Calculus

Step 1: Understanding the Concept:
The given limit is of the indeterminate form because substituting into the numerator and denominator gives . We can solve this by either factoring the numerator or using L'Hôpital's Rule.

Step 2: Key Formula or Approach:
Method 1: Factoring
Use the algebraic identity for the difference of cubes: .
Method 2: L'Hôpital's Rule
If is of the form or , then .

Step 3: Detailed Explanation:
Using Factoring:
We factor the numerator as : Now, substitute this back into the limit expression: For , we can cancel the terms: Now, we can substitute into the simplified expression: Using L'Hôpital's Rule:
Let and .
The derivatives are and .
Applying the rule: Substituting :
Step 4: Final Answer:
Both methods yield the same result. The value of the limit is 3.

Step 1: Understanding the Concept:
To find the volume of a solid generated by revolving a region between two curves about the x-axis, we use the washer method. The volume is found by integrating the difference of the areas of two circles (outer and inner radii) along the axis of revolution.

Step 2: Key Formula or Approach:
The volume by the washer method for revolution about the x-axis is given by: where is the outer radius and is the inner radius.

Step 3: Detailed Explanation:
First, we express the curves as functions of . Let's use and for variables: and .
Next, we find the points of intersection by setting the functions equal to each other: The points of intersection are at and . So, our integration interval is .
In the interval , we determine which function is on top (outer radius). Let's test : For , .
For , .
Since on , the outer radius is and the inner radius is .
Now, we set up and evaluate the integral for the volume:
Step 4: Final Answer:
The volume of the solid is cubic units.

Step 1: Understanding the Concept:
Lagrange's Mean Value Theorem (LMVT) states that if a function is continuous on the closed interval and differentiable on the open interval , then there exists at least one number in such that:
Step 2: Key Formula or Approach:
1. Find the derivative .
2. Calculate and .
3. Substitute these values into the LMVT formula and solve for .

Step 3: Detailed Explanation:
The given function is , which is a polynomial and thus continuous and differentiable everywhere. The interval is .
1. Find the derivative:
Therefore, at , we have: 2. Calculate function values at endpoints:
3. Apply the LMVT formula:
Substitute the expressions we found: Factor the numerator: Cancel the term from the numerator and denominator: Now, solve for : Since we are given , we can divide both sides by :
Step 4: Final Answer:
The value of is .

Step 1: Understanding the Concept:
This limit is in the form of the definition of a derivative. Specifically, it relates to the Fundamental Theorem of Calculus (Part 1), which connects differentiation and integration. The theorem states that if , then .

Step 2: Key Formula or Approach:
The expression can be interpreted using the definition of the derivative: Alternatively, we can use L'Hôpital's rule on the form.

Step 3: Detailed Explanation:
Method 1: Using the Fundamental Theorem of Calculus
Let and define a new function .
According to the Fundamental Theorem of Calculus, the derivative of is .
The given limit can be rewritten as: Using our definition of , the integral is : This is exactly the definition of the derivative of at the point , i.e., .
Since we know , we can evaluate it at : Method 2: Using L'Hôpital's Rule
As , the integral . The denominator is also . So we have the indeterminate form .
We can apply L'Hôpital's rule by differentiating the numerator and the denominator with respect to .
Denominator: .
Numerator: Using Leibniz integral rule (a part of the Fundamental Theorem of Calculus): So the limit becomes:
Step 4: Final Answer:
Both methods show that the value of the limit is .

Step 1: Understanding the Concept:
We need to find the area of a surface of revolution. The curve is given as a function of ( ) and is revolved about the y-axis.

Step 2: Key Formula or Approach:
The formula for the surface area generated by revolving a curve from to about the y-axis is: Note: The curve represents the right half of a circle with radius 3. The surface generated is a zone of a sphere.

Step 3: Detailed Explanation:
Let's use the standard variables and . The curve is for .
First, we find the derivative : Next, we calculate the term inside the square root in the formula: Now, we substitute this into the surface area integral. The limits of integration are from to . The term cancels out:
Step 4: Final Answer:
The area of the generated surface is square units.

Step 1: Understanding the Concept:
We are given the derivative of a function, , and a point on the function, . To find the original function , we need to integrate the derivative and then use the given point (initial condition) to find the constant of integration, .

Step 2: Key Formula or Approach:
1. Find the indefinite integral of to get : .
2. Use the power rule for integration: .
3. Use the given condition to solve for the constant of integration .

Step 3: Detailed Explanation:
We are given . We can rewrite this as .
Now, we integrate to find : Applying the power rule for integration to each term: Now, we use the initial condition to find the value of : Substituting the value of back into the expression for :
Step 4: Final Answer:
The function is , which corresponds to option (C).