (h)
+ cos(a) lim
A-0 h
Finally, use small values of h to estimate the values of the two limits in (c):
cos(h) - 1
h
and lim sin(h)
A-40
h](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff413a13f-4ccc-4089-b9ed-1c21a38ff098%2F097802e1-eb5c-4281-9ba3-d8a09fdfd456%2Fym3c0z_processed.jpeg&w=3840&q=75)
Transcribed Image Text:es
3. In this exercise, we explore how the limit definition of the derivative more
formally shows that (sin(x)] = cos(x). Letting f(x) = sin(x), note that the
limit definition of the derivative tells us that
a.
f'(x) = lim
h→0
b.
Recall the trigonometric identity for the sine of a sum of angles a and 3:
sin(a + 3) = sin(a) cos(3) + cos(a) sin(3). Use this identity and some
algebra to show that
f'(x) = lim
sin(a+h)- sin(x)
h
h→0.
sin(a) (cos(h)-1) + cos(x) sin(h)
Next, note that as h changes, a remains constant. Explain why it therefore
makes sense to say that
f'(x) = sin(x) - lim
h 0
lim
A-0
h
cos(h)-1
h
sin (h)
+ cos(a) lim
A-0 h
Finally, use small values of h to estimate the values of the two limits in (c):
cos(h) - 1
h
and lim sin(h)
A-40
h
![d. What do your results in (b) and (c) thus tell you about f'(x)?
e. By emulating the steps taken above, use the limit definition of the
derivative to argue convincingly that [cos(x)] = =sin(x).
dz](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff413a13f-4ccc-4089-b9ed-1c21a38ff098%2F097802e1-eb5c-4281-9ba3-d8a09fdfd456%2Ffcw1wb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:d. What do your results in (b) and (c) thus tell you about f'(x)?
e. By emulating the steps taken above, use the limit definition of the
derivative to argue convincingly that [cos(x)] = =sin(x).
dz
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