Science:Math Exam Resources/Courses/MATH101/April 2005/Question 01 (c)

MATH101 April 2005

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Question 01 (c)
If $\displaystyle F(x)=\int _{0}^{x^{2}}f(t)\,dt,$ where $\displaystyle f$ is a function satisfying $\displaystyle f(4)=1$, compute $\displaystyle F'(2).$

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Apply the fundmental theorem of calculus to the integral $\quad \int _{0}^{x^{2}}f(t)\,dt.$

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. By the fundamental theorem of calculus, we know that there exists a function g(x) such that $F(x)=\int _{0}^{x^{2}}f(t)\,dt=g(x^{2})-g(0)$ where g(x) is the anti-derivative of the function f(x) (i.e: g'(x) = f(x)). Taking the derivative of F(x) gives $F'(x)={\frac {d}{dx}}\left(g(x^{2})-g(0)\right)=g'(x^{2})\cdot 2x$ by the chain rule. Using the anti-derivative property of g, $F'(x)=f(x^{2})\cdot 2x.$ Plugging in x = 2 and using the fact that f(4) = 1 gives us our answer $F'(2)=f(4)\cdot 4=4$

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