Problem 47 - Precalculus with Limits

Question

In Exercises 43-50, (a) find the slope of the graph of \(f\) at the given point, (b) use the result of part (a) to find an equation of the tangent line to the graph at the point, and (c) graph the function and the tangent line. \(f(x)= \sqrt{x+1}, \quad (3, 2)\)

Step-by-Step Solution

Answer

The slope of the tangent line at (3,2) is 1/4. The equation of the tangent line is \(y = 1/4x + 5/4\).

Step 1: Finding the derivative of the function

The derivative of a function at a specific point gives the slope of the tangent at that point. Here, we have \(f(x) = \sqrt{x+1}\) which is same as \(f(x) = (x+1)^{1/2}\). By applying the power rule, the derivative \(f'(x)\) of \(f(x)\) would be \(1/2(x+1)^{-1/2} = 1/2\sqrt{x+1}\).

Step 2: Finding the slope of the tangent

Substitute \(x = 3\) into the derivative found in the previous step to find the slope of the tangent line at the point (3,2). Hence, \(f'(3) = 1/2\sqrt{3+1} = 1/4\). So, the slope of the tangent line is 1/4.

Step 3: Finding the equation of the tangent line

The equation of a line can be represented as \(y = mx + c\) where m is the slope. As the tangent line touches the curve at (3,2), we substitute \(x = 3, y = 2, m = 1/4\) into \(y = mx + c\) and solve for c. Thus, after substituting, we have, \(2 = 1/4*3 + c\), from which you get that \(c = 2 - 3/4 = 5/4\). Hence, the equation of the tangent line is \(y = 1/4x + 5/4\).

Step 4: Graph the function and the tangent line

Plot the original function \(f(x) = \sqrt{x+1}\) and the tangent line \(y = 1/4x + 5/4\). To plot these, choose various values for x and calculate the corresponding y-values.

Key Concepts

Derivative Calculation

The process of finding a derivative is known as derivative calculation. In the realm of calculus, a derivative represents the rate at which a function is changing at any given point. For the function in our exercise, the function is written as \(f(x) = \sqrt{x+1}\), which can be rewritten as a power function \(f(x) = (x+1)^{1/2}\). To calculate the derivative, we apply the power rule, which states that the derivative of \(x^n\) is \(nx^{n-1}\). Applying this rule to our function, we get \(f'(x) = \frac{1}{2}(x+1)^{-\frac{1}{2}}\), or, simplified, \(f'(x) = \frac{1}{2}\sqrt{x+1}\).

This derivative function \(f'(x)\) now allows us to find the slope at any point along the curve. For instance, at \(x = 3\), substituting this value into \(f'(x)\) gives us \(f'(3) = \frac{1}{4}\), which signifies how steep the curve is at that specific point. Derivative calculation is the foundational tool used in determining such slopes and is pivotal in understanding the behavior of functions.

Slope of the Tangent

The slope of the tangent line at a given point on a function's graph provides us with a numerical description of the function's instantaneous rate of change at that point. In the context of our textbook exercise, after finding the derivative \(f'(x)\), we evaluate it at the point \(x = 3\) to determine the slope of the tangent. We found that \(f'(3) = \frac{1}{4}\), confirming that the tangent line rises one unit for every four units it runs horizontally.

Understanding the slope of the tangent is critical for visualizing how a function behaves and for creating precise sketches of the function's graph. This value not only aids in understanding the function's instantaneous behavior but also is integral in constructing equations and models in physics, economics, and various fields of engineering. Remember, the steeper the slope, the faster the function is increasing or decreasing at that point.

Graphing Functions

Graphing functions is a visual way of understanding mathematical relationships. In this exercise, the function \(f(x) = \sqrt{x+1}\) and its tangent line at \(x = 3\), \(y = \frac{1}{4}x + \frac{5}{4}\), are plotted on a coordinate plane. To graph these, we would select a range of \(x\)-values and compute the corresponding \(y\)-values. Plot these coordinates and form a smooth curve for \(f(x)\), and a straight line for the tangent.

Graphing the function alongside its tangent line is an instructive approach to visualize where the line touches the function and how it corresponds to the slope calculated earlier. By comparing the two, one can easily see the point of tangency and the behavior of the function and the line near that point. This exercise not only solidifies understanding of the concept of slopes and tangents but also grounds abstract mathematical concepts in visual reality.