Limits And Derivatives

We are given the function for and need to find the value of guaranteed by the Mean Value Theorem (MVT).

According to the MVT, if is continuous on the closed interval and differentiable on the open interval , then there exists a point in such that:

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First, calculate and :

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Now, calculate the average rate of change:

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Next, find the derivative of :

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Now, set and solve for :

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Solving this equation will give the value of in the interval . After solving, we find that the value of is:

The correct answer is 4.

We are asked to evaluate the limit:

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We begin by expanding using the tangent addition formula:

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Since , we have:

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Next, we square both sides to find :

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Now, we compute the limit:

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We can directly differentiate the numerator and denominator using the derivative approach. The derivative of the numerator at gives us the required limit:

Using a standard result or applying L'Hopital's Rule, we find that the value of the limit is:

The correct answer is .

We are given that is the derivative of , and we need to find the value of:

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From the definition of the derivative, we know that:

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We are given that , so:

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Now, substitute into the expression for :

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The correct answer is 6.

The given limit is:

We will evaluate the limit by substituting directly into the expression, provided there are no indeterminate forms:

Substitute into the numerator and denominator:

Numerator:

Denominator:

Since both the numerator and denominator are 0, we need to simplify the expression using factorization or apply L'Hopital's Rule. Let's proceed with factorization:

Factorize the numerator and denominator:

Numerator:

Denominator:

The expression becomes:

Now cancel out the common factor of in the numerator and denominator:

Substitute into the simplified expression:

The correct value of the limit is:

We are given the limit:

To simplify this expression, we multiply both the numerator and the denominator by the conjugate of the denominator:

The denominator simplifies as follows:

So the expression becomes:

Now, as , we substitute into the simplified expression:

Answer: -6