Problem 75 - Precalculus with Limits

Question

In Exercises 71-78, find \(\lim_{h \to 0}\ \dfrac{f(x+h)-f(x)}{h} \). \(f(x) = x^2-3x\)

Step-by-Step Solution

Answer

The solution is \(2x - 3\).

Step 1: Calculation of f(x + h)

First, substitute \(x + h\) into \(f(x)\) to get \(f(x + h) = (x + h)^2 - 3(x + h)\). Apply the power rule ((a + b)^2 = a^2 + 2ab + b^2) and distribute the term -3 to get \(f(x + h) = x^2 + 2xh + h^2 - 3x - 3h\).

Step 2: Calculation of f(x + h) - f(x)

Next, subtract \(f(x)\) from \(f(x + h)\), which yields \((x^2 + 2xh + h^2 - 3x - 3h) - (x^2 - 3x) = 2xh + h^2 - 3h\).

Step 3: Simplification of the Fraction

Now, divide \(2xh + h^2 - 3h\) by \(h\), and simplify the fraction by cancelling out \(h\), to get \(2x + h - 3\). Next, apply the limit \( h \to 0 \) to this expression to get the answer, \(2x - 3\).

Key Concepts

Derivative

The derivative represents a fundamental concept in calculus, specifically showcasing how a function changes at any given point. Think of it as the function's slope or gradient at a particular point. When you compute a derivative, you're trying to understand how changes in the input of a function lead to changes in the output. This is vital for grasping more advanced math concepts.

To find a derivative, you often use the difference quotient, expressed as:

• \( \lim_{h \to 0}\ \frac{f(x+h) - f(x)}{h} \)

When you take this limit as \( h \) approaches zero, you're narrowing in on an exact rate of change at a specific point. For the function in this exercise, \( f(x) = x^2 - 3x \), calculating its derivative gives insight into how the curve of the function behaves over its entire domain. This derivative, \( 2x - 3 \), tells us the rate at which \( f(x) \) is changing at any point \( x \).

Differential Calculus

Differential calculus revolves around computing the derivative of functions, which allows us to understand and quantify change in mathematical contexts. It plays a crucial role in both theoretical and applied mathematics, forming the backbone of many scientific calculations. At its core, differential calculus is concerned with two main ideas:

• Instantaneous Rate of Change: This is essentially what a derivative is. It provides a snapshot of how rapidly a quantity is changing at any given moment, as opposed to over an interval.
• Function Behavior: Derivatives help describe the shape and behavior of function graphs, such as identifying peaks (maximums), troughs (minimums), and where the graph is increasing or decreasing.

Understanding differential calculus helps decode not just mathematical phenomena but also real-world problems like predicting trends, optimizing processes, and designing complex systems.

Limit of a Function

In calculus, the limit of a function helps us understand the behavior of the function as it gets closer to a particular point. This is pivotal when analyzing derivatives and helps form the core of many calculus concepts. A limit can be seen as a function's intended destination or goal as it approaches a specific x-value.

Mathematically, we express this as:

• \( \lim_{x \to c} f(x) = L \)

where the function \( f(x) \) draws nearer and nearer to \( L \), as \( x \) approaches \( c \). Calculating the limit is a foundational step in finding derivatives, as seen in the exercise where \( h \) approaches zero. This enables us to determine how tiny changes in the input result in variation outputs, indicating a deep understanding of the function's continuous nature at a point.

Rate of Change

The rate of change concept is a familiar idea to many, often understood as how one quantity varies in relation to another. It encapsulates the essence of a derivative quite well, shedding light on how responsive a function is to changes in its inputs. The rate of change can be constant or variable, depending on the nature of the function.

For example, a linear function has a constant rate of change, visualized as a straight line with a uniform slope. However, a function like \( f(x) = x^2 - 3x \), unfolding in this exercise, has a variable rate of change. The derivative \( 2x - 3 \) shows how the rate adapts depending on the value of \( x \). This provides critical information about the dynamic behavior of a function, aiding in predictions and rationale for systems in motion or change.