Problem 9 - Essential Calculus

Question

Identify the open intervals on which the function is increasing or decreasing. $$ f(x)=\frac{1}{x^{2}} $$

Step-by-Step Solution

Answer

The given function \(f(x)=\frac{1}{x^{2}}\) is increasing on the interval \(-\infty

Step 1: Find the derivative of the function

The derivative of a function at a point gives the rate of change of the function at that point. It is represented as \(f'(x)\). The power rule is used to find the derivative of this function. The power rule states that if \(f(x) = x^n\), then \(f'(x) = n*x^{(n-1)}\). So, the derivative of the function \(f(x)=\frac{1}{x^{2}}\) is \(f'(x)=-2*\frac{1}{x^{3}}\).

Step 2: Find the critical points

Critical points occur where the derivative is zero or undefined. We solve \(f'(x)=-2*\frac{1}{x^{3}}=0\) which yields no solution as the equation \(0=-2*\frac{1}{x^{3}}\) does not have a valid solution. The derivative is undefined at \(x=0\), so the only critical point is \(x=0\).

Step 3: Find the intervals where the function is increasing or decreasing

Having found the critical point, we choose test points from the intervals \(-\infty

Key Concepts

Calculus

Calculus is a branch of mathematics that deals with rates of change and accumulation. At its core, calculus is divided into two major parts: differential calculus, concerning rates of change and slopes of curves, and integral calculus, concerning accumulation of quantities and the areas under and between curves.

In the context of analyzing functions, calculus helps us understand the behavior of functions such as their increasing and decreasing nature over certain intervals. This involves calculating derivatives and understanding their implications, which allows us to interpret the function's rate of change at any given point. By exploring these changes, we can identify whether a function is growing or diminishing over a specific range of values.

Derivative

The derivative of a function is a central concept in calculus that represents the rate at which the function's value changes. In more visual terms, the derivative at a certain point corresponds to the slope of the tangent line to the function's graph at that point.

Finding the derivative, also known as differentiating the function, enables us to predict function behavior. It reveals where the function's graph is ascending or descending and helps locate extrema — maximum and minimum points. The process of differentiation employs various rules and formulas, with the power rule being one of the most fundamental and widely used.

Critical Points

Critical points provide invaluable information about the function's graph. These are the points on the function where the derivative is either zero or does not exist. To find critical points, we look for values of the variable where the derivative equals zero or where the derivative is undefined.

In the context of our exercise, the function given is increasing or decreasing around these critical points. By examining the signs of the derivative on either side of these points, we can conclude about the function's increasing or decreasing behavior. A change in the sign indicates a change from increasing to decreasing or vice versa, which helps us in sketching the graph of the function and understanding its overall behavior.

Power Rule

The power rule is a shortcut for finding the derivative of a function whose terms are all powers of x. According to the power rule, if a term in the function is of the form \(a \times x^n\), the derivative of that term is \(a \times n \times x^{(n-1)}\).

The application of this rule makes differentiation of polynomial-like functions much easier and more efficient, as it allows quick computation of derivatives without the need for more complicated limit processes. For instance, differentiating \(f(x)=\frac{1}{x^{2}}\) using the power rule conveniently results in a derivative that tells us at a glance about the function's decreasing interval for positive values of x and increasing interval for negative values of x.

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Problem 9

Other exercises in this chapter

Problem 9

In Exercises \(9-20\), find the points of inflection and discuss the concavity of the graph of the function. \(f(x)=x^{3}-6 x^{2}+12 x\)

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Problem 9

In Exercises 9 and \(10,\) find \(\lim _{x \rightarrow \infty} h(x),\) if possible. \(f(x)=5 x^{3}-3 x^{2}+10\) (a) \(h(x)=\frac{f(x)}{x^{2}}\) (b) \(h(x)=\frac

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Problem 9

In Exercises 9-16, find any critical numbers of the function. $$ f(x)=x^{2}(x-3) $$

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Problem 10

Find the differential \(d y\) of the given function. $$ y=\sqrt{x}+1 / \sqrt{x} $$

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Key Concepts

Calculus

Calculus is a branch of mathematics that deals with rates of change and accumulation. At its core, calculus is divided into two major parts: differential calculus, concerning rates of change and slopes of curves, and integral calculus, concerning accumulation of quantities and the areas under and between curves.

In the context of analyzing functions, calculus helps us understand the behavior of functions such as their increasing and decreasing nature over certain intervals. This involves calculating derivatives and understanding their implications, which allows us to interpret the function's rate of change at any given point. By exploring these changes, we can identify whether a function is growing or diminishing over a specific range of values.

Derivative

The derivative of a function is a central concept in calculus that represents the rate at which the function's value changes. In more visual terms, the derivative at a certain point corresponds to the slope of the tangent line to the function's graph at that point.

Finding the derivative, also known as differentiating the function, enables us to predict function behavior. It reveals where the function's graph is ascending or descending and helps locate extrema — maximum and minimum points. The process of differentiation employs various rules and formulas, with the power rule being one of the most fundamental and widely used.

Critical Points

Critical points provide invaluable information about the function's graph. These are the points on the function where the derivative is either zero or does not exist. To find critical points, we look for values of the variable where the derivative equals zero or where the derivative is undefined.

In the context of our exercise, the function given is increasing or decreasing around these critical points. By examining the signs of the derivative on either side of these points, we can conclude about the function's increasing or decreasing behavior. A change in the sign indicates a change from increasing to decreasing or vice versa, which helps us in sketching the graph of the function and understanding its overall behavior.

Power Rule

The power rule is a shortcut for finding the derivative of a function whose terms are all powers of x. According to the power rule, if a term in the function is of the form \(a \times x^n\), the derivative of that term is \(a \times n \times x^{(n-1)}\).

The application of this rule makes differentiation of polynomial-like functions much easier and more efficient, as it allows quick computation of derivatives without the need for more complicated limit processes. For instance, differentiating \(f(x)=\frac{1}{x^{2}}\) using the power rule conveniently results in a derivative that tells us at a glance about the function's decreasing interval for positive values of x and increasing interval for negative values of x.