help with calculus problem plz


Recommended Posts

19) the function defined by f(x)=(3)^(1/2)*cosx+3*sin(x) has an amplitude of

a) 3-(3)^(1/2)

b) (3)^(1/2)

c) 2(3)^(1/2)

d) 3+(3)^(1/2)

e) 3*(3)^(1/2)

i need to know how to do this problem, but i can't seem to figure it out. please show work. this is a calculus problem.

Link to comment
Share on other sites

The answer is c)

I used the old "brute force" of a spreadsheet. :shifty:

The peak occurs somewhere around 60 degrees, where cos=.5 and sin=sqrt(3)/2.

Link to comment
Share on other sites

f'(x) = 3cosx-sr(3)sinx

3cosx-sr(3)sinx=0

x=4pi/3, 7pi/3, 10pi/3

(pi/3, 4pi/3) (4pi/3, 7pi/3) (7pi/3, 10pi/3) (10pi/3, ...)

n 2pi/3 5pi/3 8pi/3 11pi/3

f'x -3 3 -3 3

min (4pi/3, -2sr(3))

max (7pi/3, 2sr(3))

maybe?

Link to comment
Share on other sites

I plotted it and the amplitude is 3.464. so the answer would be c).

And here is how you do it by hand (I used Matlab). You find the derivative, you set it to zero (to find the max of the function). You then solve for x and you put the x you found back in your original equation to find y:

>> dsolve('Dy=3^(1/2)*cos(x)+3*sin(x)','x')

ans =

3^(1/2)*sin(x)-3*cos(x)+C1 #Don't mind the C1

>> S = solve('3^(1/2)*sin(x)-3*cos(x)=0')

S =

1/3*pi

>> 3^(1/2)*cos(1/3*pi)+3*sin(1/3*pi)

ans =

3.4641

So the answer is confirmed :)

Link to comment
Share on other sites

phkhoury, what's the c1 for? there are no integration constants when taking derivatives. :)

3^1/2*cos x + 3sinx

taking derivatives:

-3^1/2*sin x + 3cos x = 0

3cos x = 3^1/2*sin x

3/3^1/2 = tan x

x = pi/3

and, as everyone said, put that back in to find the amplitude. :happy:

Link to comment
Share on other sites

mm..I'm still in high school.. I'm a senior and am in AP Calculus.. anyways, I'm pretty good at math. Freshman year, last year, and this year I have top grade in the class and I don't even try very hard. I'm terrible at English though.

Link to comment
Share on other sites

mm..I'm still in high school.. I'm a senior and am in AP Calculus.. anyways, I'm pretty good at math. Freshman year, last year, and this year I have top grade in the class and I don't even try very hard. I'm terrible at English though.

Wait until you go to college....its a lot different... ;)

Link to comment
Share on other sites

Wait until you go to college....its a lot different... ;)

Yeah, you'll only get a B for not trying real hard instead of top grade in your class.

At least that's what happened for me. :ninja:

Link to comment
Share on other sites

i'm terrible at english and good at math too, but i still dropped calc... the teacher is a pain in the ass. calc problems take me on average 3 minutes (estimating... sometimes it's alot more than that) and on average, this teacher assigns 30 problems a night. i seriously don't have time for that sometimes... i have a job, and my parents don't understand the part about how if i work 4.5 hours a night, 2 days a week, i can't do hw on those nights. so they won't let me quit the job. *sigh*

Link to comment
Share on other sites

There are two ways you can do this. The first was outlined by Dreamz above/first page of using derivitives.

The second way is to use trigonometry.

The equation you gave us:

f(x)= 3^(1/2)8cos(x) + 3*sin(x)

R*Cos(a-b)

Where "a", "R" and "b" are constants

R*Cos(a-b) = R(cos(a)*cos(b) + sin(a)*sin(b))

=R*cos(a)*cos(b) + R*Sin(a)*sin(b)

Comparing co-efficients:

i.e. R*cos(a)*cos(b) + R*Sin(a)*sin(b) = 3^(1/2)8cos(x) + 3*sin(x)

Let the "x"s be "a"s.

Therefore: R*cos(b) = 3^(1/2)

R*sin(b) = 3

R*Sin(b) / R*Cos(b) = 3/(3^(1/2))

Is: tan(b) = 3/(3^(1/2))

Draw yourself a triangle as you know that the opp. is 3 and the adj: is 3^(1/2)

Using Pythagoras's Therom: We find the hyp. is if 12^(1/2):

= (3^(1/2)) * (4^(1/2))

= 2 * (3^(1/2))

Earlier on we found that:

R * sin(b) = 3

Using the triangle:

Sin(b) = 3 / (2 * (3^(1/2))

Therefore: R * (3 / (2 * (3^(1/2))) = 3

R = 3 / (3 / (2 * (3^(1/2)))

R = 2 * (3^(1/2))

Therefore:

3^(1/2)8cos(x) + 3*sin(x) = (2 * (3^(1/2))) * cos(x-b)

We can now see that the amplitude is of 2 * (3^(1/2)) which is c) :)

It is hard to see here though.

Link to comment
Share on other sites

This topic is now closed to further replies.
  • Recently Browsing   0 members

    • No registered users viewing this page.