Fundamental Theorem Of Calculus

Step 1: Understanding the Condition for a Tangent Parallel to X-axis The equation of the given function is: For the tangent to be parallel to the X-axis, its slope must be zero. The slope of the tangent is given by the first derivative of , i.e.,

Step 2: Compute the First Derivative Setting for the required condition:

Step 3: Solve for Solving the quadratic equation: Using the quadratic formula:

Step 4: Selecting the Correct Value of Since , the valid solution is:

Solution:

A function is continuous at if and only if

In this case,

and

and , so is continuous at .

For ,

and for ,

Step 1: Define the expression. We are given the function: Let Then, we have:

Step 2: Differentiate using the chain rule. The derivative of with respect to is: Now, we need to calculate .

Step 3: Differentiate using the quotient rule. Recall the quotient rule: where: and We differentiate and : - -

Step 4: Apply the quotient rule. Using the quotient rule: Substitute the values of , , , and and simplify.

Step 5: Substitute into the formula for . After simplifying , substitute into the formula for :

Step 6: Final expression. After simplifying the final expression, we get: Thus, the correct answer is:

Step 1: Calculate (Approximate Change in ) The differential is given by: First, find : Substituting : Now, calculate : ]

Step 2: Calculate (Actual Change in ) First, compute : Now compute : Approximating , we get: Thus,

Step 3: Compute

Step 1: Compute the First Derivative To find the critical points, we first differentiate the function:

Step 2: Solve for Critical Points Set to find critical points: Dividing by 6: Factoring:

Step 3: Evaluate at Critical and Endpoint Values We evaluate at :

Step 4: Identify Maximum and Minimum Values

Step 5: Compute

Step 1: Given functional equation
The given functional equation is: where . \vspace{0.5cm}

Step 2: Differentiate with respect to
Differentiate both sides of the equation with respect to while treating as a constant. The left-hand side requires the use of the chain rule for both terms: - For the first term , the derivative is: - For the second term , use the chain rule: Thus, the left-hand side becomes: The right-hand side is simpler:

Step 3: Set up the equation
Now, equate the differentiated terms: Divide through by (assuming ):

Step 4: Substitute
Substitute into the equation to simplify: Since , this simplifies to: This is trivially true, but we need to consider the form of . The only possible function that satisfies this equation for all values of is a quadratic function. Assume: Thus, .

Step 5: Final Answer
Therefore, the derivative of the function is: Hence, the correct answer is , which corresponds to option (4).

- First, recognize as an infinite series: which is a geometric series with the first term and common ratio 1. - Therefore, the sum of the infinite series is: This gives . Now, we calculate : - Applying differentiation, you get the answer as:

Step 1: Substituting Using the Weierstrass substitution: We apply the standard transformations:

Step 2: Expressing the Denominator

Step 3: Integrating Applying logarithm integration properties:

Step 1: Using the Property Define: Using the property: Substituting , we get: Since , we rewrite: Using the property , we obtain: Multiplying numerator and denominator by , we get: Now, add the two equations for : Since the integrals are equal in magnitude, the sum simplifies and gives us: The integral of over symmetric limits is zero, so we get:

Step 1: Set Up the Integral for the Enclosed Area The enclosed area is given by the definite integral: Since changes sign at , we split the integral into two parts: For , is negative, so we take the absolute value:

Step 2: Evaluate Each Integral Computing: Evaluating both integrals: Similarly, for the second integral:

Step 3: Compute Total Area Thus, the enclosed area is:

Step 1: Checking for Exactness Given: Compute: Since , the equation is exact.

Step 2: Solving for Integrate with respect to : Differentiate with respect to : Since , equating:

Step 3: Final Equation

Step 1: Differentiate
Using the derivative of logarithm: Differentiating the expression inside: Thus, Simplifying: Since ,

Step 2: Compute
Differentiating again:

Step 3: Compute the given expression
Substituting values: Since ,

Step 4: Conclusion
Thus, the correct answer is:

Step 1: Find conditions for horizontal and vertical tangents
The given curve equation: Differentiate implicitly:

Step 2: Condition for horizontal tangents
For a horizontal tangent: Substituting in the curve equation: Thus, there are no points with horizontal tangents.

Step 3: Condition for vertical tangents
For a vertical tangent, denominator of should be zero: Substituting in the curve equation: Substituting in : So, is the only point where the tangent is vertical.

Step 4: Conclusion
Thus, the correct answer is:

- First, factorize the quadratic expressions in the numerator if possible to simplify the expression.
- Substitute in the simplified equation and check the value of the denominator, as approaches , to determine the behavior of the function.
- If the function presents an indeterminate form like , apply L'Hôpital's Rule or algebraic manipulation to resolve the indeterminacy.
- Finally, evaluate the limit to find the exact value.

Step 1: Continuity Condition at
- For to be continuous at , we require: - Therefore, the limit of as must equal , the value of at .

Step 2: Simplifying the Limit Expression
- We need to evaluate the limit: - First, note that as , so . - The expression approximates as . - Similarly, and for small .

Step 3: Substitute the Approximations
- Substitute the approximations into the limit expression: - Simplify the expression:

Step 4: Conclusion for
- Therefore, the limit of as is 1, which means: .

Step 5: Final Calculation for
- Since , we have: Thus, the correct answer is .