Q. 10 - Calculus

Question

Suppose f  is a function whose derivative f'  is given by the graph shown next. Use the given value of f, an area approximation, and the Fundamental Theorem to approximate the requested value.

Given that $f\left(0\right)=-1$, approximate $f\left(2\right)$.

Step-by-Step Solution

Answer

The approximated value is $-12.57$.

Step 1. Given information.

Consider the given question,

$f\left(0\right)=-1$

Step 2. Find the approximate value.

There are approximately $47$ boxes of $\frac{1}{4}$square unit each from $0$ to $2$.

So, area covered by the boxes from $0$ to data-custom-editor="chemistry" $2$,

$$\begin{gathered} \approx \frac{47}{4} \\ \approx 11.75 \end{gathered}$$

The value can be approximated,

$$\begin{gathered} {\int }_0^{2}f\text{'}\left(x\right)=f\left(0\right)-f\left(2\right) \\ 11.75=-1-f\left(2\right) \\ f\left(2\right)\approx -12.57 \end{gathered}$$