Q. 17 - 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): ${\int }_a^{0}f\left(x\right) dx=g\left(b\right)-g\left(a\right)$, where $g\text{'}\left(x\right)=f\left(x\right)$.

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

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

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

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 not equivalent to the fundamental theorem of calculus.

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

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

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

Consider the given statements,

$$\begin{gathered} {\int }_a^{0}f\left(x\right) dx=g\left(b\right)-g\left(a\right), \\ g\text{'}\left(x\right)=f\left(x\right) \end{gathered}$$

Then,

$$\begin{gathered} ={\int }_a^{b}f\left(x\right) dx \\ ={{\left[g\left(x\right)\right]}^b}_a \\ =g\left(b\right)-g\left(a\right) \end{gathered}$$

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,

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

This statement is false as $F\text{'}\left(x\right)=f\left(x\right)$ 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,

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

Also, $g\left(x\right)$ is an antiderivative of $h\left(x\right)$.

The statement is false as ${\int }_a^{b}h\left(x\right) dx=g\left(a\right)-g\left(b\right)$ cannot be true when $g\left(x\right)$ is an antiderivative of $h\left(x\right)$.

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

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

Consider the given statements,

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

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

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

Consider the given statements,

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

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

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