Trig plotting and direct substitution

Really stuck with this question:

The suspension in a car acts like a

damped harmonic oscillator, that is, the oscillations in

the suspension rapidly die down with time. A model for this includes both exponential

and trigonometric functions. Suppose the displacement in a car's suspension is given by

s(t)=e^{-t/2}cos(2t)

(i) Sketch the displacement of the suspension for 0 >=t>=2

and describe its behaviour (>= meaning greater than or equal to)

in a few words.

(ii) Show by direct substitution that the displacement satisfes the differential equation

4^{d^2s/dt^2}+4^{ds/dt}+17s=0

I am so lost on what to do, some suggestions whould be wonderful!

Nettie.L

Re: Trig plotting and direct substitution

Both of your questions involve taking the first and second derivative of your function.

In (i), the first derivative is useful for finding critical points and seeing where the function is increasing or decreasing. The second derivative is used to see where the function is concave up or concave down. Your teacher should have reviewed a list of steps for drawing graphs this way.

In (ii), it is as it says, direct substitution. d^2s/dt^2 is the second derivative of s, ds/dt is the first derivative of s, and s is s. Plug in on the left hand side and see if it equals 0

done. :)

Re: Trig plotting and direct substitution

Thanks for that.

With (i), is it then after I take the derivative that I figure out the amplitude, period, etc.? Do I happen to use the formula y=a sin(bx+c)+d at all?

Re: Trig plotting and direct substitution

I'm not sure if you need the formula y = a sin(bx+c) + d at all, but it should be noted that if you do know what the base function looks like, you can apply a transformation on that graph to get the desired graph of a particular function.

For example, if you know the amplitude, period, etc, of sin(x), we can derive the amplitude, period, etc, of a sin(bx+c) + d using the theory of transformations, by noting that

to transform sin(x) -> a sin(bx+c) + d we have

1) A vertical stretch by a factor of a

2) A horizontal stretch by a factor of 1/b

3) A vertical displacement by a factor of d

4) A horizontal displacement by a factor of -c/b

(1 & 2 done first, 3&4 done second)

Re: Trig plotting and direct substitution

By base function do you just mean something like s(t)=sin x ?

Re: Trig plotting and direct substitution