9. consider the function y=e^-x sin x where pie < x < pie
find dy/dx which is e^-x(cos x - sin x)
show that at stationary points, tan x = 1. - I have done this part
I don't understand this part:
How do I do this: determine the co-ordinates of the stationary points correct to 2 dp.
Could you tell me how to start off this question?
You have show that the condition for stationary points is that tanx = 1
In the next part we actually have to find those points,
that is, solve
tanx = 1
x = 45° or x = 225°
or
x = π/4 or x = 5π/4 radians
when x = π/4
y = e^-π/4 sin π/4 = .3224
when x = 5π/4
y = e^-5π/4 sin 5π/4 = -.011139
Looking at your domain, I think you made a typo, and I will assume you meant
-π < x < π
so the second point above lies outside the domain,
but there is another solution of
x = -135° or -3π/4
if x = -3π/4
y = e^-3π/4 sin 3π/4 = -.067
Within your domain, the stationary points are
(π/4, .3224) and (-3π/4, -.067)
To find the coordinates of the stationary points, we need to set the derivative of the function to zero and solve for x.
In this case, we have the derivative dy/dx = e^(-x)(cos(x) - sin(x)).
Setting this derivative equal to zero, we get:
e^(-x)(cos(x) - sin(x)) = 0
Since e^(-x) is never equal to zero, we can simplify the equation to:
cos(x) - sin(x) = 0
Next, we need to solve this equation for values of x that satisfy it.
To do this, we can transform the equation cos(x) - sin(x) = 0 into an equation involving tan(x). Recall that tan(x) = sin(x) / cos(x).
Rearranging the equation, we have sin(x) = cos(x). Dividing both sides by cos(x), we get:
tan(x) = 1
So, at stationary points, tan(x) = 1.
Now, to determine the coordinates of the stationary points, we need to find the values of x that satisfy tan(x) = 1 within the given interval (π < x < π).
To solve this equation, we can use the inverse function of tan, which is arctan:
x = arctan(1)
Using a calculator or trigonometric tables, we find that arctan(1) = π/4.
Therefore, the value of x at the stationary point is π/4.
Now, to find the y-coordinate at this point, we substitute the value of x back into the original function y = e^(-x)sin(x):
y = e^(-π/4) * sin(π/4)
Again, you can use a calculator to find the approximate value.