Question

$x$

0

1

2

3

----

----

----

----

----

$f(x)$

5

2

3

6

$f(x)$

-3

1

3

4

the derivative of the function $f$ is continuous on the closed interval $0,4$. values of $f$ and $f$ for selected values of $x$ are given in the table above. if $int_{0}^{4} f(t)dt = 8$, then $f(4) =$a 0b 3c 5d 10

Explanation:

Step1: Apply Fundamental Theorem of Calculus

$$\int_{0}^{4} f'(t)dt = f(4) - f(0)$$

Step2: Substitute known values

From the table, $f(0)=5$, and $\int_{0}^{4} f'(t)dt=8$.
$$8 = f(4) - 5$$

Step3: Solve for $f(4)$

$$f(4) = 8 + 5$$

Answer:

E. 13