Problem 51 - Calculus an Applied Approach

Question

Use a graphing utility to graph \(f\) and \(f^{\prime}\) on the interval \([-2,2] .\) $$ f(x)=x(x+1) $$

Step-by-Step Solution

Answer

The derivative \(f^{\prime}(x)\) of the function \(f(x)=x(x+1)\) is \(2*x + 1\). Both the functions are graphed with \(f(x)\) being a parabola and \(f^{\prime}(x)\) a straight line.

Step 1: Find the derivative of \(f(x)\)

The derivative of a function can be found using the power rule for differentiation, which states that the derivative of \(x^n\) with regards to \(x\) is \(n*x^{n-1}\). So, applying the power rule to each term in \(f(x) = x(x+1) = x^2 + x\), we get \(f^{\prime}(x) = 2*x + 1\).

Step 2: Graph the original function \(f(x)\)

Now, we can graph \(f(x) = x(x+1) = x^2 + x\). The graph is a parabola that crosses the x-axis at \(x = 0\) and \(x = -1\), and has its minimum point at \(x = -0.5\).

Step 3: Graph the derivative function \(f^{\prime}(x)\)

Next, we graph the derivative function \(f^{\prime}(x) = 2*x + 1\). This is a straight line with a y-intercept at \(y = 1\) and a slope of \(2\). The line crosses the x-axis at \(x=-0.5\).

Key Concepts

Graph of a Function

A graph of a function visually represents how the function behaves over a certain domain. When we graph a function, such as \(f(x) = x(x+1) = x^2 + x\), we're looking at where the points on this curve lie on the coordinate plane.

For quadratic functions like this one, the graph forms a U-shape called a parabola. It opens upwards if the coefficient of the \(x^2\) term is positive, as it is here since it's 1. The parabola for \(f(x)\) crosses the x-axis at \(x = 0\) and \(x = -1\). These are the roots or zeros of the function, meaning those are the solutions where \(f(x) = 0\).

Understanding this graph helps in visualizing the overall behavior of the function, including where it increases or decreases, as well as identifying key features such as intercepts and turning points.

Derivative of a Function

The derivative of a function gives us profound insight into the function's rate of change. Essentially, if \(f(x)\) represents a function, \(f^{\prime}(x)\) represents its derivative, or how quickly \(f(x)\) is changing at any point.

In our example, \(f(x) = x^2 + x\), the derivative, \(f^{\prime}(x)\), is computed to be \(2x + 1\) using differentiation rules. This derivative tells us that the function is increasing wherever \(f^{\prime}(x) > 0\) and decreasing where \(f^{\prime}(x) < 0\). Analyzing \(f^{\prime}(x)\) provides critical points that help in discussing the extrema or the highest and lowest points on the graph.

This information is useful in many calculus problems, including what you need to prepare if solving for optimization problems or analyzing the overall shape of the function's graph.

Power Rule for Differentiation

Differentiating functions can be efficiently done using the power rule, one of the fundamental rules in calculus. It simplifies the process by applying a straightforward formula: if you have a term \(x^n\), its derivative is \(n \cdot x^{n-1}\).

Let's see this rule in action with our function \(f(x) = x^2 + x\). Each term can be handled individually:

• For \(x^2\), applying the power rule gives us \(2 \cdot x^{2-1} = 2x\).
• For \(x\) (which is \(x^1\)), it becomes \(1 \cdot x^{0} = 1\).

The derivative of the entire function, \(f^{\prime}(x) = 2x + 1\), is the sum.

This principle makes differentiation less daunting, especially when dealing with polynomials. It helps provide tools to find rates of change quickly and is a foundational skill in calculus.

Interval Notation

Interval notation is a concise way of expressing a range or collection of numbers. It is especially handy in calculus and helps meet the need for precise expression of intervals on which a function is defined or over which an examination occurs.

For instance, in our exercise, we are interested in the interval \([-2, 2]\). This notation shows that we're considering every number between -2 and 2, inclusive of both endpoints. Here’s a breakdown:

• Brackets [ ] indicate that the endpoints -2 and 2 are included (closed interval).
• If parentheses ( ) were used, it would mean those endpoints are not included (open interval).

This allows for clarity, especially when defining domains and ranges of functions, setting up limits, and solving inequalities.