Problem 18 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the slope of the tangent line to the graph of each function at the given point and determine an equation of the tangent line. \(f(x)=-3 x+4\) at \((-1,7)\)

Step-by-Step Solution

Answer

The slope of the given function \(f(x) = -3x + 4\) is -3. Using the point-slope form with the given point \((-1, 7)\), we find the equation of the tangent line to be \(y = -3x + 4\), which is the same as the original linear function, as expected.

Step 1: Find the slope of the linear function

For a linear function in the form of \(f(x) = mx + b\), the slope (m) is the coefficient of x. In the given function, \(f(x) = -3x + 4\), the slope is -3.

Step 2: Use the point-slope form

To find the equation of the tangent line at point \((-1, 7)\), we can use the point-slope form of a linear equation, which is: \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is the given point and m is the slope. Substitute the values into the equation: \(y - 7 = -3(x - (-1))\)

Step 3: Simplify the equation

Simplify the equation by distributing the slope and combining like terms: \(y - 7 = -3(x + 1)\) => \(y - 7 = -3x - 3\) Now, add 7 to both sides to isolate y: \(y = -3x - 3 + 7\)

Step 4: Write the final equation

Write down the final equation: \(y = -3x + 4\) Notice that the equation of the tangent line is the same as the equation of the original linear function, which is expected since the tangent lines for a linear function are always the same as the function itself.

Key Concepts

Linear Functions

Imagine a straight path stretching infinitely in two directions. The mathematical version of this path is what we call a linear function. A linear function is a polynomial with a degree of one, and its graph forms a straight line. The most common form of writing a linear function is the slope-intercept form, which looks like this:

\[ y = mx + b \]

In this equation, \(m\) represents the slope, which indicates how steep the line is and which direction it goes. A positive slope means the line rises as it moves from left to right, while a negative slope means it falls. The \(b\) represents the y-intercept, the point where the line crosses the y-axis. For the given exercise, we focused on the line described by \(f(x) = -3x + 4\), which has a slope of -3 and a y-intercept of 4. Understanding the slope and y-intercept gives us a visual representation of the function and forms the basis for finding tangent lines, as they describe the rate of change and position of the linear function at any given point.

Point-Slope Form

While the slope-intercept form is useful for quickly graphing a line, another powerful tool in our mathematical toolbox is the point-slope form. Particularly handy for finding the equation of a line when we know its slope and one point on the line, the point-slope form is expressed as:

\[ y - y_1 = m(x - x_1) \]

Here, \((x_1, y_1)\) represents the coordinates of the known point and \(m\) is the slope of the line. In the context of the exercise, we applied the point-slope form using the point (-1, 7) and the slope -3 to draft the equation of the tangent line. This form helps us create the equation of a line without needing to calculate the y-intercept directly, making it exceptionally useful when dealing with tangent lines, which touch a curve at a single, precise location.

Tangent Line Equations

If you've ever slightly pushed your coffee cup to slide it closer, the movement started with just one point of the cup edge touching the table. Now, imagine this one point is where a curve meets a straight line, and you've got the essence of a tangent line. A tangent line touches a curve at exactly one point, and the slope of this line equals the slope of the curve at that point.

For non-linear functions, finding the slope of the tangent line involves calculus, specifically taking derivatives. However, for linear functions, the concept of a tangent line is somewhat simplified as the line itself is its tangent at every point. In our exercise, we found that since our given function is linear, the slope of the tangent is the same as the slope of the function, and therefore the equation of the tangent line matches the equation of the function itself. This is a unique scenario with linear functions because their rate of change is constant. Hence, the tangent line's equation is a straightforward insight into the behavior of the function at that one touching point.