Problem 14 - Precalculus: Mathematical for Calculus

Question

Find the derivative of the function at the given number. $$f(x)=2-3 x+x^{2} \quad \text { at }-1$$

Step-by-Step Solution

Answer

The derivative of the function at \( x = -1 \) is \( -5 \).

Step 1: Understand the Function

We are given the function \( f(x) = 2 - 3x + x^2 \). Our goal is to find its derivative and evaluate it at \( x = -1 \).

Step 2: Differentiate the Function

The derivative of a function \( f(x) = ax^n \) is \( f'(x) = anx^{n-1} \). Applying this rule, the derivative of \( 2 \) is \( 0 \), the derivative of \( -3x \) is \( -3 \), and the derivative of \( x^2 \) is \( 2x \). So, the derivative of the function is: \[ f'(x) = 0 - 3 + 2x = 2x - 3 \].

Step 3: Substitute the Given Number into the Derivative

We need to find the value of \( f'(x) \) at \( x = -1 \). Substitute \( -1 \) into the derivative \( f'(x) = 2x - 3 \):\[ f'(-1) = 2(-1) - 3 = -2 - 3 = -5 \].

Step 4: Compile the Result

The derivative of the function \( f(x) = 2 - 3x + x^2 \) at \( x = -1 \) is \( -5 \).

Key Concepts

Understanding Function Differentiation

Differentiation is a fundamental concept in calculus. It is all about finding the rate at which something changes. More specifically, when we differentiate a function, we are calculating its derivative. This tells us how the function's output value changes with respect to changes in its input value.
For instance, if you have a function that describes a car's position over time, the derivative of that function would give us the car's speed at any given moment.
When we say "differentiating a function," we focus on finding the derivative formula itself. This involves applying specific rules to calculate derivatives systematically:

• The power rule: If you have a term like \( x^n \), the derivative is \( nx^{n-1} \).
• Constant rule: The derivative of a constant is zero.
• Linearity of differentiation: Constants can be factored out, and derivatives of sums can be calculated separately.

When you apply these rules, you can find the derivative for more complex functions by breaking them into components differentiated separately. This is precisely what is done for the function \( f(x) = 2 - 3x + x^2 \). Each part of the function is approached individually, and through mastery of these differentiation rules, we find the overall derivative.

Deriving the Derivative Evaluation

After obtaining the derivative expression, the next step is often evaluating it at a particular point, usually required to solve a problem or understand behavior at that specific input value.
This is called derivative evaluation, where we determine the slope or rate of change at a particular point on the graph of the function. For the example \( f(x) = 2 - 3x + x^2 \), we first differentiated to achieve \( f'(x) = 2x - 3 \).
Evaluating the derivative expression means substituting the chosen x-value into this derivative formula. For instance, to find the derivative at \( x = -1 \), simply plug \( -1 \) into \( f'(x) \):

• Replace \( x \) with \( -1 \) in \( 2x - 3 \).
• Solve the arithmetic: \( 2(-1) - 3 = -2 - 3 = -5 \).

This computes the slope of the tangent line to the function at that particular x-value, providing insights into the function's behavior right at \( x = -1 \). The result of \( -5 \) signifies the rate of change at that point.

Exploring Polynomial Functions

Polynomial functions are expressions that involve variables raised to whole number powers and coefficients. They can be as simple as \( x^2 \), or far more complex like \( 4x^3 + 3x^2 - x + 7 \).
These functions are characterized by their smooth and continuous curves when graphed, with degrees determining their general shape. Each part of the polynomial, such as the constant term or the coefficient with its associated power of x, contributes uniquely to the overall behavior of the curve.
In calculus, polynomial functions are relatively easy to work with because their derivatives follow neatly from defined rules. For instance, when differentiating a polynomial, each term can be treated separately, using the power rule, to quickly arrive at the derivative:

• The derivative of \( x^n \) is \( nx^{n-1} \).
• For a constant term \( c \), its derivative is \( 0 \).
• Linear term \( ax \) becomes just \( a \) upon differentiation.

These simple rules make polynomial functions an excellent starting point for students beginning to learn calculus and working with derivatives.