Problem 56 Make a rough sketch of the curve... [FREE SOLUTION]

Chapter 2: Problem 56

Make a rough sketch of the curve $y=x^{n}(n$ an integer) for the following five cases: $$\begin{array}{ll}{\text { (i) } n=0} & {\text { (ii) } n>0, n \text { odd }} \\\\{\text { (iii) } n>0, n \text { even }} & {\text { (iv) } n<0, n \text { odd }} \\ {\text { (v) } n<0, n \text { even }}\end{array}$$ Then use these sketches to find the following limits. $$ \begin{array}{ll}{\text { (a) } \lim _{x \rightarrow 0^{+}} x^{n}} & {\text { (b) } \lim _{x \rightarrow 0^{-}} x^{n}} \\ {\text { (c) } \lim _{x \rightarrow \infty} x^{n}} & {\text { (d) } \lim _{x \rightarrow-\infty} x^{n}}\end{array} $$

Short Answer

Expert verified

Limits: (a) $ y=1,0,\infty $; (b) $ y=1,0,\infty $; (c) $ y=1,\infty,0 $; (d) $ y=1,\infty,0 $.

Step by step solution

Case (i)

For $ n=0 $, the function is $ y=x^0=1 $. The curve is a horizontal line at $ y=1 $ for all $ x $. This line doesn't depend on $ x $ being positive or negative.

Case (ii)

For $ n>0 $ and $ n $ odd, the function is $ y=x^n $. This is an odd-powered function, which means the curve passes through the origin and remains in the first and third quadrants. It has rotational symmetry about the origin. For example, for $ y=x^3 $, as $ x \to +fty $, $ y \to +fty $ and as $ x \to -fty $, $ y \to -fty $.

Case (iii)

For $ n>0 $ and $ n $ even, the function is $ y=x^n $. The curve is symmetrical about the y-axis because it is even. For example, $ y=x^2 $ creates a parabola opening upwards. As $ x \to +fty $ or $ x \to -fty $, $ y \to +fty $, and as $ x \to 0 $, $ y \to 0 $.

Case (iv)

For $ n<0 $ and $ n $ odd, the function is $ y=x^n $ (e.g., $ y=x^{-1} = \frac{1}{x} $). The curve is again in the first and third quadrants with rotational symmetry through the origin, forming a hyperbola. As $ x \to 0 $, $ y \to fty $ or $ -fty $ depending on the direction, and as $ x \to fty $ or $ -fty $, $ y \to 0 $.

Case (v)

For $ n<0 $ and $ n $ even, the function is $ y=x^n $ (e.g., $ y=x^{-2} = \frac{1}{x^2} $), forming a hyperbola that is symmetric about the y-axis. As $ x \to 0 $, $ y \to fty $, and as $ x \to fty $ or $ -fty $, $ y \to 0 $.

Solve Limits (a) and (b)

Using the sketches, determine the limits as $ x $ approaches zero. For **(a) $ \lim_{x \to 0^+} x^n $**: - $ n=0 $, $ y=1 $- $ n>0 $, $ y=0 $- **For $ n<0 $, $ y \to fty $ (even $ n $) or diverges (odd $ n $)**For **(b) $ \lim_{x \to 0^-} x^n $**:- $ n=0 $, $ y=1 $- $ n>0 $, $ y=0 $ (even $ n $) or diverges (odd $ n $)- **For $ n<0 $, $ y \to fty $ (even $ n $) or diverges (odd $ n $)**

Solve Limits (c) and (d)

Determine the limits as $ x \to \infty $ or $ x \to -\infty $: For **(c) $ \lim_{x \to \infty} x^n $:**- $ n=0 $, $ y=1 $- $ n>0 $, $ y=\infty $- $ n<0 $, $ y=0 $For **(d) $ \lim_{x \to -\infty} x^n $:**- $ n=0 $, $ y=1 $- $ n>0 $, $ y=\infty $ (even $ n $) or $ y \to -\infty $ (odd $ n $)- $ n<0 $, $ y=0 $

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Limits at Infinity

Understanding limits at infinity involves exploring how a function behaves as its variable approaches very large or very small values. In power functions such as $ y = x^n $ with integer $ n $, the limits as $ x \to \infty $ or $ x \to -\infty $ provide insight into the end behavior of the graph.
For example:

When $ n = 0 $, the function is constant and the limit is 1 for all $ x $.
For positive $ n $, the graph of the power function extends to $ \infty $ as $ x \to \infty $, as well as when $ x \to -\infty $ if $ n $ is even.
For negative $ n $, the graph approaches zero as $ x \to \pm \infty $, reflecting the decreasing nature of the function as $ x $ grows.

These observations help determine the trend and direction of polynomial graphs as they stretch into the extremes of the coordinate system.

Symmetry of Functions

Symmetry in functions can simplify the sketching of graphs. Reflecting across an axis or about a point gives information on the function's nature through algebraic properties.
For example:

An even function, such as $ y = x^2 $, maintains symmetry about the y-axis. This can be seen as the left and right sides of the graph are mirror images.
An odd function, like $ y = x^3 $, exhibits rotational symmetry about the origin. Here, substituting $ -x $ yields $ -(x^3) $, confirming this property.
Recognizing these symmetries can simplify expectations of the graph's shape without plotting multiple points.

Understanding symmetry aids in predicting function behavior and seeing patterns in mathematical models.

Polynomial Behavior

Polynomial behavior considers how the power (degree) of $ x $ truly impacts the form and flexibility of the function graph.
Here are the basics:

A zero power ($ n = 0 $) results in a constant function, which is a horizontal line, since $ y = 1 $.
Positive powers (even $ n $) like $ x^2 $ yield a parabola shape, with the ends going up as $ x \to \pm \infty $.
Odd positive powers cause curves that cross the origin, unlike the parabolic forms.
Negative powers induce divided or hyperbolic shapes, with the graphs approaching the axes but never touching them.

Mapping polynomial behavior provides a foundation for predicting and understanding more complex polynomial functions.

Graph Sketching

Graph sketching is the art of translating algebraic equations into visual representations. Power functions are excellent candidates for sketching due to their clear mathematical properties.
When sketching a graph:

Begin by identifying symmetry and behavior at limits.
Understand how the degree $ n $ impacts the location of lines and curves, particularly through the coefficients and sign of $ n $.
Use key points such as intercepts and extreme values to guide the overall shape of the curve.
Consider the presence of asymptotes in cases of negative powers, where the graph can't intersect certain lines.

This process clarifies the interaction between algebra and geometry in function representation.

Positive and Negative Powers

The distinction between positive and negative powers in functions extends beyond simple graph variance, influencing almost all aspects of mathematical interpretation.
For power functions:

Positive powers $ (x^n, n > 0) $ suggest extension and growth, with graphs rising indefinitely in certain domains.
Negative powers $ (x^n, n < 0) $ indicate reciprocal relationships, which tend towards zero as the independent variable grows but create undefined conditions at $ x = 0 $.
Even and odd factors play roles in both positive and negative powers, dictating symmetrical behavior or inclines at zero-crossing points.

Grasping these nuances provides insights into wider applications of mathematical analysis and real-world interpretations.

91影视

Short Answer

Step by step solution

Case (i)

Case (ii)

Case (iii)

Case (iv)

Case (v)

Solve Limits (a) and (b)

Solve Limits (c) and (d)

Key Concepts

Limits at Infinity

Symmetry of Functions

Polynomial Behavior

Graph Sketching

Positive and Negative Powers

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