Instructions: Please Round Your Answers To 4 Decimal Places Where Required.---Question 1:Find An Expression For The Total Function \[$ G(x) \$\] Given The Following Information:- Given The Marginal Function: $\[ 3x^5 + \frac{9}{x^2}

Question 1: Find an Expression for the Total Function

Given the Marginal Function

We are given the marginal function:

$${ 3x^5 + \frac{9}{x^2} }$$

Our goal is to find an expression for the total function $g(x)$.

Understanding the Marginal Function

The marginal function is the derivative of the total function. In other words, if we have a function $f(x)$, then its marginal function is given by $f'(x)$. Conversely, if we have a marginal function, we can find the total function by integrating it.

Finding the Total Function

To find the total function $g(x)$, we need to integrate the marginal function:

$${ g(x) = \int \left( 3x^5 + \frac{9}{x^2} \right) dx }$$

Integrating the Marginal Function

To integrate the marginal function, we can use the power rule of integration, which states that:

$${ \int x^n dx = \frac{x^{n+1}}{n+1} + C }$$

where $n$ is a constant and $C$ is the constant of integration.

Applying this rule to the first term, we get:

$${ \int 3x^5 dx = \frac{3x^6}{6} + C_1 = \frac{x^6}{2} + C_1 }$$

where $C_1$ is a constant of integration.

For the second term, we can use the rule:

$${ \int \frac{1}{x^n} dx = \frac{x^{1-n}}{1-n} + C }$$

where $n$ is a constant and $C$ is the constant of integration.

Applying this rule to the second term, we get:

$${ \int \frac{9}{x^2} dx = \frac{9x^{-1}}{-1} + C_2 = -9x^{-1} + C_2 = -\frac{9}{x} + C_2 }$$

where $C_2$ is a constant of integration.

Combining the Results

Now that we have integrated both terms, we can combine the results to get the total function:

$${ g(x) = \frac{x^6}{2} - \frac{9}{x} + C }$$

where $C$ is the constant of integration.

Conclusion

In this problem, we were given the marginal function and asked to find the total function. We used the power rule of integration to integrate the marginal function and obtained the total function. The total function is given by:

$${ g(x) = \frac{x^6}{2} - \frac{9}{x} + C }$$

where $C$ is the constant of integration.

Final Answer

The final answer is $\boxed{\frac{x^6}{2} - \frac{9}{x} + C}$.

Question 2: Find the Value of the Constant of Integration

Given the Total Function

We are given the total function:

$${ g(x) = \frac{x^6}{2} - \frac{9}{x} + C }$$

Our goal is to find the value of the constant of integration $C$.

Using the Initial Condition

We are given the initial condition:

$${ g(1) = 0 }$$

We can use this condition to find the value of $C$.

Substituting the Initial Condition

Substituting $x=1$ into the total function, we get:

$${ g(1) = \frac{1^6}{2} - \frac{9}{1} + C = 0 }$$

Simplifying this expression, we get:

$${ \frac{1}{2} - 9 + C = 0 }$$

Solving for C

Solving for $C$, we get:

$${ C = 9 - \frac{1}{2} = \frac{17}{2} }$$

Conclusion

In this problem, we were given the total function and asked to find the value of the constant of integration $C$. We used the initial condition to find the value of $C$, which is given by:

$${ C = \frac{17}{2} }$$

Final Answer

The final answer is $\boxed{\frac{17}{2}}$.

Question 3: Find the Derivative of the Total Function

Given the Total Function

We are given the total function:

$${ g(x) = \frac{x^6}{2} - \frac{9}{x} + C }$$

Our goal is to find the derivative of the total function.

Using the Power Rule of Differentiation

To find the derivative of the total function, we can use the power rule of differentiation, which states that:

$${ \frac{d}{dx} x^n = nx^{n-1} }$$

where $n$ is a constant.

Applying this rule to the first term, we get:

$${ \frac{d}{dx} \left( \frac{x^6}{2} \right) = \frac{6x^5}{2} = 3x^5 }$$

For the second term, we can use the rule:

$${ \frac{d}{dx} \left( \frac{1}{x^n} \right) = -\frac{1}{x^{n+1}} }$$

where $n$ is a constant.

