Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

${:[f^(')(x)=-7e^(x)" and "],[f(5)=24-7e^(5).],[f(0)=◻]:}$

$\begin{array}{l}f^{\prime}(x)=-7 e^{x} \text { and } f(5)=24-7 e^{5} . \\ f(0)=\square\end{array}$

View step-by-step help

View step-by-step help

Power rule with rational exponents

Full solution

Q.

\begin{array}{l}f^{\prime}(x)=-7 e^{x} \text { and } f(5)=24-7 e^{5} . \\ f(0)=\square\end{array}

Derivative of $f(x)$ : We have $f'(x) = -7e^x$ , which is the derivative of $f(x)$ . To find $f(0)$ , we need to integrate $f'(x)$ .
Integration of $f'(x)$ : The integral of $f'(x) = -7e^x$ is $f(x) = -7e^x + C$ , where $C$ is the constant of integration.
Finding Constant of Integration: We know $f(5) = 24 - 7e^5$ . Let's plug $x = 5$ into the integrated function to find $C$ .
Substitute $x = 5$ : $-7e^5 + C = 24 - 7e^5$ .
Solving for C: Solving for C, we get $C = 24$ .
Final Function $f(x)$ : Now we have the function $f(x) = -7e^x + 24$ .
Finding $f(0)$ : To find $f(0)$ , we plug in $x = 0$ into $f(x)$ .
Substitute $x = 0$ : $f(0) = -7e^0 + 24$ .
Calculate $f(0)$ : Since $e^0 = 1$ , $f(0) = -7(1) + 24$ .
Result: $f(0) = -7 + 24$ .
Result: $f(0) = -7 + 24$ . $f(0) = 17$ .

More problems from Power rule with rational exponents

Question

h(x)=\log _{2}(x)

\newline

Can we use the mean value theorem to say the equation

h^{\prime}(x)=1

has a solution where

1<x<2

\newline

1

\newline

(A) No, since the function is not differentiable on that interval.

\newline

(B) No, since the average rate of change of

h

over the interval

1 \leq x \leq 2

1

\newline

(C) Yes, both conditions for using the mean value theorem have been met.

Posted 3 months ago

Question

What is the sign of

39 \frac{11}{13}+\left(-41 \frac{1}{13}\right) \text { ? }

\newline

1

\newline

\newline

\newline

(C) Neither positive nor negative-the sum is zero.

Posted 3 months ago

Question

What is the sign of

-1042+1042 ?

\newline

1

\newline

\newline

\newline

(C) Neither positive nor negative-the sum is zero.

Posted 3 months ago

Question

What is the sign of

-18 \frac{9}{17}+\left(-18 \frac{9}{17}\right)

\newline

1

\newline

\newline

\newline

(C) Neither positive nor negative-the sum is zero.

Posted 3 months ago

Question

What is the sign of

\frac{23}{45}+\left(-\frac{23}{45}\right) ?

\newline

1

\newline

\newline

\newline

(C) Neither positive nor negative-the sum is zero.

Posted 3 months ago

Question

What is the sign of

45+(-38)

\newline

1

\newline

\newline

\newline

(C) Neither positive nor negative-the sum is zero.

Posted 3 months ago

Question

What is the sign of

-49.8+61.2

\newline

1

\newline

\newline

\newline

(C) Neither positive nor negative-the sum is zero.

Posted 3 months ago

Question

What is the sign of

-1.69+(-1.69) ?

\newline

1

\newline

\newline

\newline

(C) Neither positive nor negative-the sum is zero.

Posted 3 months ago

Question

What is the sign of

37+(-37)

\newline

1

\newline

\newline

\newline

(C) Neither positive nor negative-the sum is zero.

Posted 3 months ago

Question

\begin{array}{l} g(x)=\sqrt{2 x-9} \\ g^{\prime}(x)=? \end{array}

\newline

1

\newline

\frac{\sqrt{2 x-9}}{2}

\newline

\frac{1}{\sqrt{2 x-9}}

\newline

\frac{1}{2 \sqrt{2 x-9}}

\newline

\frac{1}{\sqrt{x}}

Posted 3 months ago