Problem 6 - Calculus of a Single Variable

Question

Finding the Slope of a Tangent Line In Exercises \(5-10\) , find the slope of the tangent line to the graph of the function at the given point. $$ g(x)=\frac{3}{2} x+1, \quad(-2,-2) $$

Step-by-Step Solution

Answer

The slope of the tangent line to the graph at the point \((-2, -2)\) is \(\frac{3}{2}\).

Step 1: Understanding the function

The given function \(g(x) = \frac{3}{2}x + 1\) represents a straight line. The slope of a line is given by the coefficient of \(x\), which in this case is \(\frac{3}{2}\). Since the function is a linear function, which represents a straight line, the slope is the same at all points.

Step 2: Finding the slope

The slope of the tangent line at any point on this graph is the same as the slope of the line itself. Thus, the slope of the tangent line to the graph of the function at the point \((-2, -2)\) is \(\frac{3}{2}\).

Key Concepts

Slope

When we talk about the slope in mathematics, especially in the context of a line, we are describing how steep that line is. Think of it as the tilt of the line. In purely numerical terms, the slope is the ratio of vertical change (rise) to horizontal change (run) between two points on the line. This can be calculated by the formula \(m = \frac{\Delta y}{\Delta x}\).
For our function \(g(x) = \frac{3}{2}x + 1\), which is a linear function, the slope is represented by the coefficient of \(x\). Here, it is \(\frac{3}{2}\). This slope tells us that for every 2 units we move horizontally, the function moves up 3 units vertically.
Slopes can be positive, negative, zero, or undefined, depending on the direction and tilt of the line.

• **Positive Slope**: The line tilts upwards from left to right.
• **Negative Slope**: The line tilts downwards from left to right.
• **Zero Slope**: The line is horizontal.
• **Undefined Slope**: The line is vertical.

Understanding slope is crucial because it gives insight into the behavior of the function, allowing predictions and interpretations of the graph's direction and steepness.

Linear Function

A linear function is a type of function that creates a straight line when graphed. It is generally expressed in the form \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept. The equation \(g(x) = \frac{3}{2}x + 1\) is a perfect example of a linear function.
Linear functions are characterized by the fact that their slope is constant, meaning the graph rises or falls at a consistent rate. This feature makes them predictable and easy to work with.
Some key characteristics of linear functions include:

• **Straight Line Graph**: Always forms a straight line graph.
• **Constant Rate of Change**: The change between any two points on the graph is the same.
• **Easy to Interpret**: Linear models are commonly used for simple predictions and interpretations in various fields like economics and engineering.

Understanding the nature of linear functions helps in recognizing patterns and establishing relationships between different variables.

Graph of a Function

The graph of a function is a visual representation of every solution to a function. It allows us to easily see important characteristics such as intercepts, slopes, and overall behavior of the function. For the linear function \(g(x) = \frac{3}{2}x + 1\), the graph is a straight line.
To plot this graph, you need:

• **Slope (\(\frac{3}{2}\))**: Tells us the tilt or steepness of the line.
• **Y-intercept (1)**: The point at which the line crosses the y-axis.

With these two pieces of information, you can easily sketch the graph by starting at the y-intercept, \((0, 1)\), then using the slope to plot the next points. Move up 3 units and across 2 units to plot one additional point, ensuring the line extends both ways.

The purpose of graphing functions is to see the behavior and relationships clearly. Whether you're analyzing data trends or solving equations, graphs are an invaluable tool to visualize solutions.