Question

In Exercises \(5-10\), find the slope of the tangent line to the graph of the function at the given point. \(f(x)=3-2 x, \quad(-1,5)\)

Short answer

The slope of the tangent line to the graph of the function \(f(x)=3-2x\) at the point (-1,5) is -2.

Step-by-step solution

Identify Function

The given function is \(f(x) = 3 - 2x\), and the point of interest is (-1,5).

Calculate the Derivative

In this case, we need to differentiate the function \(f(x)\) with respect to \(x\). For any linear function of the form \(y = mx + c\), the derivative \(f'(x)\) is equal to the coefficient of \(x\), which gives the slope of the line. Here, \(-2\) is the coefficient of \(x\). Hence, the derivative \(f'(x)\) is \(-2\).

Interpret the Result

The derivative \(-2\) is the slope of the tangent at any point on the line, including the point (-1,5). So, the slope of the tangent line to the graph of the function at the point (-1, 5) is -2.

Key concepts

Derivative of a Function

Understanding the derivative of a function is crucial for grasping the concept of the slope of a tangent line. In calculus, the derivative represents the rate of change of a function at any given point. It's like capturing a snapshot of how fast the function's value is changing at that specific instant.

For instance, if we have a function that describes the distance traveled over time, the derivative at any point in time would tell us the speed at that exact moment. In the context of the exercise, we used the derivative to determine the slope of the tangent line to the graph of the function at a particular point. The slope is equivalent to the derivative at that point because it signifies how steep the tangent line is at the point of tangency.

The process of finding the derivative is called differentiation, and for simple linear functions, like the one in the exercise, differentiation is straightforward: the derivative of a linear function is simply the coefficient of the variable x. Hence, for the function given in the exercise, the derivative was -2, indicating a constant rate of change, and consequently, a constant slope for the tangent along the graph.

Linear Function

A linear function is one of the simplest forms of functions in algebra, characterized by a straight line when graphed. Its general form is written as y = mx + c, where m is the slope of the line, and c is the y-intercept, or where the line crosses the y-axis.

A key feature of a linear function is its constant rate of change, which is the same as saying it has a constant slope. This means that for any two points on the line, the rate at which the y-value changes with respect to the x-value is consistent. This consistency makes calculating the slope straightforward—simply take any two points on the line, and divide the change in y by the change in x.

In our original exercise, we did not need to use two points to find the slope since the function provided was linear and the slope was directly given by the coefficient of x, which in this case was -2.

Differentiation Calculus

Differentiation is the fundamental operation in calculus that computes the derivative of a function. It can be applied to various types of functions beyond linear ones, including polynomial, exponential, logarithmic, and trigonometric functions. Differentiation helps to analyze the function's behavior and understand how it changes at any given point.

The principles of differentiation have vast applications across multiple fields such as physics, engineering, economics, and even biology. For example, in physics, it's used to determine velocities and accelerations. In economics, it can help find the maximum profit or minimum cost.

To differentiate a function, you apply specific differentiation rules such as the power rule, product rule, quotient rule, or chain rule depending on the complexity of the function. Remember, however, that understanding the basic concept of derivatives starts with simpler functions — and practicing differentiation with linear equations like the one from our textbook exercise provides a solid foundation for tackling more challenging calculus problems.