Given F ( X ) = 2 X − 7 F(x) = 2x - 7 F ( X ) = 2 X − 7 :(a) Find F ( X + H F(x + H F ( X + H ] And Simplify.(b) Find F ( X + H ) − F ( X ) H \frac{f(x + H) - F(x)}{h} H F ( X + H ) − F ( X ) ​ And Simplify.Part 1 Of 2:(a) F ( X + H ) = F(x + H) = F ( X + H ) = □ \square □ □ + □ \square + \square □ + □ □ \square □

Part 1: Finding $f(x + h)$ and Simplifying

Introduction to Function Composition

In this article, we will explore the concept of function composition and how it is used to find the derivative of a function. We will start by finding $f(x + h)$ and simplifying it, and then we will use this result to find $\frac{f(x + h) - f(x)}{h}$ and simplify it.

Finding $f(x + h)$

To find $f(x + h)$, we need to substitute $(x + h)$ into the function $f(x) = 2x - 7$. This means that we will replace every instance of $x$ in the function with $(x + h)$.

f(x + h) = 2(x + h) - 7

Now, let's simplify the expression by distributing the 2 to the terms inside the parentheses.

f(x + h) = 2x + 2h - 7

Simplifying $f(x + h)$

We can simplify the expression further by combining like terms. In this case, we have a constant term $-7$ that is not combined with any other terms.

f(x + h) = 2x + 2h - 7

This is the simplified form of $f(x + h)$.

Part 1 Conclusion

In this part, we found $f(x + h)$ and simplified it. We used the concept of function composition to substitute $(x + h)$ into the function $f(x) = 2x - 7$. We then simplified the expression by distributing the 2 to the terms inside the parentheses and combining like terms.

Part 2: Finding $\frac{f(x + h) - f(x)}{h}$ and Simplifying

Introduction to Derivatives

In this part, we will use the result from Part 1 to find $\frac{f(x + h) - f(x)}{h}$ and simplify it. This expression is the definition of the derivative of a function, and it will allow us to find the derivative of $f(x) = 2x - 7$.

Finding $\frac{f(x + h) - f(x)}{h}$

To find $\frac{f(x + h) - f(x)}{h}$, we need to substitute the expression for $f(x + h)$ that we found in Part 1.

\frac{f(x + h) - f(x)}{h} = \frac{(2x + 2h - 7) - (2x - 7)}{h}

Now, let's simplify the expression by combining like terms.

\frac{f(x + h) - f(x)}{h} = \frac{2x + 2h - 7 - 2x + 7}{h}

\frac{f(x + h) - f(x)}{h} = \frac{2h}{h}

Simplifying $\frac{f(x + h) - f(x)}{h}$

We can simplify the expression further by canceling out the $h$ terms.

\frac{f(x + h) - f(x)}{h} = 2

This is the simplified form of $\frac{f(x + h) - f(x)}{h}$.

Part 2 Conclusion

In this part, we used the result from Part 1 to find $\frac{f(x + h) - f(x)}{h}$ and simplify it. We used the definition of the derivative of a function to find the derivative of $f(x) = 2x - 7$. We then simplified the expression by combining like terms and canceling out the $h$ terms.

Final Conclusion

In this article, we found $f(x + h)$ and simplified it, and then we used this result to find $\frac{f(x + h) - f(x)}{h}$ and simplify it. We used the concept of function composition to substitute $(x + h)$ into the function $f(x) = 2x - 7$, and we then simplified the expression by distributing the 2 to the terms inside the parentheses and combining like terms. We also used the definition of the derivative of a function to find the derivative of $f(x) = 2x - 7$. The final answer is $\boxed{2}$.

Discussion

The concept of function composition is a powerful tool in mathematics that allows us to find the derivative of a function. By substituting $(x + h)$ into the function $f(x) = 2x - 7$, we were able to find $f(x + h)$ and simplify it. We then used this result to find $\frac{f(x + h) - f(x)}{h}$ and simplify it. The final answer is $\boxed{2}$.

References

• [1] Calculus, Michael Spivak
• [2] Calculus, James Stewart

Tags

• function composition
• derivative
• calculus
• mathematics
  Q&A: Function Composition and Derivatives

Introduction

In our previous article, we explored the concept of function composition and how it is used to find the derivative of a function. We found $f(x + h)$ and simplified it, and then we used this result to find $\frac{f(x + h) - f(x)}{h}$ and simplify it. In this article, we will answer some common questions related to function composition and derivatives.

Q1: What is function composition?

A1: Function composition is a way of combining two or more functions to create a new function. It involves substituting one function into another function, and then simplifying the resulting expression.

Q2: How do I find $f(x + h)$?

A2: To find $f(x + h)$, you need to substitute $(x + h)$ into the function $f(x)$. This means that you will replace every instance of $x$ in the function with $(x + h)$.

Q3: What is the definition of the derivative of a function?

A3: The derivative of a function is defined as the limit of the difference quotient as $h$ approaches zero. It is denoted by $\frac{dy}{dx}$ and is used to measure the rate of change of a function with respect to its input.

Q4: How do I find the derivative of a function using function composition?

A4: To find the derivative of a function using function composition, you need to find $f(x + h)$ and simplify it, and then use this result to find $\frac{f(x + h) - f(x)}{h}$ and simplify it.

Q5: What is the significance of the derivative of a function?

A5: The derivative of a function is significant because it allows us to measure the rate of change of a function with respect to its input. It is used in a wide range of applications, including physics, engineering, and economics.

Q6: How do I use the derivative of a function to solve problems?

A6: To use the derivative of a function to solve problems, you need to understand the concept of the derivative and how it is used to measure the rate of change of a function. You can then use this understanding to solve problems that involve finding the rate of change of a function.

Q7: What are some common applications of the derivative of a function?

A7: Some common applications of the derivative of a function include:

• Finding the maximum and minimum values of a function
• Determining the rate of change of a function
• Finding the slope of a tangent line to a curve
• Solving optimization problems

Q8: How do I find the maximum and minimum values of a function using the derivative?

A8: To find the maximum and minimum values of a function using the derivative, you need to find the critical points of the function by setting the derivative equal to zero and solving for $x$. You can then use the second derivative test to determine whether the critical points correspond to a maximum or minimum value.

Q9: What is the second derivative test?

A9: The second derivative test is a method used to determine whether a critical point corresponds to a maximum or minimum value. It involves finding the second derivative of the function and evaluating it at the critical point.

Q10: How do I use the second derivative test to determine whether a critical point corresponds to a maximum or minimum value?

A10: To use the second derivative test to determine whether a critical point corresponds to a maximum or minimum value, you need to find the second derivative of the function and evaluate it at the critical point. If the second derivative is positive, the critical point corresponds to a minimum value. If the second derivative is negative, the critical point corresponds to a maximum value.

Conclusion

In this article, we have answered some common questions related to function composition and derivatives. We have discussed the concept of function composition, how to find $f(x + h)$, and how to use the derivative of a function to solve problems. We have also discussed some common applications of the derivative of a function, including finding the maximum and minimum values of a function and determining the rate of change of a function.

References

• [1] Calculus, Michael Spivak
• [2] Calculus, James Stewart

Tags

• function composition
• derivative
• calculus
• mathematics
• optimization
• rate of change
• maximum and minimum values