MTH120 College Algebra Chapter 3.3

 3.3 Rates of Change and Behavior of Graphs:

Rates of change and the behavior of graphs are fundamental concepts in calculus and mathematics that help us understand how functions change with respect to their inputs (typically denoted as $�$) and how their graphs behave in various situations. These concepts are crucial for analyzing and interpreting functions, especially when dealing with real-world problems. Here are some key ideas related to rates of change and the behavior of graphs:

1. Rate of Change:

   • The rate of change of a function measures how the output of the function (typically denoted as $�$ or $�(�)$) changes as the input ($�$) changes.
   • In calculus, the rate of change at a specific point is represented by the derivative of the function. The derivative tells you the slope of the tangent line to the graph of the function at that point.
   • The average rate of change between two points on a function's graph is calculated as the change in $�$ divided by the change in $�$. This provides an average rate of change over an interval.

2. Instantaneous Rate of Change:

   • The instantaneous rate of change is the rate of change at a specific instant or point. It is precisely represented by the derivative of the function at that point.
   • The derivative measures how quickly the function is increasing or decreasing at a given value of $�$.

3. Behavior of Graphs:

   • The behavior of a function's graph refers to how the graph behaves as you move along the $�$-axis.
   • Key behaviors include increasing or decreasing trends, concavity (whether the graph is bending upwards or downwards), and any points of interest such as local maxima, minima, and inflection points.

4. Local Maxima and Minima:

   • Local maxima are points on the graph where the function reaches a peak within a small neighborhood. They correspond to places where the derivative changes from positive to negative.
   • Local minima are points on the graph where the function reaches a trough within a small neighborhood. They correspond to places where the derivative changes from negative to positive.

5. Concavity:

   • A function is concave up if its graph is bending upwards (like a cup).
   • A function is concave down if its graph is bending downwards (like a bowl).
   • Inflection points are places where the concavity changes.

6. Asymptotes:

   • Horizontal asymptotes describe the behavior of a function as $�$ approaches positive or negative infinity. They indicate where the graph levels off.
   • Vertical asymptotes indicate values of $�$ where the function approaches infinity or negative infinity.

7. Discontinuities:

   • Discontinuities are points on the graph where the function is not continuous. They can be classified as removable (a hole in the graph), jump (a jump from one value to another), or infinite (vertical asymptotes).

These concepts are essential for understanding how functions change and behave, and they are particularly important in calculus when dealing with derivatives and integrals. They are also valuable in various fields, including physics, engineering, economics, and many others, for modeling and solving real-world problems.

The average rate of change of a function over a given interval measures how the function's output (usually denoted as $�$ or $�(�)$) changes with respect to the input ($�$) over that interval. Mathematically, the average rate of change is determined by finding the difference in $�$ values (change in output) divided by the difference in $�$ values (change in input) over the specified interval.

Here are the steps to find the average rate of change of a function over an interval:

1. Identify the interval: Determine the interval of interest over which you want to find the average rate of change. This interval is typically specified as a range of values for $�$, such as $[�,�]$, where $�$ and $�$ are the endpoints of the interval.

2. Calculate the change in $�$: Evaluate the function at the endpoints of the interval to find the corresponding $�$ values. Compute the difference in $�$ values, which is the change in output, often denoted as $\mathrm{\Delta }�$:

   $\mathrm{\Delta }�=�(�)-�(�)$

3. Calculate the change in $�$: Determine the difference between the endpoints of the interval, which is the change in input, often denoted as $\mathrm{\Delta }�$:

   $\mathrm{\Delta }�=�-�$

4. Find the average rate of change: Divide the change in $�$ by the change in $�$ to obtain the average rate of change ($����$):

   $����=\frac{\mathrm{\Delta }�}{\mathrm{\Delta }�}$

The average rate of change measures the slope of the secant line (the line connecting two points on the graph) over the specified interval. It represents the average speed at which the function's output changes for each unit change in the input within that interval.

