Problem 22 - Brief Applied Calculus

Question

Evaluate each expression. $$ \left.\frac{d^{2}}{d x^{2}} x^{11}\right|_{x=-1} $$

Step-by-Step Solution

Answer

The value is -110.

Step 1: Understand the Expression

The given expression requires us to find the second derivative of the function \( x^{11} \) with respect to \( x \), and then evaluate this derivative at \( x = -1 \).

Step 2: Find the First Derivative

The first step in differentiation is to apply the power rule: if \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \). For the first derivative, we have \( \frac{d}{dx} x^{11} = 11x^{10} \).

Step 3: Find the Second Derivative

We need to differentiate the result from the previous step to obtain the second derivative. Again applying the power rule to \( 11x^{10} \), we get \( \frac{d^2}{dx^2} x^{11} = 110x^9 \).

Step 4: Evaluate the Second Derivative at x = -1

Substitute \( x = -1 \) into the second derivative. We have \( 110(-1)^9 = 110(-1) = -110 \).

Step 5: Conclude the Result

The value of the second derivative of \( x^{11} \) at \( x = -1 \) is \( -110 \).

Key Concepts

Power Rule

The power rule is a fundamental rule in calculus used to find the derivative of a function in the form of \( f(x) = x^n \). To apply the power rule, you shift the exponent down to the coefficient and reduce the original exponent by one.
In simpler terms, if you have \( x^n \), the derivative is given by \( nx^{n-1} \). This means you multiply the variable by its power and then subtract one from the power.

• Example: For the function \( x^{11} \), using the power rule gives \( 11x^{10} \).
• This rule makes differentiation straightforward for polynomials.

Understanding the power rule is essential because it simplifies the process of taking derivatives, making it much quicker and easier for more complex expressions.

Differentiation

Differentiation is the process of finding the derivative of a function, which measures how the function changes as its input changes. When you differentiate a function like \( x^{11} \), it means you're examining how its rate of change behaves.
The first differentiation step results in the first derivative. Applying the power rule to \( x^{11} \) gives \( 11x^{10} \). This derivative shows the rate at which \( x^{11} \)'s value changes.

• The second differentiation step provides what's called the second derivative, which gives information on how the rate of change itself changes.
• For the function \( 11x^{10} \), further differentiation results in \( 110x^9 \).

Each step of differentiation peels back another layer of understanding about a function's behavior. It's like investigating a function's acceleration, giving deeper insight into the nature of a function's growth or decline.

Evaluate Derivative at a Point

After finding the derivatives, sometimes it's crucial to evaluate them at specific points to gain exact insight into a function's behavior at those coordinates. Evaluating at a point is simply about substituting the specified \( x \) value into the derived expression.
In the exercise, we've calculated the second derivative as \( 110x^9 \). To evaluate it at \( x = -1 \), substitute -1 in place of \( x \):

• This substitution turns the expression into \( 110(-1)^9 \).
• Since \((-1)^9 = -1\), this evaluation results in \( 110(-1) = -110 \).

Evaluating a derivative at a specific point is a powerful process, enabling us to determine the slope or curvature at that exact location, which can inform conclusions about a function's real-world behavior or stability at that point.