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        riddles >> easy >> Calling all Calculus Genius'!!!!
        
(Message started by: killacal on May 24^th, 2006, 1:13pm)

Title: Calling all Calculus Genius'!!!!
Post by killacal on May 24^th, 2006, 1:13pm

Solve this one!!!

Graph y= 3+2cos (x/2 - 3.14/4)

I've been trying since yesterday!! I can't stand this stuff!

HELP!!! ???

Title: Re: Calling all Calculus Genius'!!!!
Post by SMQ on May 24^th, 2006, 2:01pm

Well, let's see: first off, we need to know some values of the cosine function by itself:

cos(0) = cos(2pi) = 1
cos(pi/6) = cos(11pi/6) = sqrt(3)/2
cos(pi/4) = cos(7pi/4) = sqrt(2)/2
cos(pi/3) = cos(5pi/3) = 1/2
cos(pi/2) = cos(3pi/2) = 0
cos(2pi/3) = cos(4pi/3) = -1/2
cos(3pi/4) = cos(5pi/4) = -sqrt(2)/2
cos(5pi/6) = cos(7pi/6) = -sqrt(3)/2
cos(pi) = -1

Hopefully with those you could graph y = cos(x) over 0 <= x <= 2pi. If x < 0 or x > 2pi, the pattern just repeats. And if you can do that, you can graph y = 3 + 2cos(x) just as easily -- put in the known points and connect the dots with a smooth curve.

Now, for the problem at hand, we need to find some points where x/2 - pi/4 is one of our known values, right? So for any of our known points, p, we'd like to find the value of x so that x/2 - pi/4 = p. A little algebra, and we have x = 2(p + pi/4). Looking at each of our known points, we get

p = 0:
x = 2(p+pi/4) = 2(0+pi/4) = 2(pi/4) = pi/2
y = 3+2cos(x/2-pi/4) = 3+2cos(p) = 3+2cos(0) = 3+2(1) = 5

p = pi/6:
x = 2(p+pi/4) = 2(pi/6+pi/4) = 2(5pi/12) = 5pi/6
y = 3+2cos(x/2-pi/4) = 3+2cos(p) = 3+2cos(pi/6) = 3+2(sqrt(3)/2) = 3+sqrt(3)

...and so on. Plot the points and connect the dots.

--SMQ

Title: Re: Calling all Calculus Genius'!!!!
Post by killacal on May 24^th, 2006, 2:14pm

WOW!! Thanks SMQ!

You really broke it down. I'm going to connect those dots, see what I come up with and get back at you.

I appreciate the help to the fullest. Bless your heart!! :'(

on 05/24/06 at 14:01:50, SMQ wrote:

Title: Re: Calling all Calculus Genius'!!!!
Post by rmsgrey on May 24^th, 2006, 5:05pm

on 05/24/06 at 13:13:13, killacal wrote:

Solve this one!!!

Graph y= 3+2cos (x/2 - 3.14/4)

I've been trying since yesterday!! I can't stand this stuff!

HELP!!! ???

Another approach (rather than SMQ's perfectly viable find-and-plot-points approach) is to look at the function as made up of a simpler function with transformations applied to it...

Breaking it down, I'd look at it starting from cos(x):

cos(x)

-> cos (x-Pi/2) by shifting the cos graph by Pi/2 to the right.

-> cos [ (x-Pi/2) /2 ] which is effectively cos(u) -> cos (u/2) (where u is a linear function of x) so stretches the graph by a factor of 2 parallel to the x axis.

-> 2{cos [ (x-Pi/2) /2 ]} which is just stretching by 2 parallel to the y-axis.

-> 3+ (2{cos [ (x-Pi/2) /2 ]}) and finally shift everything up by 3.

If you sketch each graph in turn, you'll see that the transformations I describe move you from one to the next...