Problem 3 - Precalculus: Mathematics for Calculus

Question

Find the slope of the tangent line to the graph of \(f\) at the given point. $$f(x)=3 x+4, \quad \text { at }(1,7)$$

Step-by-Step Solution

Answer

The slope of the tangent line at (1, 7) is 3.

Step 1: Understanding the Formula for Slope of Tangent Line

The function given is a linear function: \(f(x) = 3x + 4\). For linear functions, the slope of the tangent line at any point is simply the coefficient of \(x\). This is because the derivative of a linear function at any point gives its constant slope.

Step 2: Identify the Slope from the Linear Function

Observing the function \(f(x) = 3x + 4\), the coefficient of \(x\) is 3. For any point on the line given by a linear equation, the slope is this coefficient. Therefore, the slope of the tangent line at the point \((1, 7)\) is 3.

Step 3: Verification by Calculating Derivative

To verify, let's calculate the derivative of \(f(x) = 3x + 4\). The derivative \(f'(x)\) of \(3x + 4\) is 3, as the derivative \(d/dx\) of \(ax + b\) is \(a\). Thus, the slope of the tangent line is indeed 3.

Key Concepts

Slope of Tangent Line

The slope of a tangent line at a specific point on the function provides the rate of change of that function at that exact point. For linear functions like the one in this exercise, the concept is relatively straightforward. Since a linear function is a straight line, its slope remains the same at all points along the line.

• The slope tells us how steep the line is.
• In a linear function, the slope doesn't change, meaning the tangent slope is constant everywhere on the function.

In the example of the linear function \(f(x) = 3x + 4\), the slope at any point, including the specified point \((1, 7)\), is the coefficient of \(x\), which is \(3\). Understanding the slope of a tangent line helps in many real-world applications, such as predicting trends or determining speed.

Derivative of Linear Function

The derivative is a fundamental tool in calculus that allows us to find how fast something is changing. Differentiation helps us find the slope of a function at a given point, giving crucial insight for various analyses. For a linear function, the derivative is straightforward.

• The derivative of a linear function, like \(f(x) = ax + b\), is simply its slope \(a\).
• Linear functions have constant derivatives, meaning their slope never changes.

When we differentiate \(f(x) = 3x + 4\), the derivative \(f'(x)\) equals \(3\). This derivative tells us that no matter which point on the line we choose, the slope will always be \(3\). This concept simplifies tracing the rate of change across linear relationships.

Linear Equations

Linear equations are algebraic expressions where each term is a constant or product of a constant and a single variable. They are often seen as the simplest form of algebraic equations, expressing relationships where a change in one variable results in a constant change in another.

• A standard form of a linear equation is \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept.
• Linear equations form straight lines when graphically represented.

In our problem, \(f(x) = 3x + 4\) is a linear equation. Here, \(3\) represents the slope, and \(4\) is the y-intercept. Linear equations allow us to predict outcomes, make connections across data points, and understand simpler causal relationships. This kind of mathematical tool is invaluable in disciplines ranging from economics to engineering.