Math Problem Statement

What is the difference quotient of f if f(x)=x^2 +5x -1?

Solution

The difference quotient of a function f(x)f(x) is given by:

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

where hh is a small increment. Given the function f(x)=x2+5x1f(x) = x^2 + 5x - 1, we need to find f(x+h)f(x+h) and then substitute it into the difference quotient formula.

  1. Calculate f(x+h)f(x+h):

f(x+h)=(x+h)2+5(x+h)1f(x+h) = (x+h)^2 + 5(x+h) - 1

Expand (x+h)2(x+h)^2 and 5(x+h)5(x+h):

f(x+h)=x2+2xh+h2+5x+5h1f(x+h) = x^2 + 2xh + h^2 + 5x + 5h - 1

  1. Find the difference f(x+h)f(x)f(x+h) - f(x):

f(x+h)f(x)=(x2+2xh+h2+5x+5h1)(x2+5x1)f(x+h) - f(x) = \left( x^2 + 2xh + h^2 + 5x + 5h - 1 \right) - \left( x^2 + 5x - 1 \right)

Simplify by canceling out x2x^2, 5x5x, and 1-1:

f(x+h)f(x)=2xh+h2+5hf(x+h) - f(x) = 2xh + h^2 + 5h

  1. Form the difference quotient:

f(x+h)f(x)h=2xh+h2+5hh\frac{f(x+h) - f(x)}{h} = \frac{2xh + h^2 + 5h}{h}

Simplify by dividing each term by hh:

f(x+h)f(x)h=2x+h+5\frac{f(x+h) - f(x)}{h} = 2x + h + 5

So, the difference quotient for f(x)=x2+5x1f(x) = x^2 + 5x - 1 is:

f(x+h)f(x)h=2x+h+5\frac{f(x+h) - f(x)}{h} = 2x + h + 5

Would you like further details on this process or have any questions?


Here are 5 related questions for further exploration:

  1. How does the difference quotient relate to the derivative of a function?
  2. What happens to the difference quotient as hh approaches 0?
  3. How would you find the derivative of f(x)=x2+5x1f(x) = x^2 + 5x - 1 using limits?
  4. Can you calculate the difference quotient for a different function, such as f(x)=sin(x)f(x) = \sin(x)?
  5. What is the significance of the difference quotient in calculus?

Tip: The difference quotient is the foundation for understanding derivatives, which measure the rate of change of a function.

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Math Problem Analysis

Mathematical Concepts

Functions
Difference Quotient
Algebra

Formulas

Difference quotient formula

Theorems

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Suitable Grade Level

Grades 11-12