Question

\mathrm{\\{} T a n g e n t ~ L i n e s ~ \(\quad\) The graph of \(f(x)=-\sin x\) has infinitely many tangent lines that pass through the origin. Use Newton's Method to approximate the slope of the tangent line having the greatest slope to three decimal places.

Short answer

The slope of the tangent line with the greatest positive slope will be obtained by plugging the approximation of x, obtained from Newton's method, into the derivative function \( f'(x) \).

Step-by-step solution

Write Down the Given Function and its Derivative

The given function is \(f(x) = -sin(x)\). The derivative of the function, representing the slope of the line at any given point x, is \(f'(x) = -cos(x)\).

Define the Equation for Tangent Line through the Origin

Any line passing through the origin has the formula \(y = mx\), where m is the slope of the line. Since the line is also tangent to the function, \(y\) can be replaced by \(f(x)\) and \(m\) by \(f'(x)\). So, the equation becomes \( f(x) = f'(x)*x \). Which simplifies to, \( -sin(x) = -cos(x)*x \).

Define the Equation to Apply Newton's Method

To apply Newton's method, the equation should be written in the form of \( g(x) = 0 \). Thus rewrite the equation from step 2 as \( g(x) = sin(x) + cos(x)*x = 0 \).

Apply Newton's Method

Newton's method is an iterative method which uses the formula \( x_{n+1} = x_n - g(x_n) / g'(x_n) \), where \( x_{n+1} \) is the next guess and \( x_n \) is the previous guess. The derivative of g(x) is \( g'(x) = cos(x) - sin(x)*x \). Starting with a guess \( x_0 \), usually 0, you can calculate \( x_1, x_2, x_3, ... \) until the desired level of precision is reached.

Find the Maximum Slope

The derivative of the function, \( f'(x) \), will give the slope of the tangent at any point x. Plug the approximated value of x using Newton's method into \( f'(x) \) to get the maximum slope to three decimal places.

Key concepts

Tangent Lines

Imagine a smooth curve and a line that just touches it at a single point without cutting through. This line is known as a tangent line. In calculus, determining a tangent line to a function's graph at a point involves finding its slope at that point. This concept is crucial for understanding how rates of change work within the function.
A key relationship exists between a function and its tangent lines: the slope of the tangent line at any point is equal to the derivative of the function at that point. This ties the abstract concept of the derivative to a clear geometrical interpretation. If a problem asks for the slope of the tangent line with the greatest slope through a certain point, like the origin, it’s essentially seeking the maximum rate of change of the function when passing through that specific point.

Derivative of a Function

A cornerstone of calculus, the derivative of a function at any point epitomizes the rate at which the function's value is changing at that point. For a function represented as a graph, this rate can be visualized as the slope of a tangent line to the graph at a specific point.
In the context of our exercise, the derivative of the function, noted as \( f'(x) \), equates to the slope of a tangent line at any point \( x \) along the function \( f(x) = -\text{sin}(x) \). Computationally, given a known function, its derivative expresses how rapidly the function's output changes as its input varies slightly. This might all seem quite theoretical, but derivatives have practical applications in fields such as engineering, physics, and economics where changes and rates are essential.

Maximum Slope

Finding the maximum slope of a function's tangent lines is a particular task in calculus. In our example, the maximum slope represents the steepest tangent line passing through the origin on the graph of \( f(x) = -\text{sin}(x) \). To identify this, we look for the peak rate of change of the function, which occurs where its derivative reaches an extreme value.
In practice, when we analyze the derivative of the function, we are essentially exploring the behavior of the tangent lines' slopes. Once we have the derivative, \( f'(x) = -\text{cos}(x) \), we can employ various methods to ascertain where this derivative has its maximum (or minimum) value. The maximum value of the slope corresponds to the sharpest angle at which the tangent line ascends or descends the curve.

Iterative Methods

Iterative methods constitute a category of algorithms that solve problems through repetition, gradually honing in on a solution by iterations. Newton's Method, a famous iterative technique in calculus, solves equations by repeatedly updating an initial guess. It can quickly zero in on a solution, given a function that we can differentiate.
When applying Newton’s Method, we iteratively estimate the zero of a function—meaning, we're looking for the input value that would give an output of zero. The sequence of approximations is generated using the formula \( x_{n+1} = x_n - g(x_n) / g'(x_n) \), with each iteration ideally moving closer to the actual zero. This method is particularly useful for finding extreme points of a function, like maximum slopes, by setting up an equation where the zero corresponds to an extreme value of the derivative.