Questions: Suppose F(3)=-5, F(8)=8, and F'(x)=f(x). ∫ from 3 to 8 of f(x) dx=13

Transcript text: Suppose $F(3)=-5, F(8)=8$, and $F^{\prime}(x)=f(x)$. \[ \int_{3}^{8} f(x) d x=13 \]

Solution

Solution Steps

To solve this problem, we need to understand the relationship between the function $ F(x) $, its derivative $ f(x) $, and the definite integral of $ f(x) $ from 3 to 8. The Fundamental Theorem of Calculus tells us that the integral of $ f(x) $ from 3 to 8 is equal to $ F(8) - F(3) $. We can use this information to verify the given integral value.

Step 1: Given Information

We are provided with the following values:

$ F(3) = -5 $
$ F(8) = 8 $
The integral $ \int_{3}^{8} f(x) \, dx = 13 $

Step 2: Apply the Fundamental Theorem of Calculus

According to the Fundamental Theorem of Calculus, we have: \[ \int_{3}^{8} f(x) \, dx = F(8) - F(3) \] Substituting the known values: \[ F(8) - F(3) = 8 - (-5) = 8 + 5 = 13 \]

Step 3: Verify the Integral Value

We calculated $ F(8) - F(3) $ and found it to be $ 13 $, which matches the given integral value. Therefore, the relationship holds true.