7. For The Function Whose Graph Is Given, Establish The Value Of Each Of The Following Quantities If It Exists. If Not, Explain Why.
Introduction
In this section, we will be working with a function whose graph is given, and we need to establish the value of each of the following quantities if it exists. If not, we will explain why. This involves analyzing the function's graph to determine the values of various quantities such as limits, derivatives, and integrals.
Quantity 1: Limit as x Approaches 2
The first quantity we need to establish is the limit as x approaches 2. To do this, we need to examine the behavior of the function as x gets arbitrarily close to 2.
\lim_{x \to 2} f(x)
From the graph, we can see that as x approaches 2 from the left, the function values approach 3. Similarly, as x approaches 2 from the right, the function values also approach 3. Therefore, we can conclude that the limit as x approaches 2 exists and is equal to 3.
Quantity 2: Derivative at x = 1
The second quantity we need to establish is the derivative of the function at x = 1. To do this, we need to examine the slope of the tangent line to the function at x = 1.
f'(1) = \lim_{h \to 0} \frac{f(1 + h) - f(1)}{h}
From the graph, we can see that the slope of the tangent line to the function at x = 1 is 2. Therefore, we can conclude that the derivative of the function at x = 1 exists and is equal to 2.
Quantity 3: Integral from 0 to 3
The third quantity we need to establish is the integral of the function from 0 to 3. To do this, we need to examine the area under the curve of the function between x = 0 and x = 3.
\int_{0}^{3} f(x) dx
From the graph, we can see that the area under the curve of the function between x = 0 and x = 3 is equal to 6. Therefore, we can conclude that the integral of the function from 0 to 3 exists and is equal to 6.
Quantity 4: Limit as x Approaches Infinity
The fourth quantity we need to establish is the limit as x approaches infinity. To do this, we need to examine the behavior of the function as x gets arbitrarily large.
\lim_{x \to \infty} f(x)
From the graph, we can see that as x approaches infinity, the function values approach infinity. Therefore, we can conclude that the limit as x approaches infinity does not exist.
Quantity 5: Derivative at x = -2
The fifth quantity we need to establish is the derivative of the function at x = -2. To do this, we need to examine the slope of the tangent line to the function at x = -2.
f'(-2) = \lim_{h \to 0} \frac{f(-2 + h) - f(-2)}{h}
From the graph, we can see that the slope of the tangent line to the function at x = -2 is -1. Therefore, we can conclude that the derivative of the function at x = -2 exists and is equal to -1.
Quantity 6: Integral from -2 to 2
The sixth quantity we need to establish is the integral of the function from -2 to 2. To do this, we need to examine the area under the curve of the function between x = -2 and x = 2.
\int_{-2}^{2} f(x) dx
From the graph, we can see that the area under the curve of the function between x = -2 and x = 2 is equal to 0. Therefore, we can conclude that the integral of the function from -2 to 2 exists and is equal to 0.
Conclusion
In this section, we established the value of each of the following quantities if it exists. If not, we explained why. We analyzed the function's graph to determine the values of various quantities such as limits, derivatives, and integrals. We found that the limit as x approaches 2 exists and is equal to 3, the derivative at x = 1 exists and is equal to 2, the integral from 0 to 3 exists and is equal to 6, the limit as x approaches infinity does not exist, the derivative at x = -2 exists and is equal to -1, and the integral from -2 to 2 exists and is equal to 0.