Answer to Question #278274 in Calculus for Sara

Question

Evaluate F(x)=∫ √t

2 + 9

x

4

.

a) F(4) b) F’(4) c) F’’(4)

Answer 1

a)

$$F(4)=\displaystyle\int_{4}^{4}\sqrt{t^2+9}dt=0$$

b)

By the Fundamental Theorem of Calculus (part 1)

$$F'(x)=\sqrt{x^2+9}$$

$$F'(4)=\sqrt{4^2+9}=5$$

c)

$$F''(x)=\dfrac{2x}{2\sqrt{x^2+9}}=\dfrac{x}{\sqrt{x^2+9}}$$

$$F''(4)=\dfrac{4}{\sqrt{4^2+9}}=\dfrac{4}{5}$$