FUNDAMENTAL THEOREM OF CALCULUS

Problems:

1. State ”Fundamental Theorem of Calculus”.

2. (1) Suppose that f is a continuous function and g is a differentiable function. Let

H (x) = ∫_{a}^{g(x)} f (t)dt.

Find H′(x).

(2) Suppose that f is a continuous function and g_{1}, g_{2} are differentiable function. Let

K(x) = ∫_{g_{1}(x)}^{g_{2}(x)} f(t)dt.

Find K′(x).

3. f(x) = \{ \begin{matrix}x + 1  if x < 1 \\ 3  if x = 1 \\ x^{2}  if 1 < x \end{matrix}

Let

F (x) = ∫_{−1}^{x} f(t)dt

Find F (x) and F′(x).

4. f(x) = \{ \begin{matrix}x + 1  if x < 1 \\ 3  if x = 1 \\ x^{2} + 1 if 1 < x \end{matrix}

Let

F (x) = ∫_{−1}^{x} f(t)dt

Find F (x) and F′(x).