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Sagot :
To evaluate [tex]\(\frac{f(x+h) - f(x)}{h}\)[/tex] given that [tex]\(f(x) = x^2 - 5\)[/tex], follow these steps:
1. Determine [tex]\(f(x)\)[/tex]:
Given [tex]\(f(x) = x^2 - 5\)[/tex].
2. Determine [tex]\(f(x+h)\)[/tex]:
Substitute [tex]\(x + h\)[/tex] into the function:
[tex]\[ f(x+h) = (x + h)^2 - 5 \][/tex]
3. Expand [tex]\(f(x+h)\)[/tex]:
[tex]\[ f(x+h) = (x + h)^2 - 5 = x^2 + 2xh + h^2 - 5 \][/tex]
4. Substitute [tex]\(f(x)\)[/tex] and [tex]\(f(x+h)\)[/tex] into the difference quotient:
The difference quotient is:
[tex]\[ \frac{f(x+h) - f(x)}{h} \][/tex]
Plug in [tex]\(f(x)\)[/tex] and [tex]\(f(x+h)\)[/tex]:
[tex]\[ \frac{(x^2 + 2xh + h^2 - 5) - (x^2 - 5)}{h} \][/tex]
5. Simplify the numerator:
[tex]\[ (x^2 + 2xh + h^2 - 5) - (x^2 - 5) = x^2 + 2xh + h^2 - 5 - x^2 + 5 = 2xh + h^2 \][/tex]
6. Combine like terms:
[tex]\[ \frac{2xh + h^2}{h} \][/tex]
7. Factor out [tex]\(h\)[/tex] from the numerator:
[tex]\[ \frac{h(2x + h)}{h} \][/tex]
8. Cancel the [tex]\(h\)[/tex] in the numerator and the denominator:
[tex]\[ 2x + h \][/tex]
Therefore, the evaluated difference quotient [tex]\(\frac{f(x+h) - f(x)}{h}\)[/tex] is:
[tex]\[ \frac{f(x+h) - f(x)}{h} = \frac{(x^2 + 2xh + h^2 - 5) - (x^2 - 5)}{h} = 2x + h \][/tex]
1. Determine [tex]\(f(x)\)[/tex]:
Given [tex]\(f(x) = x^2 - 5\)[/tex].
2. Determine [tex]\(f(x+h)\)[/tex]:
Substitute [tex]\(x + h\)[/tex] into the function:
[tex]\[ f(x+h) = (x + h)^2 - 5 \][/tex]
3. Expand [tex]\(f(x+h)\)[/tex]:
[tex]\[ f(x+h) = (x + h)^2 - 5 = x^2 + 2xh + h^2 - 5 \][/tex]
4. Substitute [tex]\(f(x)\)[/tex] and [tex]\(f(x+h)\)[/tex] into the difference quotient:
The difference quotient is:
[tex]\[ \frac{f(x+h) - f(x)}{h} \][/tex]
Plug in [tex]\(f(x)\)[/tex] and [tex]\(f(x+h)\)[/tex]:
[tex]\[ \frac{(x^2 + 2xh + h^2 - 5) - (x^2 - 5)}{h} \][/tex]
5. Simplify the numerator:
[tex]\[ (x^2 + 2xh + h^2 - 5) - (x^2 - 5) = x^2 + 2xh + h^2 - 5 - x^2 + 5 = 2xh + h^2 \][/tex]
6. Combine like terms:
[tex]\[ \frac{2xh + h^2}{h} \][/tex]
7. Factor out [tex]\(h\)[/tex] from the numerator:
[tex]\[ \frac{h(2x + h)}{h} \][/tex]
8. Cancel the [tex]\(h\)[/tex] in the numerator and the denominator:
[tex]\[ 2x + h \][/tex]
Therefore, the evaluated difference quotient [tex]\(\frac{f(x+h) - f(x)}{h}\)[/tex] is:
[tex]\[ \frac{f(x+h) - f(x)}{h} = \frac{(x^2 + 2xh + h^2 - 5) - (x^2 - 5)}{h} = 2x + h \][/tex]
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