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Newton's Quotient Help

By — McGraw-Hill Professional
Updated on Sep 6, 2011

Introduction to Newton's Quotient

Very early in an introductory calculus course, students use function evaluation to evaluate an important formula called Newton’s Quotient.

Introduction to Functions Examples

When evaluating Newton’s Quotient, we will be given a function such as f ( x ) = x 2 + 3. We need to find f ( a + h ) and f ( a ). Once we have these two quantities, we will put them into the quotient and simplify. Simplifying the quotient is usually the messiest part. For f ( x ) = x 2 + 3, we have f ( a + h ) = ( a + h ) 2 + 3 = ( a + h )( a + h ) + 3 = a 2 + 2 ah + h 2 + 3, and f ( a ) = a 2 + 3. We will substitute a 2 + 2 ah + h 2 + 3 for f ( a + h ) and a 2 + 3 for f ( a ).

Introduction to Functions Examples

Now we need to simplify this fraction.

Introduction to Functions Examples

Examples

Evaluate Newton’s Quotient for the given functions.

  • f ( x ) = 3 x 2

Introduction to Functions Examples

 

  • f ( x ) = x 2 − 2 x + 5

Introduction to Functions Examples

  • Introduction to Functions Examples

Introduction to Functions Examples

Do not worry—you will not spend a lot of time evaluating Newton’s Quotient in calculus, there are formulas that do most of the work. What is Newton’s Quotient, anyway? It is nothing more than the slope formula where x 1 = a, y 1 = f ( a ), x 2 = a + h , and y 2 = f ( a + h ).

Introduction to Functions Examples

Practice problems for this concept can be found at: Functions Practice Test.

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