Problem 2 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Suppose we want to use a tangent line approximation of \(f(x)=\sqrt{x}\) at \(x=a\) to approximate a particular square root numerically. Which values of \(a\) should we choose to approximate each of the following? (a) \(\sqrt{102}\) (b) \(\sqrt{8}\) (c) \(\sqrt{18}\) (d) \(\sqrt{115.5}\)

Step-by-Step Solution

Answer

For \(\sqrt{102}\), use \(a = 100\), for \(\sqrt{8}\), use \(a = 9\), for \(\sqrt{18}\), use \(a = 16\), and for \(\sqrt{115.5}\), use \(a = 121\).

Step 1: Understanding and Identifying the Concept

To solve this problem, use the tangent line approximation concept which is a method to approximate the values of a function using the tangent line at a certain point. The method relies on the tangent line as a close approximation to the function around the point of tangency. The function here is the square root function \(f(x) = \sqrt{x}\), and we should select the tangent point (\(a\), \(f(a)\)) so that it's close to the point at which we want to approximate the function.

Step 2: Calculate the Approximate Value for (a)

The point at which to draw the tangent line should be the closest perfect square to the given values. This provides a good approximation because the function is most like its tangent line near the point of tangency. Let's calculate each part of the question: (a) For \(\sqrt{102}\), use \(a = 100\) because it is the closest perfect square to 102. (b) For \(\sqrt{8}\), use \(a = 9\) as it is the closest perfect square to 8. (c) For \(\sqrt{18}\), use \(a = 16\) because it is the nearest perfect square to 18. (d) For \(\sqrt{115.5}\), use \(a = 121\) as it is the nearest perfect square to 115.5.

Key Concepts

Square Root Function

The square root function is one of the fundamental mathematical operations that helps us find a number which, when multiplied by itself, gives the original number. In mathematical terms, the square root of a number \(x\) is denoted by \(\sqrt{x}\). For example, if \(x = 9\), then \(\sqrt{9} = 3\) because \(3 \times 3 = 9\).
The square root function is commonly graphed as a curve, starting from the origin and increasing steadily. It's important because it arises in various scientific and engineering problems, often relating to areas and volumes. Understanding the square root function helps in topics such as quadratic equations and geometry.
Moreover, the square root function is non-linear, which means it doesn't have a constant rate of change. This characteristic makes it more challenging to work with, especially in approximation. By using tools like calculus or approximation methods, we can estimate values of the square root function even when they aren’t perfect squares.

Perfect Squares

Perfect squares are essential for understanding the approximation of non-perfect square roots. A perfect square is a number that is the square of an integer. For example, \(1, 4, 9, 16, 25\), and so on are perfect squares because they are the square of \(1, 2, 3, 4, 5\) respectively.
Calculating the square root of a perfect square is straightforward since the result is a whole number. However, things get trickier when dealing with non-perfect squares. Suppose you need to find \(\sqrt{10}\), knowing that \(\sqrt{9} = 3\) and \(\sqrt{16} = 4\), 10 is not exactly a perfect square. In such situations, knowing the close perfect squares provides a good baseline which is helpful to approximate the non-perfect square root.
By identifying the nearest perfect square, we simplify the calculations involved in approximations like the tangent line method, as illustrated in solving square root approximation problems.

Numerical Approximation

Numerical approximation is a method used to estimate the values of functions at points where exact calculation is difficult. For example, when computing the square root of numbers that are not perfect squares, numerical methods like tangent line approximation can be very useful.
Tangent line approximation makes use of the tangent line of a function at a particular point to estimate the value of the function at nearby points. This is based on the assumption that, near the point of tangency, the function behaves similarly to its tangent line, making the tangent a good local approximation. The method becomes more accurate as the point of approximation moves closer to the point of tangency.
For instance, when approximating \(\sqrt{102}\), we select 100 as the point of tangency because it is a perfect square and very close to 102, allowing us to use its tangent line for approximation. This approach helps us get quick and useful estimates, especially when exact computation is not straightforward.

Function Approximation

Function approximation is a broad mathematical concept of finding simpler functions that closely resemble more complex functions at specified points. This technique is pivotal when computations need to be simplified or exact solutions are difficult to ascertain.
The tangent line approximation is a form of linear approximation used frequently for estimating function values. If we have a curve, the tangent line at a point provides a straight-line approximation which is easier to work with. For the square root function \(f(x) = \sqrt{x}\), finding the tangent line involves determining the derivative, which tells us the slope of the tangent line.
Once the slope is known, we can construct a line equation that approximates the square root function near the chosen point. This is particularly useful in scenarios involving small deviations around a perfect square, offering insights into how the function values change gradually and allowing us to make educated guesses about function behaviors near these points.