Math / Calculus

QuestionSuppose that $F^{\prime}(x)=f(x)$ and $F(0)=3, F(2)=7$ . a. What is the area under $y=f(x)$ over $[0,2]$ if $f(x) \geq 0$ ? b. What is the graphical interpretation of $F(2)-F(0)$ if $f(x)$ takes on both positive and negative values?

Studdy Solution

STEP 1

1. $F'(x) = f(x)$ implies that $F(x)$ is an antiderivative of $f(x)$ .
2. The Fundamental Theorem of Calculus relates the definite integral of a function to its antiderivative.
3. The area under the curve $y = f(x)$ from $x = a$ to $x = b$ is given by the definite integral $\int_a^b f(x) \, dx$ .

STEP 2

1. Calculate the area under $y = f(x)$ over $[0,2]$ when $f(x) \geq 0$ .
2. Interpret $F(2) - F(0)$ graphically when $f(x)$ takes on both positive and negative values.

STEP 3

To find the area under $y = f(x)$ over $[0,2]$ when $f(x) \geq 0$ , we use the Fundamental Theorem of Calculus:
$\int_0^2 f(x) \, dx = F(2) - F(0)$
Given $F(0) = 3$ and $F(2) = 7$ , we substitute these values:
$\int_0^2 f(x) \, dx = 7 - 3 = 4$
Therefore, the area under $y = f(x)$ over $[0,2]$ is $\boxed{4}$ .

STEP 4

The graphical interpretation of $F(2) - F(0)$ when $f(x)$ takes on both positive and negative values is as follows:
- $F(2) - F(0)$ represents the net area between the curve $y = f(x)$ and the x-axis over the interval $[0,2]$ . - The net area is the total area above the x-axis minus the total area below the x-axis. - If $f(x)$ is positive, it contributes positively to the net area; if $f(x)$ is negative, it contributes negatively.
Thus, $F(2) - F(0)$ gives the net signed area under the curve $y = f(x)$ from $x = 0$ to $x = 2$ .
The area under $y = f(x)$ over $[0,2]$ is $\boxed{4}$ , and $F(2) - F(0)$ represents the net signed area under the curve when $f(x)$ has both positive and negative values.

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