Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Answer the following True or False. $\newline$ If $\int_{a}^{b} f(x) \, dx$ converges and a < c < b , then $\int_{a}^{c} f(x) \, dx$ also converges. $\newline$ True $\newline$ False $\newline$

View step-by-step help

View step-by-step help

Properties of matrices

Full solution

Q. Answer the following True or False.

\newline

If

\int_{a}^{b} f(x) \, dx

converges and

a < c < b

, then

\int_{a}^{c} f(x) \, dx

also converges.

\newline

True

\newline

False

\newline

Consider Integral Convergence: Let's consider the given integral from $a$ to $b$ of $f(x)$ . If this integral converges, it means that the area under the curve of $f(x)$ from $a$ to $b$ is finite.
Subset Interval Analysis: Now, let's consider the integral from $a$ to $c$ of $f(x)$ , where a < c < b. Since $c$ is between $a$ and $b$ , the interval from $a$ to $c$ is a subset of the interval from $a$ to $b$ .
No Singularities or Divergence: The convergence of the integral from $a$ to $b$ of $f(x)$ implies that the function $f(x)$ does not have any singularities or behaviors that would cause the integral to diverge in the interval $[a, b]$ .
Convergence Implication: Since the interval from $a$ to $c$ is contained within the interval from $a$ to $b$ , and we know that there are no issues in the larger interval that would prevent convergence, the integral from $a$ to $c$ must also converge.
True Statement Validation: Therefore, the statement is true: if the integral from $a$ to $b$ of $f(x)$ converges, then the integral from $a$ to $c$ of $f(x)$ also converges for any $c$ such that a < c < b.

More problems from Properties of matrices

Question

Is this an identity matrix?

\newline

\begin{bmatrix} 0 & 1 1 & 0 \end{bmatrix}

\newline

\newline

\text{yes}

\newline

\text{no}

Posted 5 months ago

Question

Consider this matrix transformation:

\newline

\left[\begin{array}{ll} -3 & 4 \\ & \\ -3 & 2 \end{array}\right]

\newline

What is the image of

\left[\begin{array}{c}-2 \\ 2\end{array}\right]

under this transformation?

Posted 4 months ago

Question

Consider this matrix transformation:

\newline

\left[\begin{array}{cc} 2 & 3 \\ & \\ -3 & -1 \end{array}\right]

\newline

What is the image of

\left[\begin{array}{l}4 \\ 1\end{array}\right]

under this transformation?

Posted 9 months ago

Question

Consider this matrix transformation:

\newline

\left[\begin{array}{cc} -4 & -2 \\ 2 & 3 \end{array}\right]

\newline

What is the image of

\left[\begin{array}{c}3 \\ -1\end{array}\right]

under this transformation?

Posted 3 months ago

Question

Answer the following True or False.

\newline

\frac{d}{dx}3x^{\frac{1}{3}}=x^{-\frac{2}{3}}

, the fundamental theorem of calculus tells us that

\int_{-1}^{1}x^{-\frac{2}{3}}dx=3(1^{\frac{1}{3}})-3(-1)^{\frac{1}{3}}=6

\newline

\newline

Posted 8 months ago

Question

Answer the following True or False.

\newline

\int 3^{x}\,dx = \frac{3^{x} + C}{\ln 3}

\newline

\newline

Posted 8 months ago

Question

Answer the following True or False.

\newline

\int \frac{1}{\sec x}\,dx = \ln(\sec x) + C

\newline

\newline

Posted 8 months ago

Question

Answer the following True or False.

\newline

a

b

are both positive real numbers. Then

\int_{a}^{b}\frac{1}{x^{2}}dx=\int_{-b}^{-a}\frac{1}{x^{2}}dx.

\newline

\newline

\newline

Posted 8 months ago

Question

Answer the following True or False.

\newline

\int_{-a}^{a}(3x^{3}+9x)dx=0

\newline

\newline

Posted 8 months ago

Question

x

makes the equation below true?

\newline

10 x+8=78

\newline

4

\newline

7

\newline

8

\newline

9

Posted 8 months ago

Related topics

Algebra - Order of Operations Algebra - Distributive Property `X` and `Y` Axes Geometry - Scalene Triangle Common Multiple Geometry - Quadrant