Sure, let's solve the given problem step-by-step:
Given the function [tex]\( f(x) = 2x + 3 \)[/tex]:
### (a) Calculate [tex]\(\frac{f(3+h) - f(3)}{h}\)[/tex]
1. Evaluate [tex]\( f(3+h) \)[/tex]:
- Substitute [tex]\( 3 + h \)[/tex] into the function [tex]\( f(x) \)[/tex].
- [tex]\( f(3 + h) = 2(3 + h) + 3 \)[/tex]
- Simplify the expression:
[tex]\[
f(3 + h) = 2 \cdot 3 + 2 \cdot h + 3 = 6 + 2h + 3 = 9 + 2h
\][/tex]
2. Evaluate [tex]\( f(3) \)[/tex]:
- Substitute [tex]\( 3 \)[/tex] into the function [tex]\( f(x) \)[/tex].
- [tex]\( f(3) = 2 \cdot 3 + 3 \)[/tex]
- Simplify the expression:
[tex]\[
f(3) = 6 + 3 = 9
\][/tex]
3. Calculate the Difference Quotient:
- The difference quotient is given by:
[tex]\[
\frac{f(3 + h) - f(3)}{h}
\][/tex]
- Substitute the values we found for [tex]\( f(3 + h) \)[/tex] and [tex]\( f(3) \)[/tex]:
[tex]\[
\frac{(9 + 2h) - 9}{h}
\][/tex]
- Simplify the numerator:
[tex]\[
\frac{9 + 2h - 9}{h} = \frac{2h}{h}
\][/tex]
- Simplify the fraction:
[tex]\[
\frac{2h}{h} = 2
\][/tex]
4. Final Answer:
- The difference quotient simplifies to:
[tex]\[
\frac{f(3 + h) - f(3)}{h} = 2
\][/tex]
So, the detailed step-by-step solution shows that:
[tex]\[
\frac{f(3+h) - f(3)}{h} = 2
\][/tex]
