Q. 16 - Calculus

Question

Determine whether or not each statement that follows is equivalent to the Fundamental Theorem of Calculus. Assume that all functions here are integrable.

Part (a): If $f\text{'}\left(x\right)=F\left(x\right)$, then ${\int }_a^{b}F\left(x\right) dx={{\left[f\left(x\right)\right]}^b}_a$.

Part (b): ${\int }_a^{b}G\left(x\right) dx={{\left[G\text{'}\left(x\right)\right]}^b}_a$.

Part (c): If $h\left(x\right)$ is the derivative of $g\left(x\right)$, then ${\int }_a^{b}h\left(x\right) dx=g\left(b\right)-g\left(a\right)$.

Part (d): ${{\left[\int w\text{'}\left(x\right) dx\right]}^b}_a={\int }_a^{b}w\left(x\right) dx$.

Part (e): ${{\left[\int s\text{'}\left(x\right) dx\right]}^b}_a={{\left[s\text{'}\text{'}\left(x\right)\right]}^b}_a$.

Step-by-Step Solution

Answer

Part (a): The statement is equivalent to the fundamental theorem of calculus.

Part (b): The statement is not equivalent to the fundamental theorem of calculus. 

Part (c): The statement is equivalent to the fundamental theorem of calculus. 

Part (d): The statement is not equivalent to the fundamental theorem of calculus. 

Part (e): The statement is not equivalent to the fundamental theorem of calculus. 

Part (a) Step 1. Determine whether equivalent to the Fundamental Theorem of Calculus.

Consider the given statements,

$$\begin{gathered} {\int }_a^{b}F\left(x\right) dx={{\left[f\left(x\right)\right]}^b}_a, \\ f\text{'}\left(x\right)=F\left(x\right) \end{gathered}$$

Hence, the statement is equivalent to the fundamental theorem of calculus.

Part (b) Step 1. Determine whether equivalent to the Fundamental Theorem of Calculus.

Consider the given statements,

${\int }_a^{b}G\left(x\right) dx={{\left[G\text{'}\left(x\right)\right]}^b}_a$

This statement is false as $G\text{'}\left(x\right)=G\left(x\right)$ is not possible.

Hence, the statement is not equivalent to the fundamental theorem of calculus.

Part (c) Step 1. Determine whether equivalent to the Fundamental Theorem of Calculus.

Consider the given statements,

${\int }_a^{b}h\left(x\right) dx=g\left(b\right)-g\left(a\right)$

Also, it has been said $g\text{'}\left(x\right)=h\left(x\right)$.

Hence, the statement is equivalent to the fundamental theorem of calculus.

Part (d) Step 1. Determine whether equivalent to the Fundamental Theorem of Calculus.

Consider the given statements,

${{\left[\int w\text{'}\left(x\right) dx\right]}^b}_a={\int }_a^{b}w\left(x\right) dx$

The statement is false.

For the statement to be true, it should've been $\left[{\int }_a^{b}w\text{'}\left(x\right) dx\right]={{\left[w\left(x\right)\right]}^b}_a$.

Hence, the statement is not equivalent to the fundamental theorem of calculus.

Part (e) Step 1. Determine whether equivalent to the Fundamental Theorem of Calculus.

Consider the given statements,

${{\left[\int s\text{'}\left(x\right) dx\right]}^b}_a={{\left[s\text{'}\text{'}\left(x\right)\right]}^b}_a$

The statement is false.

For the statement to be true, it should've been ${{\left[{\int }_a^{b}s\text{'}\left(x\right) dx\right]}^b}_a={{\left[s\left(x\right)\right]}^b}_a$.

Hence, the statement is not equivalent to the fundamental theorem of calculus.