Question

Describe a process for finding the slope of the line tangent to the graph of \(f\) at \((a, f(a).\)

Short answer

Answer: The slope of the tangent line to the graph of a function f at the point (a, f(a)) is given by the value of the derivative of the function, f'(a), at that point.

Step-by-step solution

Understand the concept of a derivative

The derivative of a function represents the rate of change of the function with respect to its input variable. In geometric terms, it gives us the slope of the tangent line to the graph of the function at a given point. The derivative of a function \(f(x)\) is denoted by \(f'(x)\) or \(\frac{d}{dx}f(x)\).

Differentiate the function

In order to find the slope of the tangent line to the graph of \(f\) at point \((a, f(a))\), we first need to find the derivative of \(f\). Differentiation can be done using various rules such as power rule, product rule, quotient rule, and chain rule, depending on the form of the function \(f(x)\). Since the function is not provided in this case, we will just represent the derivative of \(f\) as \(f'(x)\).

Evaluate the derivative at the given point

Once we have the derivative \(f'(x)\), we can find the slope of the tangent line at the given point \((a, f(a))\) by evaluating the derivative at \(x=a\). In other words, we need to find \(f'(a)\), which represents the slope of the tangent line at that point.

Interpret the result

The value \(f'(a)\) represents the slope of the tangent line to the graph of the function \(f\) at the point \((a, f(a))\). This value can be used to construct the equation of the tangent line or to analyze the behavior of the function at that point, such as whether the function is increasing or decreasing and the rate at which it is doing so.

Key concepts

Derivative

A derivative is a fundamental concept in calculus, crucial for understanding how functions change. Think of it as the tool that helps us measure how a function's output value (y) changes as its input value (x) varies. In simpler terms, the derivative tells us the rate at which one quantity changes in relation to another. This is especially handy when you want to determine how steep a graph is at a specific point, which relates directly to the slope of a tangent line.

Mathematically, the derivative of a function \(f(x)\) is denoted as \(f'(x)\) or \(\frac{d}{dx}f(x)\). Calculating the derivative involves applying specific rules and techniques from calculus, which can vary depending on the type of function you're working with. These may include, for example, the power rule or the chain rule.

Understanding derivatives helps us analyze changes, identify trends, and comprehend the mechanics of how different quantities interact. Mastering this concept is essential when exploring more complex mathematical principles.

Differentiation

Differentiation is the process used to find the derivative of a function. It's like stepping into the toolkit of calculus to extract the information you need about a function's behavior. By differentiating a function, you figure out how its values change, capturing this change as a mathematical expression.

When we differentiate a function \(f(x)\), we're essentially performing a set of mathematical operations that tell us how to obtain its derivative—\(f'(x)\). This process uses well-known rules of differentiation, which include:

• Power Rule: Useful when dealing with expressions like \(x^n\).
• Product Rule: Applies when multiplying two functions.
• Quotient Rule: Handy for dividing one function by another.
• Chain Rule: Helpful in tackling composite functions.

By applying these rules, you can systematically find the derivative, transforming abstract mathematical equations into practical insights about the function's behavior across its domain.

Slope of the Tangent

The slope of the tangent line to a curve at a specific point provides critical insight into that curve's behavior at that point. Geometrically, the tangent line is a straight line that just "touches" the curve at one point, without crossing it. The slope of this line tells us how steep the curve is, indicating whether it is rising or falling at that precise location.

To find the slope of the tangent line at a particular point, we evaluate the function's derivative at that point. If \(f(x)\) is our function, we first determine its derivative \(f'(x)\). Then, to get the slope of the tangent to the graph of \(f\) at the point \((a, f(a))\), we calculate \(f'(a)\).

Think of \(f'(a)\) as the slope, or gradient, of this tangent line, providing a snapshot of how the curve behaves right around \(a\). This slope can also hint at the function's overall trend at that point, indicating whether it’s increasing, decreasing, or flat. Understanding how to find and interpret tangent slopes is vital for deeper analytic insights in calculus.