Problem 31 - Calculus: Early Transcendentals

Question

Find \( f'(a) \). \( f(x) = 3x^2 - 4x + 1 \)

Step-by-Step Solution

Answer

\( f'(a) = 6a - 4 \).

Step 1: Identify the function and point

The function given to us is \( f(x) = 3x^2 - 4x + 1 \). We need to find the derivative \( f'(a) \), which means we'll calculate the derivative of the function and then evaluate it at a specific point \( x = a \).

Step 2: Differentiate the function

To find \( f'(x) \), differentiate each term of \( f(x) \) using the power rule. The power rule states that \( \frac{d}{dx}[x^n] = nx^{n-1} \). Applying this:- For \( 3x^2 \), the derivative is \( 6x \).- For \( -4x \), the derivative is \( -4 \).- For the constant \( 1 \), the derivative is \( 0 \).Therefore, the derivative \( f'(x) \) is \( 6x - 4 \).

Step 3: Evaluate the derivative at x = a

Substitute \( x = a \) into the derivative \( f'(x) = 6x - 4 \) to find \( f'(a) \).Thus, \( f'(a) = 6a - 4 \).

Key Concepts

Power Rule

When dealing with polynomial functions, the power rule is a very handy tool for finding derivatives. The rule states: if you have a term like \( x^n \), its derivative is \( nx^{n-1} \). This means you multiply the exponent by the coefficient and then subtract one from the exponent.

For example, if you have a term \( 3x^2 \), apply the power rule by multiplying the exponent 2 by the coefficient 3, which gives you 6. Then, reduce the exponent by 1, resulting in \( 6x^{2-1} = 6x \).

If a term is linear like \( -4x \), think of it as \( -4x^1 \). Using the power rule, you'll find the derivative is \( -4 \times 1 \times x^{1-1} = -4 \times x^0 = -4 \), since any value raised to the power of zero is 1.

• Power rule empowers you to differentiate polynomials term by term.
• Remember to adjust the exponent and multiply correctly.
• It's especially useful for simplifying complex expressions.

Differentiation

Differentiation is the process we use to find the derivative of a function. The derivative is simply a way to describe how a function changes at any point.

When you differentiate each term in a polynomial, you're essentially finding the slope of the function at each point. Learning differentiation is like learning to read a map of the function's behavior.

In our example, the function is \( f(x) = 3x^2 - 4x + 1 \). By differentiating, we found \( f'(x) = 6x - 4 \). This expression tells us the rate of change for different values of \( x \).

• Differentiation helps us determine increasing or decreasing trends.
• It is a core concept in calculus with real-world applications from physics to economics.
• Clear understanding of differentiation aids in mastering advanced calculus topics.

Differentiation can be performed on functions of various complexities, but it all starts with understanding basic rules like the power rule.

Evaluating Derivatives

The process of evaluating derivatives involves taking the derivative function \( f'(x) \) and plugging in a specific value for \( x \). This tells you how fast the original function \( f(x) \) is changing at that particular point.

For instance, in the problem, we found the derivative \( f'(x) = 6x - 4 \). To evaluate this at \( x = a \), the expression becomes \( f'(a) = 6a - 4 \).

By calculating this, you gain insight into the behavior of the function \( f(x) \) at \( x = a \). In practical terms, if \( a \) was a specific time or point, \( f'(a) \) would give the rate of change at that exact moment.

• Evaluating derivatives helps solve real-world problems where change matters.
• It is useful in physics for finding velocities and accelerations.
• Understanding how to evaluate gives predictive power over a function's behavior.

Grasping the concept of evaluating derivatives is vital, as it converts abstract calculations into tangible results.