Math Problem Statement

Calculate and simplify the Difference Quotient (f(x+h)-f(x))/h for f(x) = x^5.

Solution

We are tasked with calculating and simplifying the difference quotient for the function f(x)=x5f(x) = x^5. The difference quotient formula is:

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

Substituting f(x)=x5f(x) = x^5, we get:

(x+h)5x5h\frac{(x+h)^5 - x^5}{h}

Step-by-Step Solution:

  1. Expand (x+h)5(x+h)^5 using the binomial theorem:

(x+h)5=x5+5x4h+10x3h2+10x2h3+5xh4+h5(x+h)^5 = x^5 + 5x^4h + 10x^3h^2 + 10x^2h^3 + 5xh^4 + h^5

  1. Substitute the expansion back into the difference quotient:

x5+5x4h+10x3h2+10x2h3+5xh4+h5x5h\frac{x^5 + 5x^4h + 10x^3h^2 + 10x^2h^3 + 5xh^4 + h^5 - x^5}{h}

  1. Simplify the expression by canceling out x5x^5:

5x4h+10x3h2+10x2h3+5xh4+h5h\frac{5x^4h + 10x^3h^2 + 10x^2h^3 + 5xh^4 + h^5}{h}

  1. Factor out hh from the numerator:

h(5x4+10x3h+10x2h2+5xh3+h4)h\frac{h(5x^4 + 10x^3h + 10x^2h^2 + 5xh^3 + h^4)}{h}

  1. Cancel the hh in the numerator and denominator (assuming h0h \neq 0):

5x4+10x3h+10x2h2+5xh3+h45x^4 + 10x^3h + 10x^2h^2 + 5xh^3 + h^4

This is the simplified form of the difference quotient for f(x)=x5f(x) = x^5. The hint that the fraction reduces into a polynomial expression is confirmed, as the final result is a polynomial in terms of hh.

Would you like more details on any step? Here are five related questions:

  1. How does the binomial theorem help in expanding expressions like (x+h)5(x + h)^5?
  2. What is the significance of the difference quotient in calculus?
  3. How does this process relate to finding the derivative of f(x)=x5f(x) = x^5?
  4. What happens to the terms involving hh as h0h \to 0?
  5. How would this process change for a different function, such as f(x)=x3f(x) = x^3?

Tip: The difference quotient is a critical concept as it lays the foundation for understanding derivatives and rates of change in calculus.

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Math Problem Analysis

Mathematical Concepts

Difference Quotient
Polynomial Expansion
Binomial Theorem

Formulas

Difference Quotient formula: (f(x+h) - f(x))/h
Binomial Expansion: (x+h)^5 = x^5 + 5x^4h + 10x^3h^2 + 10x^2h^3 + 5xh^4 + h^5

Theorems

Binomial Theorem

Suitable Grade Level

Grade 11-12 or introductory Calculus