Find f(4) if S (0 to x) dt=xcospix.
Take the derivative of each side:
I assume the left side is written incorrectly, you really have the integral of f(x) dt.
f(x) = d/dx (x cospi x)
= cosPIx - PIxsinPIx
Now, what is f(4)?
how did you get cosPIx - PIxsinPIx ?
d/dx (x cosPIx)= cosPIx * d/dx x + x d/dx (cosPIx)
= cosPIx + x(-PI sin pi x)
check my thinking.
Why would you take the derivative? The interval is from 0 to x. How does that contribute into solving the equation. I get -1 when I plug 4 into cosPIx -PIxsinPIx.
To find the value of f(4), we need to differentiate the given expression and then substitute x=4 into the derivative.
Let's start by differentiating the expression S(0 to x) dt = x*cos(pi*x) with respect to x.
Using the Fundamental Theorem of Calculus, the derivative of the definite integral gives us the integrand evaluated at the upper limit, x. So, differentiating both sides, we have:
d/dx [S(0 to x) dt] = d/dx [x*cos(pi*x)]
Now, let's differentiate the left side of the equation. The derivative of an integral with respect to its upper limit is simply the function being integrated. Thus, the left side can be rewritten as f(x):
f(x) = d/dx [x*cos(pi*x)]
Next, we apply the product rule to differentiate the right side. The product rule states that if we have a function u(x) multiplied by v(x), then the derivative is given by:
d/dx [u(x)*v(x)] = u'(x)*v(x) + u(x)*v'(x)
In this case, u(x) = x and v(x) = cos(pi*x), so we have:
f(x) = d/dx [x] * cos(pi*x) + x * d/dx [cos(pi*x)]
Applying the derivatives, we get:
f(x) = 1 * cos(pi*x) + x * (-sin(pi*x)) * pi
Simplifying further, we have:
f(x) = cos(pi*x) - pi*x*sin(pi*x)
Now we can find f(4) by substituting x=4 into the expression for f(x):
f(4) = cos(pi*4) - pi*4*sin(pi*4)
Evaluating this expression gives us the value of f(4).