To evaluate and simplify the difference quotient for the function [tex]\( f(x) = 4x + 10 \)[/tex], we'll go through the following steps:
1. Definition of the Difference Quotient:
The difference quotient is given by the formula:
[tex]\[
\frac{f(x + h) - f(x)}{h}
\][/tex]
where [tex]\( h \)[/tex] is a small increment and [tex]\( f(x + h) \)[/tex] is the function evaluated at [tex]\( x + h \)[/tex].
2. Evaluate [tex]\( f(x + h) \)[/tex]:
We need to find the value of the function when [tex]\( x \)[/tex] is replaced by [tex]\( x + h \)[/tex]. For our function [tex]\( f(x) = 4x + 10 \)[/tex]:
[tex]\[
f(x + h) = 4(x + h) + 10
\][/tex]
Simplifying this, we get:
[tex]\[
f(x + h) = 4x + 4h + 10
\][/tex]
3. Form the Difference Quotient:
Substitute [tex]\( f(x + h) \)[/tex] and [tex]\( f(x) \)[/tex] into the difference quotient formula:
[tex]\[
\frac{f(x + h) - f(x)}{h} = \frac{(4x + 4h + 10) - (4x + 10)}{h}
\][/tex]
Simplifying the numerator:
[tex]\[
\frac{(4x + 4h + 10) - 4x - 10}{h} = \frac{4h}{h}
\][/tex]
4. Simplify the Difference Quotient:
Simplify the expression by canceling out [tex]\( h \)[/tex]:
[tex]\[
\frac{4h}{h} = 4
\][/tex]
Therefore, the simplified difference quotient for the function [tex]\( f(x) = 4x + 10 \)[/tex] is:
[tex]\[
4
\][/tex]
The evaluated and simplified difference quotient confirms that the rate of change of the function [tex]\( f(x) \)[/tex] is consistently 4.