Here's an example to illustrate the concept:

Example: Suppose you have a function $�(�)=2{�}^2+3�-1$ and you want to find the average rate of change over the interval $[-1,2]$.

1. Identify the interval: The interval is $[-1,2]$.

2. Calculate the change in $�$: $\mathrm{\Delta }�=�(2)-�(-1)=(2(2{)}^2+3(2)-1)-(2(-1{)}^2+3(-1)-1)=12-(-4)=16$

3. Calculate the change in $�$: $\mathrm{\Delta }�=2-(-1)=3$

4. Find the average rate of change: $����=\frac{\mathrm{\Delta }�}{\mathrm{\Delta }�}=\frac{16}{3}$

So, the average rate of change of the function $�(�)$ over the interval $[-1,2]$ is $\frac{16}{3}$, which means that, on average, the function's output increases by $\frac{16}{3}$ units for each unit increase in the input within that interval.

The rate of change measures how one quantity changes in relation to another quantity. In mathematics, it's often used to describe how a function's output (usually denoted as $�$ or $�(�)$) changes concerning its input (usually denoted as $�$).

The rate of change can be calculated in various contexts, and here are a few common ones:

1. Average Rate of Change:

   • The average rate of change of a function over an interval measures the average rate at which the function's output changes concerning its input within that interval.
   • It is calculated as the difference in the function's values (change in output) divided by the difference in input values (change in input) over the interval.
   • Symbolically, the average rate of change ($����$) over an interval $[�,�]$ is given by: $����=\frac{�(�)-�(�)}{�-�}$

2. Instantaneous Rate of Change:

   • The instantaneous rate of change of a function measures how quickly the function's output changes at a specific point.
   • In calculus, this is precisely represented by the derivative of the function at that point.
   • Symbolically, the instantaneous rate of change at a point $�=�$ is given by: $����={�}^{\mathrm{\prime }}(�)$
   • The derivative ${�}^{\mathrm{\prime }}(�)$ represents the slope of the tangent line to the graph of the function at the point $(�,�(�))$.

3. Rate of Change in Real-World Applications:

   • In real-world applications, the rate of change is used to describe how one variable changes concerning another.
   • For example, in physics, velocity is the rate of change of displacement with respect to time, and acceleration is the rate of change of velocity.
   • In economics, marginal cost represents the rate of change of total cost concerning the quantity produced.
   • In biology, growth rate measures how a population changes concerning time.

Understanding and calculating rates of change is fundamental in various fields, including mathematics, physics, engineering, economics, and the sciences. It helps us analyze how quantities change over time or in relation to other factors, leading to valuable insights and predictions. Calculus, in particular, provides powerful tools for studying rates of change through derivatives and integrals.

To calculate the average rate of change of a function for the interval between two values ${�}_1$ and ${�}_2$, you'll need the values of the function at these two points. Here's the formula and an example:

Formula for Average Rate of Change (AROC): $����=\frac{�({�}_2)-�({�}_1)}{{�}_2-{�}_1}$

Where:

• $�({�}_2)$ is the value of the function at ${�}_2$.
• $�({�}_1)$ is the value of the function at ${�}_1$.
• ${�}_2$ and ${�}_1$ are the two points defining the interval.

Example: Let's say you have a function $�(�)=2{�}^2+3�-1$, and you want to calculate the average rate of change of this function between ${�}_1=1$ and ${�}_2=3$.

1. Find the value of the function at ${�}_1$ and ${�}_2$:

   • $�({�}_1)=2(1{)}^2+3(1)-1=2+3-1=4$
   • $�({�}_2)=2(3{)}^2+3(3)-1=18+9-1=26$

2. Plug these values into the formula: $����=\frac{�(3)-�(1)}{3-1}=\frac{26-4}{3-1}=\frac{22}{2}=11$

So, the average rate of change of the function $�(�)$ between $�=1$ and $�=3$ is 11. This means that, on average, the function's output increases by 11 units for each unit increase in the input within that interval.

You can use this same formula to calculate the average rate of change for any function over any interval as long as you know the values of the function at the endpoints of the interval.

