To evaluate the difference quotient for the function [tex]\( f(x) = 2x - 9 \)[/tex], we will use the definition of the difference quotient, which is:
[tex]\[
\frac{f(x + h) - f(x)}{h}
\][/tex]
Here's the step-by-step solution:
1. Identify the function:
[tex]\[
f(x) = 2x - 9
\][/tex]
2. Calculate [tex]\( f(x + h) \)[/tex]:
Substitute [tex]\( x + h \)[/tex] into the function [tex]\( f(x) \)[/tex]:
[tex]\[
f(x + h) = 2(x + h) - 9
\][/tex]
Simplify the expression:
[tex]\[
f(x + h) = 2x + 2h - 9
\][/tex]
3. Substitute [tex]\( f(x + h) \)[/tex] and [tex]\( f(x) \)[/tex] into the difference quotient formula:
[tex]\[
\frac{f(x + h) - f(x)}{h}
\][/tex]
4. Substitute the expressions we found:
[tex]\[
\frac{(2x + 2h - 9) - (2x - 9)}{h}
\][/tex]
5. Simplify the numerator:
[tex]\[
(2x + 2h - 9) - (2x - 9) = 2x + 2h - 9 - 2x + 9 = 2h
\][/tex]
6. Form the difference quotient:
[tex]\[
\frac{2h}{h}
\][/tex]
7. Simplify the result:
[tex]\[
\frac{2h}{h} = 2
\][/tex]
This expression [tex]\( 2 \)[/tex] is the simplified form of the difference quotient for any [tex]\( h \neq 0 \)[/tex]. To provide a specific example, if we choose [tex]\( x = 1 \)[/tex] and [tex]\( h = 0.01 \)[/tex]:
- The difference quotient would still be [tex]\( 2 \)[/tex], as the calculation doesn't depend on the specific values of [tex]\( x \)[/tex] and [tex]\( h \)[/tex] other than [tex]\( h \)[/tex] needing to be non-zero.
Therefore, the evaluated difference quotient for the function [tex]\( f(x) = 2x - 9 \)[/tex] is:
[tex]\[
2
\][/tex]
However, to compare, if we look at a numerical example closer to what was made:
For [tex]\( x = 1 \)[/tex] and [tex]\( h = 0.01 \)[/tex]:
[tex]\[
\frac{f(1 + 0.01) - f(1)}{0.01}
\][/tex]
Calculate [tex]\( f(1.01) \)[/tex]:
[tex]\[
f(1.01) = 2(1.01) - 9 = 2.02 - 9 = -6.98
\][/tex]
Calculate [tex]\( f(1) \)[/tex]:
[tex]\[
f(1) = 2(1) - 9 = 2 - 9 = -7
\][/tex]
Now substitute into the difference quotient:
[tex]\[
\frac{-6.98 - (-7)}{0.01} = \frac{-6.98 + 7}{0.01} = \frac{0.02}{0.01} = 2
\][/tex]
Given the high precision of the original Python result, we'd find:
[tex]\[
1.9999999999999574
\][/tex]
which is very close to mathematically exact [tex]\(2\)[/tex].
