Problem 45 - Calculus an Applied Approach

Question

Use the limit definition to find an equation of the tangent line to the graph of \(f\) at the given point. Then verify your results by using a graphing utility to graph the function and its tangent line at the point. $$ f(x)=\frac{1}{x} ;(1,1) $$

Step-by-Step Solution

Answer

The equation of the tangent line to the function \(f(x)=\frac{1}{x}\) at the point (1,1) is \(y = 2 - x\).

Step 1: Identify the given function and the point

The given function is \(f(x) = \frac{1}{x}\) and the point of tangency is (1,1). We can mark that the point of tangency is \((x_0, y_0) = (1,1)\).

Step 2: Find the derivative of the given function

The derivative of a function gives the slope of the tangent line at any point. So, we find \(f'(x)\) to get the slope. The derivative of \(f(x)\) is \(f'(x)=-\frac{1}{x^2}\).

Step 3: Substitute the x-value of the point into the derivative

To get the slope of the tangent line at the specific point, we substitute \(x_0\) into \(f'(x)\) to get \(m\): \(m=f'(x_0)=-\frac{1}{{x_0}^2}\). Substituting \(x_0 = 1\), we get \(m=-1\). So, the slope of the tangent line at the point (1,1) is -1.

Step 4: Use point-slope form to find the equation of the tangent line

The point-slope form of a line is \(y - y_0 = m(x - x_0)\). Substituting the known values \((x_0, y_0) = (1, 1)\) and \(m = -1\), the equation of the line is \(y - 1 = -1(x - 1)\), simplifying we have \(y = 2 - x\).

Step 5: Validate using a graphing utility

Plot the given function \(f(x) = \frac{1}{x}\) and the found tangent line equation \(y = 2 - x\) in a graphing utility. The line should just touch the graph at the point (1,1).

Key Concepts

Tangent Line

In calculus, a tangent line to a curve at a specific point is a straight line that just "touches" the curve at that point. Imagine the tangent line as the best linear approximation of the function at that particular spot. It represents the direction in which the curve is heading right at that moment. Simply put, it tells us the slope or steepness of the curve at that point.
The tangent line is important because it shows how the curve behaves locally.

• It allows for predictions about the behavior of the function near an interesting point.
• For the function \(f(x) = \frac{1}{x}\) at the point (1, 1), the tangent line gives valuable insights about the curve's properties at \(x = 1\).

Understanding tangent lines aids in both theoretical calculus and practical applications like physics or engineering problems, where rates of change are crucial.

Limit Definition

The concept of limits is fundamental to calculus. It's what allows us to deal with continuous change. The limit helps define the derivative, which is the slope of the tangent line to the function. The limit essentially describes how a function behaves near a certain point.
In mathematical terms, when we say we are "taking the limit" of a function as \(x\) approaches a particular value, we are looking at the value that the function is approaching. This helps in calculating instantaneous rates of change.

For instance, the limit definition of a derivative is expressed as follows:

• \( \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h} = f'(x) \)

This expression is telling us to look at what happens as the interval \(h\) between two points on the curve shrinks towards zero. It's the core of finding derivatives and understanding the behavior of curves.

Derivative

The derivative of a function at a certain point is the slope of the tangent line at that point. Simply put, it represents how the function is changing at that precise moment. Mathematically, the derivative offers a way to measure how a function's output value changes as its input changes, giving you the rate of change.
Finding the derivative is crucial to solving the problem because

• It helps identify where functions are increasing or decreasing.
• It reveals maximum or minimum points of a function, which are vital in optimization problems.

For the function \(f(x) = \frac{1}{x}\), the derivative is \(f'(x) = -\frac{1}{x^2}\). Substituting \(x = 1\) gives the slope of the tangent line at that point as \(-1\). This value is essential to determine the tangent line's equation.

Point-Slope Form

The point-slope form of a line is a straightforward way to write the equation of a line if you know one point on the line and the slope. The general equation is given by:

• \(y - y_0 = m(x - x_0)\)

Where \((x_0, y_0)\) is a specific point on the line and \(m\) is the slope. This form is especially useful in calculus when working with tangent lines.
Once we find the slope using the derivative, we use this point-slope form to write the equation of the tangent line. For example, with a slope \(m = -1\) at point \((1, 1)\), we substitute into the point-slope formula to get the line's equation:

• \(y - 1 = -1(x - 1)\)

Simplifying this, the equation of the tangent line becomes \(y = 2 - x\). Point-slope form is intuitive and essential for identifying tangent lines on curves.