To compute the average rate of change of a function over a specific interval, you need to follow these steps:

1. Identify the interval: Determine the interval or range of values for the independent variable $�$ over which you want to calculate the average rate of change. The interval is usually denoted as $[{�}_1,{�}_2]$, where ${�}_1$ and ${�}_2$ are the endpoints of the interval.

2. Find the function values: Evaluate the function at the endpoints of the interval to find the corresponding values of the dependent variable $�$ or $�(�)$. This gives you $�({�}_1)$ and $�({�}_2)$.

3. Calculate the change in $�$: Find the difference in the function values at the two endpoints. This represents the change in the dependent variable $�$ over the interval.

   $\mathrm{\Delta }�=�({�}_2)-�({�}_1)$

4. Calculate the change in $�$: Find the difference between the two endpoints of the independent variable $�$. This represents the change in $�$ over the interval.

   $\mathrm{\Delta }�={�}_2-{�}_1$

5. Compute the average rate of change: Divide the change in $�$ by the change in $�$ to obtain the average rate of change ($����$).

   $����=\frac{\mathrm{\Delta }�}{\mathrm{\Delta }�}=\frac{�({�}_2)-�({�}_1)}{{�}_2-{�}_1}$

Here's a simple example:

Example: Suppose you have a function $�(�)=2{�}^2+3�-1$ and you want to find the average rate of change over the interval $[1,3]$.

1. Identify the interval: The interval is $[1,3]$.

2. Find the function values at the endpoints:

   • $�(1)=2(1{)}^2+3(1)-1=2+3-1=4$
   • $�(3)=2(3{)}^2+3(3)-1=18+9-1=26$

3. Calculate the change in $�$: $\mathrm{\Delta }�=�(3)-�(1)=26-4=22$

4. Calculate the change in $�$: $\mathrm{\Delta }�=3-1=2$

5. Compute the average rate of change: $����=\frac{\mathrm{\Delta }�}{\mathrm{\Delta }�}=\frac{22}{2}=11$

So, the average rate of change of the function $�(�)$ over the interval $[1,3]$ is 11. This means that, on average, the function's output increases by 11 units for each unit increase in the input within that interval.

You can compute the average rate of change of a function from a table of values by following these steps:

1. Identify the Interval: Look at the table and determine the interval (range of $�$ values) over which you want to calculate the average rate of change.

2. Find the Function Values: Locate the values of the function (usually denoted as $�$ or $�(�)$) corresponding to the $�$ values in the interval. You will need the function values at the beginning and end of the interval.

3. Calculate the Change in $�$: Find the difference in the function values at the two endpoints of the interval. This represents the change in the dependent variable ($�$) over the interval.

4. Calculate the Change in $�$: Determine the difference between the two $�$ values that correspond to the endpoints of the interval. This represents the change in the independent variable ($�$) over the interval.

5. Compute the Average Rate of Change: Divide the change in $�$ by the change in $�$ to obtain the average rate of change ($����$).

   $����=\frac{\mathrm{\Delta }�}{\mathrm{\Delta }�}$

1. Identify the interval: The interval is $[2,6]$.

2. Find the function values at the endpoints of the interval:

   • $�(2)=8$
   • $�(6)=32$

3. Calculate the change in $�$: $\mathrm{\Delta }�=�(6)-�(2)=32-8=24$

4. Calculate the change in $�$: $\mathrm{\Delta }�=6-2=4$

5. Compute the average rate of change: $����=\frac{\mathrm{\Delta }�}{\mathrm{\Delta }�}=\frac{24}{4}=6$

So, the average rate of change of the function $�(�)$ over the interval $[2,6]$ is 6. This means that, on average, the function's output increases by 6 units for each unit increase in the input within that interval.

You can determine where a function is increasing, decreasing, or constant by examining its graph. Here are the key steps to do this:

1. Understand the Concept:

   • A function is considered "increasing" on an interval if, as you move from left to right along the x-axis, its graph rises or goes up within that interval.
   • A function is considered "decreasing" on an interval if, as you move from left to right along the x-axis, its graph falls or goes down within that interval.
   • A function is "constant" on an interval if its graph remains flat or does not rise or fall significantly within that interval.

