suppose
x
S f(t) dt= x^2 - 2x + 1. Find f(x).
1
S = integral and 1 = lower level, x = upper
i don't understand what i'm supposed to find/what to do
and i didn't make any typos.
To find the function f(x), we need to solve the equation:
∫[1tox] f(t) dt = x^2 - 2x + 1
Start by taking the derivative with respect to x on both sides of the equation:
d/dx [∫[1tox] f(t) dt] = d/dx (x^2 - 2x + 1)
Now, let's use the Fundamental Theorem of Calculus. According to this theorem, if F(x) is the antiderivative of f(x), then:
d/dx [∫[a to b] f(t) dt] = F(b) - F(a)
This means that the derivative of an integral with respect to x is just the function itself (in this case, f(x)). So, we can rewrite the equation as follows:
f(x) = d/dx (x^2 - 2x + 1)
Now, let's differentiate the right-hand side of the equation. Apply the power rule and the constant rule:
f(x) = 2x - 2
Therefore, the function f(x) is f(x) = 2x - 2.