Calculus: Early Transcendentals, Chapter 4, 4.2, Section 4.2, Problem 14

Mean Value Theorem states a function f(x) that the satisfies the following hypotheses:
1. a function f(x) that is continuous on the closed interval [a,b]
2. differentiable on the open interval (a,b)

Then there is a number “c” such that a ltcltb and
f'(c) =(f(b)-f(a))/(b-a)
or f(b) – f(a) = f’(c) (b-a).
Mean Value Theorem indicates that at least one number “c” will exists within the said function. To be able to solve the value of “c”, we consider the special case of Mean Value theorem where we assumed f(a)= f(b) (Rolle’s Theorem).

A. For the given function f(x) = e ^(-x) with closed interval [0,2], we solve first for the endpoints.
Plug-in x= 0 in f(x)=e^(-x) :
f(0) = e^(-0)
=e^(0)
=1
First endpoint: (a, f(a)) = (0,1)

Plug-in x= 2 in f(x)=e^(-x):
f(0) = e^(-2)
= (1)/(e^ (2)) or 0.135335
then (b, f(b)) = (2,(1)/(e^(2)) )
Second endpoint: (b, f(b)) = (2,(1)/(e^(2)) )
Law of Exponent: x^(-n) = (1)/(x^(n))
B. Solve for the slope of the secant line using the formula:
f'(c) = (f(b)-f(a))/(b-a)
Plug-in the two endpoints:
f'(c) = ((1)/(e^(2))-1)/(2-0)
f'(c) = ((1)/(e^(2))-1)/(2)
f'(c) = (1)/(2e^(2)) - (1)/(2) or -0.432332358
C. Solve for the derivative function f'(x).
f'(x) = e^(-x) * (-1 dx)
= -e^(-x) dx
D. Solve for (c, f(c))
Recall the slope of the tangent line = f'(c).
Equate the answers from part B and C:
((1)/(e^(2))-1)/(2) = -e^(-x)
((1)/(e^(2))-1)/(2)= -(1)/(e^(x))
Multiply both sides by -1:
(((1)/(e^(2))-1)/(2))*(-1) = ((-1)/(e^(x))*(-1)
(1-(1)/(e^(2)))/(2)= (1)/(e^(x))
Cross-multiply to isolate e^(x) :
e^(x) = (2)/(1-(1)/(e^(2)))
Take LN from both sides:
ln(e^(x)) = ln ((2)/(1-(1)/(e^(2))))
x = ln(2) - ln(1-(1)/(e^(2)))
x = ln(2) - ln((e^(2)-(1))/(e^(2)))
x = ln(2) - (ln(e^(2)-1)-ln(e^(2)))
x = ln(2) - ln(e^(2)-1) + ln(e^(2))
x = ln(2) - ln(e^(2)-1) + ln(e^(2))
x = ln((2)/(e^(2)-1)) + 2
or x= 0.8385606384 rounded off to c= x =0.8386.
Solve for f(c):
Plug-in x=0.8386 in f(x) = e^(-x)
f(0.8386) = e^(-0.8386)
= 0.4323
Then (c, f(c)) =( 0.8386 , 0.4323)
E. Find tangent line equation.
slope of tangent line = f'(c) =-0.4323
point (c, f(c)) =( 0.8386 , 0.4323)
It shows that secant line and tangent line are parallel.
Using the formula: y=mx+b to solve for b.
0.4323 = (-0.4323) *(0.8386) +b
0.4323 =-0.3625 +b
0.4323 +0.3625 = -0.3625 +0.3625 +b
b=0.7948
Tangent line equation: y = -0.4323x +0.7948
Please see the attached file for the graph.
Blue line is the secant line using the two endpoints: (a, f(a)) and (b, f(b)).
Green line is the tangent line using line equation: y = -0.4323x +0.7948.
Red curve is the graph of f(x) = e^(-x) .

Comments

Post a Comment

Popular posts from this blog

How does Bilbo show leadership and courage in The Hobbit?

Bilbo’s increase in confidence and courage is the defining aspect of his character arc. At the beginning of the novel, he is shut away content in his Hobbit hole, happy to remain separate and coddled from the rest of the world. In the movie, Bilbo chooses to join the dwarves on their quest out of a sense for adventure. In the book, however, Bilbo wants nothing to do with them and only joins because he had been duped into signing a contract. Despite being on the adventure somewhat against his will, Bilbo still proves himself time and again, saving the dwarves from the spiders, the trolls, and a number of other dangers they encounter along the way. Bilbo’s character arc has even been cited as a sort of Christian metaphor for the growth of the soul, though how much Tolkien intended on this front is up for debate. From a thematic standpoint, the ring of invisibility also plays into Bilbo’s brand of bravery. If Hobbits are the overlooked creatures of Middle Earth, then it is noteworthy that...
Read more

In “Goodbye to All That,” Joan Didion writes that the “lesson” of her story is that “it is distinctly possible to remain too long at the fair.” What does she mean? How does the final section of the essay portray how she came to this understanding, her feelings about it, and the consequences of it?

In this classic essay, Didion describes moving to New York City from her native Sacramento at the age of twenty-one, and, as she freely admits, a somewhat naive twenty-one, her mind, "programmed by all the movies I had ever seen and all the songs I had ever heard about New York," with the typical feeling, at that age, that, "nothing like this...has ever happened to anyone before." Intending to stay for only six months, she falls in love with the city. As she says, "I do not mean 'love' in any colloquial way, I mean that I was in love with the city the way you love the first person who ever touches you and never love anyone quite that way again." This initial love led her to prolong her residency, since, "I still believed in possibilities then, still had the sense, so peculiar to New York, that something extraordinary would happen any minute, any day, any month." But as the next years roll past, and the "new faces" she encounters...
Read more

Why does the poet say "all the men and women merely players"?

It's one of the most famous lines in literature: All the world's a stage,And all the men and women merely players . . . This famous saying comes from Shakespeare's play As You Like It, and it's spoken by Jacques. What he's saying here is that simply going about our daily lives is a type of performance. Just like actors on a stage, we all have roles to perform: Daughter. Student. Sister. Captain of the soccer team. The fact that Shakespeare writes that men and women are merely players suggests that the roles are preordained and that a person's choice in being a mother, for example, or a doctor or starting point guard for the Brooklyn Nets is out of their hands. What roles a person performs over the course of their lifetime is up to fate. The notion that "all the world's a stage" has entered pop culture, but within the context of the play, this idea does not sit well with Jacques. He's a pessimist, and the idea that everything is up to fate makes...
Read more