Working a difference quotient involving a square root
Working a difference quotient involving a square root
Suppose f (x) = x and suppose we want to simplify the differnce quotient
f (x + h) - f (x) h
as much as possible (say, to eliminate the h in the denominator).
Substituting the definition of f into the quotient, we have
f (x + h) - f (x) x + h - x
=
h
h
at which point we are stuck, as far as basic algebraic manipulations go.
To the rescue, however, comes the conjugate.
For any expression of the form A- B, we say its conjugate is A+ B, and vice versa: the conjugate of
the latter is the former: we get to the expressions conjugate by simply changing the sign of the operation
between the two square root expressions (plus to minus, or minus to plus).
By writing the number 1 as the expression's conjugate divided by itself, we get a powerful tool for manipulating these types of expressions.
With
x+h- x
,
h
the conjugate we want to use is x + h + x, so we multiply our expression by the conjugate over itself:
x+h- x x+h- x x+h+ x
h
=
h
?
.
x+h+ x
The key idea is that the numerators multiply in a nice way. Note that the two numerators together have the form
(A - B) ? (A + B)
which is equal to A2 - B2 (you might recall the phrase difference of squares). The squaring eliminates the square roots from the numerator.
As a result, our expression above becomes
x+h- h
x
?
x x
+ +
h h
+ +
x
x
=
x
+
h
-
x
h( x + h - x)
=
h( x
h +h
-
x)
=
h h
x
+
1 h
-
x
=
x
+
1 h
-
. x
Thus, we have shown that
x+h- x =
1 .
h
x+h- x
This is as simplified as we can make it, and it has the advantage over the original expression in that it has no h in the denominator (which will be a consideration when you see this sort of thing again in Calculus).
................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.
Related download
- derivatives math 120 calculus i
- table of integrals
- vector matrix and tensor derivatives
- properties of the trace and matrix derivatives
- chapter 4 fourier series and integrals
- working a difference quotient involving a square root
- derivation of the inverse hyperbolic trig functions
- section 14 5 3 23 08 directional derivatives and
- truncation errors using taylor series to approximation
- derivatives instantaneous velocity
Related searches
- find the difference quotient solver
- square root without calculator
- how to figure out square root problems
- square root of a number
- square root chart
- square root table 1 100 pdf
- 3 square root x
- how to make a square root symbol
- square root cube root chart
- calculator square root 10 square root 5
- how to simplify a square root expression
- how to use the difference quotient formula