Problem 7 - Active Calculus

Question

For each of the following prompts, sketch a graph on the provided axes of a function that has the stated properties. a. \(y=f(x)\) such that - \(f(-2)=2\) and \(\lim _{x \rightarrow-2} f(x)=1\) \- \(f(-1)=3\) and \(\lim _{x \rightarrow-1} f(x)=3\) \- \(f(1)\) is not defined and \(\lim _{x \rightarrow 1} f(x)=0\) \- \(f(2)=1\) and \(\lim _{x \rightarrow 2} f(x)\) does not exist. b. \(y=g(x)\) such that \- \(\text { - } g(-2)=3, g(-1)=-1, g(1)=-2, \text { and } g(2)=3\) \- At \(x=-2,-1,1\) and \(2, g\) has a limit, and its limit equals the value of the function at that point. \- \(g(0)\) is not defined and \(\lim _{x \rightarrow 0} g(x)\) does not exist.

Step-by-Step Solution

Answer

From given points and limits, y=f(x) has discontinuities at -2, 1, and 2 with specific limits; y=g(x) is continuous at -2, -1, 1, and 2 but discontinuous at 0.

Step 1: Analyze given points and limits of y=f(x)

First, identify the points: 1. At \(x = -2, f(-2) = 2\), but \(\lim _{x \rightarrow -2} f(x) = 1\). 2. At \(x = -1, f(-1) = 3\) and \(\lim _{x \rightarrow -1} f(x) = 3\). 3. At \(x = 1\), \(f(1)\) is not defined, and \(\lim _{x \rightarrow 1} f(x) = 0\). 4. At \(x = 2, f(2) = 1\) and \(\lim _{x \rightarrow 2} f(x)\) does not exist.

Step 2: Sketch y=f(x) for f(-2) and its limit

Mark a point at \(x = -2\) with \(f(-2) = 2\). Also, indicate that as \(x\) approaches \( -2\), the function approaches \(y = 1\) by drawing a point at \(x = -2, y = 1\) and a small open circle around it.

Step 3: Sketch y=f(x) for f(-1) and its limit

At \(x = -1\), draw a point at \(f(-1) = 3\). Since the limit as \(x\) approaches \(-1\) is \(3\), there is no discontinuity here.

Step 4: Sketch y=f(x) for f(1) and its limit

For \(x = 1\), indicate that \(f(1)\) is not defined by leaving a blank space or an open circle around \(x = 1\). However, the limit as \(x\) approaches \(1\) is \(y = 0\), so draw the approaching lines or curve towards \( (1,0) \) but leave \( (1,0) \) open or undefined.

Step 5: Sketch y=f(x) for f(2) and its limit

At \(x = 2\), draw a point at \(f(2) = 1\). Since the limit does not exist, show that approaching from different sides toward \(x = 2\) yields different values. Draw two different functions heading to that \(x\) point.

Step 6: Summary for y=f(x)

Combine all details to complete the sketch of \(y = f(x)\) with discontinuities at \(x = -2\), \(x = 1\), and different limits at \(x = 2\), ensuring the function aligns with given values and limits.

Step 7: Analyze given points and limits of y=g(x)

Identify the points: 1. At \(x = -2, g(-2) = 3\). 2. At \(x = -1, g(-1) = -1\). 3. At \(x = 1, g(1) = -2\). 4. At \(x = 2, g(2) = 3\). At these points, the limit equals the value of the function.

Step 8: Sketch y=g(x) for each point

Plot points at \(x = -2, y = 3\); \(x = -1, y = -1\); \(x = 1, y = -2\); and \(x = 2, y = 3\). Indicate that the limits at these points equal the function values by making smooth connections or dots indicating no discontinuity at these points.

Step 9: Sketch y=g(x) around x=0

For \(x = 0\), indicate that \(g(0)\) is not defined and the limit does not exist. Leave an open space or no connection around \(x = 0\), showing a discontinuity or break in the graph.

Step 10: Summary for y=g(x)

Combine all plotted points and the discontinuity at \(x = 0\) to complete the graph. Ensure it meets all specified conditions.

Key Concepts

limit of a function

In calculus, the limit of a function is the value that the function approaches as the input (or the independent variable) approaches a certain value. Limits are essential for defining concepts like continuity, derivatives, and integrals. The notation for the limit of a function as x approaches a specific value is written as \(\text{lim}_{x \to c} f(x) = L\). Here, if x gets closer to c, then f(x) gets closer to L.
For example, in the given exercise, let's consider the limit for the function y = f(x) at x = -2. The function value is f(-2) = 2, but the limit as x approaches -2 is 1. This tells us that as x gets very close to -2, the nearby values of the function f(x) are near 1, not 2.
Limits are fundamental for understanding more complex calculus topics and play a crucial role in graph sketching.

discontinuous functions

A function is discontinuous where it isn’t smooth or uninterrupted. Discontinuities are points on the graph where the function is not defined or takes a sudden jump. There are various types of discontinuities, including point discontinuities, jump discontinuities, and infinite discontinuities.
In the provided exercise, y = f(x) has discontinuities at specified points:

• At x = -2: the function value is 2, but the limit is 1. This jump from 2 to 1 represents a discontinuity.
• At x = 1: the function is not defined, but the limit as x approaches 1 is 0. This creates a gap in the graph.
• At x = 2: the function value is 1 but the limit does not exist indicating another type of discontinuity.

Understanding these discontinuities helps in accurately sketching the graph.

function properties

A function's properties refer to the characteristics and behaviors of the function. These include continuity, limits, derivatives, and more. For graph sketching, key properties are:

• Points and intervals where the function is continuous or discontinuous.
• Where the function has specific values and how it behaves near those points.

For example, in the exercise’s y = g(x):

• At x = -2, g(-2) = 3 and the limit is also 3, indicating no discontinuity.
• At x = 0, the function is not defined, and the limit does not exist, creating a notable gap in the graph.

Being aware of these properties aids in sketching and interpreting graphs accurately, ensuring all given conditions are met.