2. Locate Critical Points:

   • Identify the critical points, which are places on the graph where the function's behavior changes. These points include:
     • Local maxima: Points where the graph reaches a peak.
     • Local minima: Points where the graph reaches a trough.
     • Points where the graph changes from increasing to decreasing or vice versa (these points may not always correspond to maxima or minima).
     • Endpoints of the interval you are interested in.

3. Examine the Graph Around Critical Points:

   • Around each critical point, examine the behavior of the graph.
     • If the graph is rising to the right of a critical point, it's increasing.
     • If the graph is falling to the right of a critical point, it's decreasing.
     • If the graph is flat to the right of a critical point, it's constant.

4. Consider Intervals Between Critical Points:

   • For intervals between critical points or endpoints, analyze the overall trend of the graph. Does it rise or fall over the entire interval, or is it relatively flat?

5. Use the First Derivative Test:

   • If you have access to calculus, you can use the first derivative test to determine the sign of the derivative at specific points. If the derivative is positive, the function is increasing. If it's negative, the function is decreasing. If it's zero, the function may have a local extremum.

Here's an example to illustrate these steps:

Example: Consider the function $�(�)={�}^3-3{�}^2-9�+5$.

1. Understand the Concept: We want to determine where this function is increasing, decreasing, or constant.

2. Locate Critical Points:

   • Find the critical points by taking the derivative and setting it equal to zero. In this case, the critical points are $�=-1$ and $�=3$.

3. Examine the Graph Around Critical Points:

   • At $�=-1$, if you examine the graph to the right, it rises, so the function is increasing.
   • At $�=3$, if you examine the graph to the right, it falls, so the function is decreasing.

4. Consider Intervals Between Critical Points:

   • Between -1 and 3, the graph is increasing (you can see this by examining the graph).

5. Use the First Derivative Test:

   • The derivative of $�(�)$ is ${�}^{\mathrm{\prime }}(�)=3{�}^2-6�-9$.
   • Plug in values into ${�}^{\mathrm{\prime }}(�)$ to determine where it's positive (increasing) or negative (decreasing):
     • ${�}^{\mathrm{\prime }}(-2)=-15$ (negative, so decreasing).
     • ${�}^{\mathrm{\prime }}(0)=-9$ (negative, so decreasing).
     • ${�}^{\mathrm{\prime }}(2)=9$ (positive, so increasing).
     • ${�}^{\mathrm{\prime }}(4)=27$ (positive, so increasing).

By following these steps and examining the graph and derivative, you can determine where the function $�(�)$ is increasing, decreasing, or constant.

Local minima and local maxima are important concepts in calculus and optimization, particularly when analyzing the behavior of functions. They represent points on a function's graph where the function reaches its lowest (minimum) or highest (maximum) values within a specific neighborhood of that point. Here's a more detailed explanation of local minima and local maxima:

Local Minimum (Minima):

• A local minimum of a function occurs at a point $�$ in its domain if there exists a neighborhood around $�$ such that the function's value at $�$ is less than or equal to its values at all other points within that neighborhood.
• Symbolically, $�(�)$ is a local minimum if there exists a neighborhood $(�,�)$ around $�$ such that $�(�)\le �(�)$ for all $�$ in $(�,�)$.

Local Maximum (Maxima):

• A local maximum of a function occurs at a point $�$ in its domain if there exists a neighborhood around $�$ such that the function's value at $�$ is greater than or equal to its values at all other points within that neighborhood.
• Symbolically, $�(�)$ is a local maximum if there exists a neighborhood $(�,�)$ around $�$ such that $�(�)\ge �(�)$ for all $�$ in $(�,�)$.

Key points to note:

• Local minima and maxima are sometimes referred to as "relative" minima and maxima because they are relative to the surrounding points within a neighborhood.
• These points may or may not be global minima or maxima, which are the lowest and highest points on the entire function's domain.
• A point that is neither a local minimum nor a local maximum is called a "saddle point."

