Find The Value Of The Constant $c$ That Makes The Following Function Continuous On $(-\infty, \infty$\].$\[ F(y) = \begin{cases} cy + 5 & \text{if } Y \in (-\infty, 2] \\ cy^2 - 5 & \text{if } Y \in (2, \infty) \end{cases}

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Introduction

In calculus, a function is considered continuous if it can be drawn without lifting the pencil from the paper. In mathematical terms, a function $f(x)$ is continuous at a point $x=a$ if the following conditions are met:

1. The function is defined at $x=a$.
2. The limit of the function as $x$ approaches $a$ exists.
3. The limit of the function as $x$ approaches $a$ is equal to the value of the function at $x=a$.

In this article, we will discuss how to find the value of the constant $c$ that makes the following function continuous on the interval $(-\infty, \infty)$.

The Function

The given function is defined as:

${ f(y) = \begin{cases} cy + 5 & \text{if } y \in (-\infty, 2] \\ cy^2 - 5 & \text{if } y \in (2, \infty) \end{cases} }$

This function is a piecewise function, meaning that it is defined differently on different intervals. In this case, the function is defined as a linear function on the interval $(-\infty, 2]$ and as a quadratic function on the interval $(2, \infty)$.

Continuity at $y=2$

To find the value of $c$ that makes the function continuous on the interval $(-\infty, \infty)$, we need to find the value of $c$ that makes the function continuous at $y=2$. This means that we need to find the value of $c$ that makes the following conditions true:

1. The function is defined at $y=2$.
2. The limit of the function as $y$ approaches $2$ exists.
3. The limit of the function as $y$ approaches $2$ is equal to the value of the function at $y=2$.

Limit of the Function as $y$ Approaches $2$

To find the limit of the function as $y$ approaches $2$, we need to find the limit of the two parts of the function separately. The limit of the linear part of the function as $y$ approaches $2$ is:

$$\lim_{y \to 2^-} (cy + 5) = 2c + 5$$

The limit of the quadratic part of the function as $y$ approaches $2$ is:

$$\lim_{y \to 2^+} (cy^2 - 5) = 4c - 5$$

Equating the Limits

Since the function is continuous at $y=2$, the two limits must be equal. Therefore, we can set up the following equation:

$$2c + 5 = 4c - 5$$

Solving for $c$

To solve for $c$, we can add $5$ to both sides of the equation:

$$2c + 10 = 4c$$

Then, we can subtract $2c$ from both sides of the equation:

$$10 = 2c$$

Finally, we can divide both sides of the equation by $2$ to solve for $c$:

$$c = 5$$

Conclusion

In this article, we discussed how to find the value of the constant $c$ that makes the following function continuous on the interval $(-\infty, \infty)$. We found that the value of $c$ that makes the function continuous is $c=5$. This means that the function is continuous on the interval $(-\infty, \infty)$ when $c=5$.

Example Use Case

Suppose we want to find the value of the function at $y=3$ when $c=5$. We can plug in $y=3$ and $c=5$ into the function:

$$f(3) = 5(3)^2 - 5 = 40$$

Therefore, the value of the function at $y=3$ when $c=5$ is $40$.

Further Reading

For further reading on continuity and piecewise functions, we recommend the following resources:

• Wikipedia: Continuity
• Khan Academy: Piecewise functions
• Mathway: Piecewise functions

References

• Calculus by Michael Spivak
• Calculus by James Stewart

Note: The references provided are for further reading and are not required for the solution to the problem.

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Q: What is the main goal of finding the value of $c$ for a continuous function?

A: The main goal of finding the value of $c$ for a continuous function is to ensure that the function is continuous on the interval $(-\infty, \infty)$. This means that the function must be defined at all points on the interval, and the limit of the function as $x$ approaches any point on the interval must exist and be equal to the value of the function at that point.

Q: What is the significance of the point $y=2$ in the given function?

A: The point $y=2$ is significant because it is the point where the function changes from a linear function to a quadratic function. To ensure that the function is continuous on the interval $(-\infty, \infty)$, we need to find the value of $c$ that makes the function continuous at $y=2$.

Q: How do we find the limit of the function as $y$ approaches $2$?

A: To find the limit of the function as $y$ approaches $2$, we need to find the limit of the two parts of the function separately. The limit of the linear part of the function as $y$ approaches $2$ is $2c + 5$, and the limit of the quadratic part of the function as $y$ approaches $2$ is $4c - 5$.

Q: Why do we need to equate the two limits?

A: We need to equate the two limits because the function is continuous at $y=2$. This means that the limit of the function as $y$ approaches $2$ must exist and be equal to the value of the function at $y=2$.

Q: How do we solve for $c$?

A: To solve for $c$, we can set up an equation using the two limits and solve for $c$. In this case, we can add $5$ to both sides of the equation $2c + 5 = 4c - 5$ to get $2c + 10 = 4c$. Then, we can subtract $2c$ from both sides of the equation to get $10 = 2c$. Finally, we can divide both sides of the equation by $2$ to solve for $c$.

Q: What is the value of $c$ that makes the function continuous on the interval $(-\infty, \infty)$?

A: The value of $c$ that makes the function continuous on the interval $(-\infty, \infty)$ is $c=5$.

Q: How do we use the value of $c$ to find the value of the function at a specific point?

A: To use the value of $c$ to find the value of the function at a specific point, we can plug in the value of $y$ and $c$ into the function. For example, if we want to find the value of the function at $y=3$ when $c=5$, we can plug in $y=3$ and $c=5$ into the function to get $f(3) = 5(3)^2 - 5 = 40$.

Q: What are some common mistakes to avoid when finding the value of $c$ for a continuous function?

A: Some common mistakes to avoid when finding the value of $c$ for a continuous function include:

• Not checking if the function is defined at all points on the interval
• Not checking if the limit of the function as $x$ approaches any point on the interval exists
• Not equating the two limits when the function changes from one type of function to another
• Not solving for $c$ correctly

Q: What are some real-world applications of finding the value of $c$ for a continuous function?

A: Some real-world applications of finding the value of $c$ for a continuous function include:

• Modeling population growth and decline
• Modeling the spread of diseases
• Modeling the behavior of physical systems, such as springs and pendulums
• Modeling the behavior of economic systems, such as supply and demand

Q: What are some resources for further reading on finding the value of $c$ for a continuous function?

A: Some resources for further reading on finding the value of $c$ for a continuous function include:

• Wikipedia: Continuity
• Khan Academy: Piecewise functions
• Mathway: Piecewise functions
• Calculus by Michael Spivak
• Calculus by James Stewart