How do you use the fundamental theorem of calculus to find F'(x) given #F(x)=int t^2sqrt(1-t^3)dt# from [0,x]?
1 Answer
Mar 3, 2017
# F'(x) = x^2sqrt(1-x^3) #
Explanation:
If asked to find the derivative of an integral using the fundamental theorem of Calculus, you should not evaluate the integral
The Fundamental Theorem of Calculus tells us that:
# d/dx \ int_a^x \ f(t) \ dt = f(x) #
(ie the derivative of an integral gives us the original function back).
We are asked to find:
# F'(x) = d/dx F(x) #
# " " = d/dx \ int_0^x \ t^2sqrt(1-t^3) \ dt #
# " " = f(x)# , where#f(t)=t^2sqrt(1-t^3) \ \ \ # (using FTOC)
# " " = x^2sqrt(1-x^3) #