To find local minima and maxima for a given function, you typically follow these steps:

1. Find the critical points: Determine where the derivative of the function is zero or undefined. These are potential locations for local minima and maxima.

2. Use the first or second derivative test: To determine whether each critical point corresponds to a local minimum, local maximum, or neither, you can use the first or second derivative test:

   • First Derivative Test: Analyze the sign changes of the derivative on either side of each critical point. A sign change from negative to positive indicates a local minimum, and a sign change from positive to negative indicates a local maximum.
   • Second Derivative Test: Compute the second derivative at each critical point. If the second derivative is positive, it's a local minimum. If the second derivative is negative, it's a local maximum. If the second derivative is zero, the test is inconclusive.

Local minima and maxima are crucial in optimization, curve sketching, and understanding the behavior of functions in various real-world applications, including economics, physics, engineering, and more.

Toolkit functions are a set of common functions used in calculus to understand and analyze various concepts. When analyzing toolkit functions for increasing or decreasing intervals, you are interested in determining the intervals of $�$ where the function is increasing, decreasing, or constant. Here's how you can analyze some common toolkit functions:

1. Linear Function: A linear function $�(�)=��+�$ has a constant rate of change, meaning it is either always increasing (if $�>0$) or always decreasing (if $�<0$). There are no local maxima or minima unless it's a horizontal line.

2. Quadratic Function: A quadratic function $�(�)=�{�}^2+��+�$ can have different behaviors:

   • If $�>0$, the parabola opens upward, and the function is increasing on the intervals outside of any real roots and decreasing in between the real roots (if they exist).
   • If $�<0$, the parabola opens downward, and the function is decreasing on the intervals outside of any real roots and increasing in between the real roots (if they exist).
   • If $�=0$, it's a horizontal line, so the function is neither increasing nor decreasing.

3. Exponential Function: An exponential function $�(�)={�}^{�}$ where $�>1$ is always increasing for all real values of $�$. On the other hand, if $0<�<1$, it's always decreasing.

4. Logarithmic Function: A logarithmic function $�(�)={\log }_{�}(�)$ where $�>1$ is always increasing for all $�>0$. If $0<�<1$, it's always decreasing.

5. Trigonometric Functions:

   • The sine function, $�(�)=\sin (�)$, has periodic behavior between -1 and 1, but it's increasing on some intervals and decreasing on others.
   • The cosine function, $�(�)=\cos (�)$, also has periodic behavior between -1 and 1 and alternates between increasing and decreasing intervals.

6. Absolute Value Function: The absolute value function $�(�)=\mathrm{\mid }�\mathrm{\mid }$ is increasing for $�\ge 0$ and decreasing for $�\le 0$.

When analyzing these functions, you can identify intervals of increasing and decreasing behavior by considering the sign of the derivative (slope of the tangent line) or by examining the behavior of the graph. Here are the general steps:

• Find the critical points (where the derivative is zero or undefined).
• Use the first derivative test: Determine the sign changes of the derivative on either side of each critical point.
  • If the derivative changes from negative to positive, the function is increasing.
  • If the derivative changes from positive to negative, the function is decreasing.
• Examine the intervals between critical points and the behavior near the endpoints of the domain.

These techniques will help you identify where toolkit functions are increasing, decreasing, or constant, which is valuable for understanding their behavior and solving various problems in calculus and mathematics.

To locate the absolute maximum and absolute minimum of a function using a graph, follow these steps:

1. Identify the Interval: Determine the interval over which you want to find the absolute maximum and minimum. This interval should be within the domain of the function.

2. Plot the Graph: Plot the graph of the function within the specified interval. Ensure that you have a clear and accurate representation of the function's behavior.

3. Examine Critical Points:

   • Critical points are where the derivative of the function is zero or undefined. Identify these points on the graph. They could potentially be candidates for extreme values.
   • Note that not all critical points are guaranteed to be extreme values (maximum or minimum).

