**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*.

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* ).

Now we need to simplify this fraction.

**Examples**

Evaluate Newton’s Quotient for the given functions.

*f*(*x*) = 3*x*^{2}

*f*(*x*) =*x*^{2}− 2*x*+ 5

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* ).

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

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