Problem 21 - Calculus: Early Transcendentals

Question

If an equation of the tangent line to the curve \( y = f(x) \) at the point where \( a = 2 \) is \( y = 4x - 5 \), find \( f(2) \) and \( f'(2) \).

Step-by-Step Solution

Answer

\( f(2) = 3 \) and \( f'(2) = 4 \).

Step 1: Interpret the Tangent Line Equation

The equation of the tangent line at a given point provides two pieces of information. The slope of the tangent line is the derivative of the function at that point, and the line itself passes through the point on the curve.

Step 2: Identify the Slope as the Derivative

Since the equation of the tangent line is given as \( y = 4x - 5 \), the slope of this line is 4. Therefore, the derivative of \( f(x) \) at \( x = 2 \) is 4. Thus, \( f'(2) = 4 \).

Step 3: Find the Point on the Tangent Line

To find which point \((2, y)\) on the tangent line corresponds to \( f(2) \), substitute \( x = 2 \) into the line equation. This gives us \( y = 4(2) - 5 = 8 - 5 = 3 \).

Step 4: Conclude the Value of the Function

Since the tangent line passes through the point where \( x = 2 \) and \( y = f(2) \), and since we found that \( y = 3 \) at \( x = 2 \), it follows that \( f(2) = 3 \).

Key Concepts

Derivative

A derivative represents the rate at which a function changes as its input changes. It is the fundamental concept in calculus that describes how a function behaves at any given point. When dealing with a function\( f(x) \), the derivative at a specific point\( x = a \) tells us the slope of the tangent line to the curve at that point. This slope indicates how steep the curve is or, in simpler terms, how quickly the function's value is increasing or decreasing at\( x = a \).
In the context of the given problem, the equation of the tangent line is\( y = 4x - 5 \). This tells us that the slope of the tangent line at the point where\( x = 2 \) is 4. Thus, the derivative of the function at\( x = 2 \), denoted as\( f'(2) \), is equal to 4.
Understanding derivatives is key to solving problems related to rates of change, such as speed in physics or profit growth in economics. It provides insight into the local behavior of functions.

Slope

The slope of a line conveys its steepness and direction. In mathematical terms, the slope is defined as the ratio of the change in the\( y \) values to the change in the\( x \) values between two points on the line. For a linear equation like\( y = 4x - 5 \), the coefficient of\( x \) is the slope, which is 4 in this case.
This means for every unit increase in\( x \), the\( y \) value increases by 4. Positive slopes indicate lines ascending as you move from left to right, while negative slopes indicate lines descending.
When concerned with the tangent line to a curve at a specific point, the slope of this tangent line is especially significant. It not only tells us how the line is oriented but also reflects the instantaneous rate of change of the function, captured by the derivative at that point.

Function Value

The function value\( f(a) \) at a certain point\( a \) refers to the output or result of the function when\( x = a \). In other words, it is the\( y \) coordinate of the point on the function's graph at\( x = a \).
In the problem at hand, we are given that the tangent line passes through the point where\( x = 2 \). By substituting\( x = 2 \) into the equation of the tangent line\( y = 4x - 5 \), we find\( y = 3 \). Therefore, since the tangent line intersects the curve at this point, we conclude that\( f(2) = 3 \). This value reflects the height of the curve at\( x = 2 \), giving us a precise measurement of the function's behavior at that point.
Understanding function values helps in interpreting the position of points on the curve, which is crucial for visualizing the shape and form of the graph.

Equation of the Line

The equation of a line provides a mathematical representation of a straight line on a graph and is typically expressed in the form\( y = mx + b \). Here,\( m \) is the slope and\( b \) is the y-intercept, which is the point where the line crosses the y-axis.
In our example problem, the tangent line to the curve\( y = f(x) \) at\( x = 2 \) is described by the equation\( y = 4x - 5 \).
Understanding the equation of a line is crucial in calculus, especially when working with tangent lines. These equations allow us to predict the\( y \) values for any\( x \) on a line. This is particularly useful in determining where a tangent line touches a curve at just one point, providing insight into the function's instantaneous height and the direction of its slope at that exact spot.
Equations of lines, and specifically tangent lines, help us analyze functions in detail, ultimately broadening our understanding of the function's properties and behavior.