4. Examine Endpoints: If your interval includes endpoints, evaluate the function at those endpoints. These are also potential candidates for extreme values.

5. Determine Maximum and Minimum:

   • Compare the function values at critical points, endpoints, and any other significant points in your interval.
   • Identify the largest value as the absolute maximum and the smallest value as the absolute minimum within the specified interval.

6. Consider Local Maxima and Minima: Look for local maxima (relative maxima) and minima (relative minima) by examining where the function reaches peaks or valleys within the interval. Local extrema may or may not be the absolute extrema.

7. Verify Your Findings: Double-check your results to ensure that you have indeed found the absolute maximum and minimum within the specified interval.

Here's an example:

Example: Find the absolute maximum and minimum of the function $�(�)={�}^3-3{�}^2-9�+5$ over the interval $[-1,4]$.

1. Identify the Interval: The interval is $[-1,4]$.

2. Plot the Graph: Plot the graph of $�(�)$ over the interval $[-1,4]$.

3. Examine Critical Points: Find critical points by taking the derivative and setting it equal to zero. In this case, the critical points are $�=-1$ and $�=3$. Identify these points on the graph.

4. Examine Endpoints: Evaluate the function at the endpoints of the interval:

   • $�(-1)=8$
   • $�(4)=-35$

5. Determine Maximum and Minimum:

   • The function values are: $�(-1)=8$, $�(3)=-20$, $�(4)=-35$.
   • The absolute maximum is -20 at $�=3$.
   • The absolute minimum is 8 at $�=-1$.

6. Consider Local Maxima and Minima: Check the graph for any local maxima or minima. In this case, there are no local extrema within the interval.

7. Verify Your Findings: Ensure that the values you identified as the absolute maximum and minimum are indeed the largest and smallest values within the interval.

By following these steps and carefully analyzing the graph, you can locate the absolute maximum and minimum of a function over a specified interval.

Let's work through a few examples of finding the absolute maximum and minimum of functions using their graphs and the specified intervals:

Example 1: Find the absolute maximum and minimum of the function $�(�)={�}^2-4�+5$ over the interval $[-1,5]$.

1. Identify the Interval: The interval is $[-1,5]$.

2. Plot the Graph: Plot the graph of $�(�)$ over the interval $[-1,5]$. The graph is a parabola opening upward.

3. Examine Critical Points: Find the critical points by taking the derivative and setting it equal to zero. In this case, the critical point is $�=2$. Identify this point on the graph.

4. Examine Endpoints: Evaluate the function at the endpoints of the interval:

   • $�(-1)=10$
   • $�(5)=10$

5. Determine Maximum and Minimum:

   • The function values are: $�(-1)=10$, $�(2)=1$, $�(5)=10$.
   • The absolute maximum is 10 at both $�=-1$ and $�=5$.
   • The absolute minimum is 1 at $�=2$.

So, the absolute maximum is 10, occurring at $�=-1$ and $�=5$, and the absolute minimum is 1 at $�=2$.

Example 2: Find the absolute maximum and minimum of the function $�(�)=\sin (�)$ over the interval $[0,�]$.

1. Identify the Interval: The interval is $[0,�]$.

2. Plot the Graph: Plot the graph of $�(�)=\sin (�)$ over the interval $[0,�]$. The graph is a sinusoidal curve.

3. Examine Critical Points: The critical points for $\sin (�)$ occur where its derivative, $\cos (�)$, equals zero. In this case, there are no critical points within the interval.

4. Examine Endpoints: Evaluate the function at the endpoints of the interval:

   • $�(0)=0$
   • $�(�)=0$

5. Determine Maximum and Minimum:

   • The function values are: $�(0)=0$, $�(�)=0$.
   • The absolute maximum and minimum are both 0.

In this case, both the absolute maximum and minimum occur at the endpoints of the interval $[0,�]$, and they are both 0.

These examples illustrate how to find the absolute maximum and minimum of functions over specified intervals by examining their graphs and evaluating the function at critical points and endpoints.