Applying this rule to the second term, we get:

$${ \frac{d}{dx} \left( -\frac{9}{x} \right) = -\frac{9}{x^2} }$$

Combining the Results

Now that we have differentiated both terms, we can combine the results to get the derivative of the total function:

$${ g'(x) = 3x^5 - \frac{9}{x^2} }$$

Conclusion

In this problem, we were given the total function and asked to find the derivative of the total function. We used the power rule of differentiation to find the derivative, which is given by:

$${ g'(x) = 3x^5 - \frac{9}{x^2} }$$

Final Answer

The final answer is $\boxed{3x^5 - \frac{9}{x^2}}$.

Question 4: Find the Second Derivative of the Total Function

Given the Total Function

We are given the total function:

$${ g(x) = \frac{x^6}{2} - \frac{9}{x} + C }$$

Our goal is to find the second derivative of the total function.

Using the Power Rule of Differentiation

To find the second derivative of the total function, we can use the power rule of differentiation, which states that:

$${ \frac{d}{dx} x^n = nx^{n-1} }$$

where $n$ is a constant.

Applying this rule to the first term, we get:

$${ \frac{d}{dx} \left( \frac{x^6}{2} \right) = \frac{6x^5}{2} = 3x^5 }$$

For the second term, we can use the rule:

$${ \frac{d}{dx} \left( \frac{1}{x^n} \right) = -\frac{1}{x^{n+1}} }$$

where $n$ is a constant.

Applying this rule to the second term, we get:

$${ \frac{d}{dx} \left( -\frac{9}{x} \right) = -\frac{9}{x^2} }$$

Combining the Results

Now that we have differentiated both terms, we can combine the results to get the first derivative of the total function:

$${ g'(x) = 3x^5 - \frac{9}{x^2} }$$

To find the second derivative, we can differentiate the first derivative:

$${ g''(x) = \frac{d}{dx} \left( 3x^5 - \frac{9}{x^2} \right) }$$

Using the power rule of differentiation, we get:

$${ g''(x) = 15x^4 + \frac{18}{x^3} }$$

Conclusion

In this problem, we were given the total function and asked to find the second derivative of the total function. We used the power rule of differentiation to find the second derivative, which is given by:

$${ g''(x) = 15x^4 + \frac{18}{x^3} }$$

Final Answer

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Question 1: What is the Marginal Function?

Answer

The marginal function is the derivative of a total function. It represents the rate of change of the total function with respect to the variable.

Question 2: How Do I Find the Total Function from the Marginal Function?

Answer

To find the total function from the marginal function, you need to integrate the marginal function. This can be done using the power rule of integration, which states that:

$${ \int x^n dx = \frac{x^{n+1}}{n+1} + C }$$

where $n$ is a constant and $C$ is the constant of integration.

Question 3: What is the Power Rule of Differentiation?

Answer

The power rule of differentiation is a rule in calculus that states that if $f(x) = x^n$, then $f'(x) = nx^{n-1}$. This rule can be used to find the derivative of a function that is a power function.

Question 4: How Do I Find the Second Derivative of a Function?

Answer

To find the second derivative of a function, you need to differentiate the first derivative of the function. This can be done using the power rule of differentiation.

Question 5: What is the Constant of Integration?

Answer

The constant of integration is a constant that is added to the result of an integration. It represents the fact that the integral of a function can be shifted by a constant amount.

Question 6: How Do I Use the Initial Condition to Find the Constant of Integration?

Answer

To use the initial condition to find the constant of integration, you need to substitute the initial condition into the total function and solve for the constant of integration.

Question 7: What is the Marginal Function of a Total Function?

Answer

The marginal function of a total function is the derivative of the total function. It represents the rate of change of the total function with respect to the variable.

Question 8: How Do I Find the Marginal Function of a Total Function?

Answer

To find the marginal function of a total function, you need to differentiate the total function. This can be done using the power rule of differentiation.

Question 9: What is the Second Derivative of a Function?

Answer

The second derivative of a function is the derivative of the first derivative of the function. It represents the rate of change of the rate of change of the function.

Question 10: How Do I Use the Second Derivative to Analyze a Function?

Answer

To use the second derivative to analyze a function, you need to examine the sign and magnitude of the second derivative. A positive second derivative indicates that the function is concave up, while a negative second derivative indicates that the function is concave down.

Conclusion

In this article, we have discussed various concepts in calculus and mathematical functions, including the marginal function, total function, power rule of differentiation, constant of integration, and second derivative. We have also provided answers to frequently asked questions about these concepts. We hope that this article has been helpful in understanding these concepts and how to apply them in real-world problems.