Find The Difference Quotient $\frac{f(x+h)-f(x)}{h}$, Where $h \neq 0$, For The Function Below.$f(x)=4x^2-x+4$Simplify Your Answer As Much As Possible.$\frac{f(x+h)-f(x)}{h} = \square$

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Introduction

In calculus, the difference quotient is a fundamental concept used to find the derivative of a function. It is defined as the limit of the difference quotient as the change in the input variable approaches zero. In this article, we will find the difference quotient of a quadratic function and simplify our answer as much as possible.

The Quadratic Function

The given quadratic function is f(x)=4x2−x+4f(x) = 4x^2 - x + 4. To find the difference quotient, we need to substitute (x+h)(x+h) into the function and simplify.

Substituting (x+h)(x+h) into the Function

We will substitute (x+h)(x+h) into the function f(x)=4x2−x+4f(x) = 4x^2 - x + 4.

f(x+h) = 4(x+h)^2 - (x+h) + 4

Expanding the Expression

We will expand the expression (x+h)2(x+h)^2 using the binomial theorem.

(x+h)^2 = x^2 + 2xh + h^2

Substituting this expression into the function, we get:

f(x+h) = 4(x^2 + 2xh + h^2) - (x+h) + 4

Simplifying the Expression

We will simplify the expression by distributing the 4 and combining like terms.

f(x+h) = 4x^2 + 8xh + 4h^2 - x - h + 4

Finding the Difference Quotient

Now that we have found f(x+h)f(x+h), we can find the difference quotient by substituting it into the formula:

f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}

We will substitute f(x+h)f(x+h) and f(x)f(x) into the formula.

\frac{f(x+h)-f(x)}{h} = \frac{(4x^2 + 8xh + 4h^2 - x - h + 4) - (4x^2 - x + 4)}{h}

Simplifying the Difference Quotient

We will simplify the difference quotient by combining like terms.

\frac{f(x+h)-f(x)}{h} = \frac{4x^2 + 8xh + 4h^2 - x - h + 4 - 4x^2 + x - 4}{h}

Canceling Out the Like Terms

We will cancel out the like terms in the numerator.

\frac{f(x+h)-f(x)}{h} = \frac{8xh + 4h^2 - h}{h}

Factoring Out the Common Term

We will factor out the common term hh from the numerator.

\frac{f(x+h)-f(x)}{h} = \frac{h(8x + 4h - 1)}{h}

Canceling Out the Common Term

We will cancel out the common term hh from the numerator and denominator.

\frac{f(x+h)-f(x)}{h} = 8x + 4h - 1

Conclusion

In this article, we found the difference quotient of a quadratic function and simplified our answer as much as possible. The difference quotient is a fundamental concept used to find the derivative of a function, and it is essential to understand how to find it for various types of functions.

Final Answer

Introduction

In our previous article, we found the difference quotient of a quadratic function and simplified our answer as much as possible. In this article, we will provide a comprehensive guide to the difference quotient, including its definition, formula, and examples.

Q&A: Difference Quotient

Q: What is the difference quotient?

A: The difference quotient is a fundamental concept in calculus that is used to find the derivative of a function. It is defined as the limit of the difference quotient as the change in the input variable approaches zero.

Q: What is the formula for the difference quotient?

A: The formula for the difference quotient is:

f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}

Q: How do I find the difference quotient of a function?

A: To find the difference quotient of a function, you need to substitute (x+h)(x+h) into the function and simplify. Then, you need to divide the result by hh.

Q: What is the difference between the difference quotient and the derivative?

A: The difference quotient is the limit of the difference quotient as the change in the input variable approaches zero, while the derivative is the limit of the difference quotient as the change in the input variable approaches zero.

Q: Can I use the difference quotient to find the derivative of a function?

A: Yes, you can use the difference quotient to find the derivative of a function. However, you need to take the limit of the difference quotient as the change in the input variable approaches zero.

Q: What are some examples of functions that can be used to find the difference quotient?

A: Some examples of functions that can be used to find the difference quotient include:

  • Linear functions: f(x)=mx+bf(x) = mx + b
  • Quadratic functions: f(x)=ax2+bx+cf(x) = ax^2 + bx + c
  • Polynomial functions: f(x)=anxn+an−1xn−1+…+a1x+a0f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0

Q: How do I simplify the difference quotient?

A: To simplify the difference quotient, you need to combine like terms and cancel out any common factors.

Q: What is the final answer for the difference quotient of a quadratic function?

A: The final answer for the difference quotient of a quadratic function is:

f(x+h)−f(x)h=8x+4h−1\frac{f(x+h)-f(x)}{h} = 8x + 4h - 1

Conclusion

In this article, we provided a comprehensive guide to the difference quotient, including its definition, formula, and examples. We also answered some frequently asked questions about the difference quotient.

Final Answer

The final answer is 8x+4h−1\boxed{8x + 4h - 1}.

Additional Resources

  • Calculus: A First Course by Michael Sullivan
  • Calculus: Early Transcendentals by James Stewart
  • Difference Quotient: A Comprehensive Guide by [Author]

Related Articles

  • Finding the Derivative of a Function
  • The Limit of a Function
  • The Chain Rule

Comments

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