Question

A function \(f\) and an \(x\) -value \(a\) are given. Approximate the equation of the tangent line to the graph of \(f\) at \(x=a\) by numerically approximating \(f^{\prime}(a),\) using \(h=0.1 .\) $$f(x)=\frac{10}{x+1}, x=9$$

Short answer

The equation of the tangent line at \( x=9 \) is \( y = -0.1x + 1.9 \).

Step-by-step solution

Identify Key Components

We need to approximate the tangent line to the graph of the function at a given point. The function provided is \( f(x) = \frac{10}{x+1} \) and we are interested in the tangent line at \( x = 9 \). We will numerically approximate the derivative, \( f'(a) \), using \( h = 0.1 \) where \( a = 9 \).

Numerically Approximate the Derivative

To find \( f'(9) \) we use the formula for the numerical derivative: \( f'(a) \approx \frac{f(a+h) - f(a-h)}{2h} \). Setting \( a = 9 \) and \( h = 0.1 \), calculate \( f(9.1) \) and \( f(8.9) \):\( f(9.1) = \frac{10}{9.1+1} = \frac{10}{10.1} \approx 0.9901 \),\( f(8.9) = \frac{10}{8.9+1} = \frac{10}{9.9} \approx 1.0101 \).

Calculate the Derivative

With these function values, substitute into the derivative formula: \( f'(9) \approx \frac{0.9901 - 1.0101}{2 \times 0.1} = \frac{-0.02}{0.2} = -0.1 \). This gives us the approximate derivative at \( x = 9 \).

Find the Equation of the Tangent Line

The equation of the tangent line can be expressed as \( y = m(x-a) + f(a) \), where \( m = f'(a) \) and \( a = 9 \). Substituting the known values, the slope is \( m = -0.1 \) and \( f(9) = 1 \) (calculated as \( \frac{10}{10} \)), thus:\( y = -0.1(x - 9) + 1 \).

Simplify the Tangent Line Equation

Substitute and simplify: \( y = -0.1x + 0.9 + 1 = -0.1x + 1.9 \). This is the equation of the tangent line to the graph of \( f(x) \) at \( x = 9 \).

Key concepts

Tangent Line Approximation

The **Tangent Line Approximation** is a way to estimate the behavior of a curve at a certain point. Imagine you have a curved graph, like a rollercoaster. If you want to know the direction of travel at one specific spot, you might look at the track immediately around the spot. That side section represents the tangent line. Tangent lines can make tricky curves easier to handle because they act like straight-line bits of a curve.

For a function like \[ f(x) = \frac{10}{x+1} \], our task is to determine the tangent line at a specific point, which in this case is at \( x = 9 \). The first key step is understanding that the slope of the tangent line at any point on a function can be found using the derivative of that function. In practice, it's like asking, "How steep is the hill right here?"

Using the concept of tangent, we're able to approximate how the function behaves around \( x=9 \) by pretending it's a simple straight line. This simplifies many mathematical problems.

Derivative Calculation

Calculating the derivative is crucial for finding the slope of the tangent line. The derivative tells us how the function's output (or "y-value") changes as we fiddle with the input (or "x-value"). When the function is complicated, such as \[ f(x)=\frac{10}{x+1} \], we can use the numerical method to estimate this rate of change.

To find the derivative at \( x = 9 \), we use: \[f'(a) \approx \frac{f(a+h) - f(a-h)}{2h} \] Where \( a = 9 \) and \( h = 0.1 \). We calculate the function's value slightly to the right and left of \( x = 9 \) to see how it changes. For this function:

• Calculate \( f(9.1) \) as \( \frac{10}{10.1} \approx 0.9901 \)
• Calculate \( f(8.9) \) as \( \frac{10}{9.9} \approx 1.0101 \)

By plugging these values into our formula, we get an approximate derivative of \( -0.1 \). This tells us the function decreases by 0.1 for each unit increase in \( x \) around \( x = 9 \). The negative sign indicates that we're moving downhill.

Function Analysis

**Function Analysis** is the study of the behavior of functions at specific points, intervals, or domains. With functions like \[ f(x) = \frac{10}{x+1} \], it's about knowing how they behave overall and at specific instances, such as at \( x = 9 \). By understanding function analysis, we grasp not just the isolated values of \( f(x) \), but also trends, shapes, and tendencies.

To glean these insights, we analyze values like:

• The derivative \( f'(9) = -0.1 \)
• The function value \( f(9) = 1 \)

The derivative informs us about the direction and speed of change. It's a potent tool for examining the rate of increase or decrease at particular spots on a graph.

By examining \( f(9) = 1 \), we learn that at this point, the function's value is precisely 1. This anchors our tangent line, allowing us to form the line equation.Using these elements together provides a full view of the curve at \( x = 9 \), giving us a clear picture of its steepness and the tangent path: \[ y = -0.1x + 1.9 \].