Problem 30 - Technical Mathematics with Calculus

Question

Find the slope of the tangent to the curve \(y=1 /(x+1)\) at \(x=2\).

Step-by-Step Solution

Answer

The slope of the tangent to the curve at \( x=2 \) is \( -\frac{1}{9} \).

Step 1: Determine the derivative of the function

To find the slope of the tangent to the curve at a specific point, we first need to determine the derivative of the function with respect to x. In this case, the function is given by \( y = \frac{1}{x+1} \). Using the power rule, the derivative of \( y \) with respect to \( x \) is \( y' = -\frac{1}{(x+1)^2} \).

Step 2: Evaluate the derivative at the given point

Now, we need to find the value of the derivative at \( x=2 \). By plugging \( x=2 \) into the derivative function \( y' = -\frac{1}{(x+1)^2} \), we get \( y'(2) = -\frac{1}{(2+1)^2} = -\frac{1}{3^2} = -\frac{1}{9} \).

Step 3: Conclude the slope of the tangent

The slope of the tangent to the curve at the point where \( x=2 \) is equal to the value of the derivative at that point. Hence, the slope is \( -\frac{1}{9} \).

Key Concepts

Derivative Calculus

When discussing calculus, one of the fundamental concepts is the derivative. It measures how a function's output changes as its input changes. In simpler terms, if you imagine a curve on a graph, the derivative at any point on the curve tells us the slope of the tangent line to the curve at that point.

The process of finding a derivative is known as differentiation. For a function like \( y=f(x) \), the derivative is often denoted as \( f'(x) \) or \( \frac{dy}{dx} \), and it represents the rate of change of \( y \) with respect to \( x \). This concept is incredibly powerful because it allows us to understand the behavior of functions, optimize real-world problems, and even predict future outcomes in various scientific fields.

To calculate a derivative, you can use several rules and techniques, such as the power rule, the product rule, or the chain rule depending on the complexity of the function you have.

Power Rule Differentiation

The power rule is one of the simplest and most useful tools in differentiation. It is used for finding the derivative of a function in the form of \( x^n \), where \( n \) is any real number. The rule states that the derivative of \( x^n \) is \( nx^{n-1} \).

For instance, if we have a function \( f(x) = x^4 \), according to the power rule, the derivative \( f'(x) = 4x^3 \). This rule significantly simplifies the process of differentiation, especially when dealing with polynomial functions or when they are a part of more complex expressions.

In the textbook exercise we're considering, the power rule is used in a slightly more sophisticated version because the function \( y = \frac{1}{x+1} \) can be rewritten as \( y = (x+1)^{-1} \). Applying the power rule, the derivative is \( y' = -1(x+1)^{-1-1} = -\frac{1}{(x+1)^2} \), which is exactly what we need to find the slope of the tangent.

Slope of a Curve

The slope of a curve at a particular point is essentially the steepness and direction of a line that just touches the curve at that point, this line is what we call a 'tangent'. The concept of a slope is critical when it comes to analyzing the behavior of curves, whether they're representing journeys, growth trends, or any variable changes.

A positive slope means the line is rising as it moves from left to right, while a negative slope indicates it's falling. The steeper the slope, the greater the change in the Y-direction for a given change in the X-direction. If the slope is zero, the tangent line is horizontal, indicating that there is no change in \( y \) as \( x \) changes at that point.

To find the slope of the curve at a specific point, you compute the derivative and then evaluate it at that point, as illustrated in the textbook exercise for \( y=\frac{1}{x+1} \) at \( x=2 \). By evaluating the derivative at this point, we determined the slope of the tangent to be \( -\frac{1}{9} \), indicating a slight decline in the curve at this point.