Answer :
To find the difference quotient of the function [tex]\( f(x) = 2x^2 - x + 3 \)[/tex], we need to compute and simplify [tex]\(\frac{f(x+h) - f(x)}{h}\)[/tex], where [tex]\( h \neq 0 \)[/tex].
### Step-by-Step Solution:
1. Compute [tex]\( f(x + h) \)[/tex]:
We need to substitute [tex]\( x + h \)[/tex] into the function [tex]\( f(x) \)[/tex].
[tex]\[ f(x + h) = 2(x + h)^2 - (x + h) + 3 \][/tex]
2. Expand [tex]\( f(x + h) \)[/tex]:
[tex]\[ f(x + h) = 2(x^2 + 2xh + h^2) - x - h + 3 \][/tex]
[tex]\[ f(x + h) = 2x^2 + 4xh + 2h^2 - x - h + 3 \][/tex]
3. Write [tex]\( f(x) \)[/tex]:
We already have [tex]\( f(x) \)[/tex] from the given function:
[tex]\[ f(x) = 2x^2 - x + 3 \][/tex]
4. Find [tex]\( f(x + h) - f(x) \)[/tex]:
[tex]\[ f(x + h) - f(x) = \left(2x^2 + 4xh + 2h^2 - x - h + 3\right) - \left(2x^2 - x + 3\right) \][/tex]
Distribute the subtraction:
[tex]\[ f(x + h) - f(x) = 2x^2 + 4xh + 2h^2 - x - h + 3 - 2x^2 + x - 3 \][/tex]
5. Simplify [tex]\( f(x + h) - f(x) \)[/tex]:
Combine like terms:
[tex]\[ f(x + h) - f(x) = 4xh + 2h^2 - h \][/tex]
6. Form the difference quotient:
[tex]\[ \frac{f(x + h) - f(x)}{h} = \frac{4xh + 2h^2 - h}{h} \][/tex]
7. Simplify the expression:
Factor [tex]\( h \)[/tex] out of the numerator:
[tex]\[ \frac{f(x + h) - f(x)}{h} = \frac{h(4x + 2h - 1)}{h} \][/tex]
Since [tex]\( h \neq 0 \)[/tex], we can cancel [tex]\( h \)[/tex] in the numerator and the denominator:
[tex]\[ \frac{f(x + h) - f(x)}{h} = 4x + 2h - 1 \][/tex]
Therefore, the simplified difference quotient is:
[tex]\[ \boxed{4x + 2h - 1} \][/tex]
### Step-by-Step Solution:
1. Compute [tex]\( f(x + h) \)[/tex]:
We need to substitute [tex]\( x + h \)[/tex] into the function [tex]\( f(x) \)[/tex].
[tex]\[ f(x + h) = 2(x + h)^2 - (x + h) + 3 \][/tex]
2. Expand [tex]\( f(x + h) \)[/tex]:
[tex]\[ f(x + h) = 2(x^2 + 2xh + h^2) - x - h + 3 \][/tex]
[tex]\[ f(x + h) = 2x^2 + 4xh + 2h^2 - x - h + 3 \][/tex]
3. Write [tex]\( f(x) \)[/tex]:
We already have [tex]\( f(x) \)[/tex] from the given function:
[tex]\[ f(x) = 2x^2 - x + 3 \][/tex]
4. Find [tex]\( f(x + h) - f(x) \)[/tex]:
[tex]\[ f(x + h) - f(x) = \left(2x^2 + 4xh + 2h^2 - x - h + 3\right) - \left(2x^2 - x + 3\right) \][/tex]
Distribute the subtraction:
[tex]\[ f(x + h) - f(x) = 2x^2 + 4xh + 2h^2 - x - h + 3 - 2x^2 + x - 3 \][/tex]
5. Simplify [tex]\( f(x + h) - f(x) \)[/tex]:
Combine like terms:
[tex]\[ f(x + h) - f(x) = 4xh + 2h^2 - h \][/tex]
6. Form the difference quotient:
[tex]\[ \frac{f(x + h) - f(x)}{h} = \frac{4xh + 2h^2 - h}{h} \][/tex]
7. Simplify the expression:
Factor [tex]\( h \)[/tex] out of the numerator:
[tex]\[ \frac{f(x + h) - f(x)}{h} = \frac{h(4x + 2h - 1)}{h} \][/tex]
Since [tex]\( h \neq 0 \)[/tex], we can cancel [tex]\( h \)[/tex] in the numerator and the denominator:
[tex]\[ \frac{f(x + h) - f(x)}{h} = 4x + 2h - 1 \][/tex]
Therefore, the simplified difference quotient is:
[tex]\[ \boxed{4x + 2h - 1} \][/tex]
