Problem 65 - Contemporary Precalculus

Question

Do Exercise 63 with \(f(x)=x^{12}\) in place of \(f(x)=x^{5}\)

Step-by-Step Solution

Answer

Answer: The derivative of the function \(f(x) = x^{12}\) at \(x=2\) is 24,576.

Step 1: Find the derivative of \(f(x) = x^{12}\)

To find the derivative of the given function \(f(x) = x^{12}\), we will apply the power rule. The power rule states that if \(f(x) = x^n\), then \(f'(x) = nx^{n-1}\). So for our function, the derivative will be: $$ f'(x) = 12x^{12-1} $$ Simplifying the expression, we have: $$ f'(x) = 12x^{11} $$

Step 2: Evaluate the derivative at the specific point

Now we need to evaluate the derivative at a specific point. The exercise doesn't mention any specific point, but let's assume we have to find the derivative at \(x=2\). To do this, we will substitute the value of \(x\) in our calculated derivative: $$ f'(2) = 12(2)^{11} $$ Now, compute the value: $$ f'(2) = 12 \times 2048 $$ Finally, we have the value of the derivative at \(x=2\): $$ f'(2) = 24576 $$ The derivative of the function \(f(x) = x^{12}\) at \(x=3\) is 24576.

Key Concepts

Derivative

The concept of a derivative is central to calculus and represents the rate at which a function changes. In simpler terms, it tells us how a function behaves as its input values change. When we find the derivative of a function, we are determining this rate of change.

• The derivative measures the slope of the tangent line to the curve of a function at any point.
• It provides insight into whether a function is increasing or decreasing, and by how much, at a particular point.

Derivatives have practical applications in various fields such as physics, engineering, and economics for solving real-world problems. Calculating the derivative involves different methods, such as limits, differentiation rules, and the power rule, which we will learn about next.

Power Rule

The power rule is a quick and straightforward method to find the derivative of polynomial functions of the form \(f(x) = x^n\), where \(n\) is any real number. Here’s how it works:- The derivative of the function \(x^n\) is given by \(nx^{n-1}\).- You multiply the exponent \(n\) by the base \(x\) raised to the power of \(n-1\).In our example using \(f(x) = x^{12}\), by applying the power rule, we derived:\[f'(x) = 12x^{12-1} = 12x^{11}\]The power rule simplifies the process of differentiation by providing a formulaic way to find the derivative of any term with a power. Remember:- The power rule applies directly as long as the term is a simple power of \(x\).- For more complex terms, other rules in combination with the power rule might be required.

Polynomial Function

A polynomial function is a sum of terms, each consisting of a variable raised to a non-negative integer power multiplied by a coefficient. They can be expressed as:\[f(x) = a_nx^n + a_{n-1}x^{n-1} + \, ... \, + a_1x + a_0\]where \(a_n, a_{n-1}, ..., a_1, a_0\) are coefficients, and \(n\) is the degree of the polynomial that represents the highest power of \(x\).

• Polynomial functions are smooth and continuous, meaning they don't have any breaks or holes.
• They can represent simple linear relationships when the degree is 1 or complex behaviors when the degree is higher.
In our exercise, \(f(x) = x^{12}\) is a polynomial function with a single term and degree 12. When deriving such functions, each term is differentiated individually using straightforward methods like the power rule, making derivatives of polynomial functions simple to calculate.