Marsh Chandler
08/09/2024 · Primary School
3) \( \lim _{\Delta x \rightarrow 0} \frac{F(x-\Delta x)-F(x)}{\Delta x} \) a partir da \( F(x)=2 x^{3}-5 x^{2}+c x-a \)
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Para calcular o limite \( \lim _{\Delta x \rightarrow 0} \frac{F(x-\Delta x)-F(x)}{\Delta x} \) a partir da função \( F(x) = 2x^3 - 5x^2 + cx - a \), vamos primeiro encontrar \( F(x - \Delta x) \).
Substituindo \( x - \Delta x \) na função \( F \):
\[
F(x - \Delta x) = 2(x - \Delta x)^3 - 5(x - \Delta x)^2 + c(x - \Delta x) - a
\]
Agora, vamos expandir cada termo:
1. **Expansão de \( (x - \Delta x)^3 \)**:
\[
(x - \Delta x)^3 = x^3 - 3x^2\Delta x + 3x(\Delta x)^2 - (\Delta x)^3
\]
Portanto,
\[
2(x - \Delta x)^3 = 2x^3 - 6x^2\Delta x + 6x(\Delta x)^2 - 2(\Delta x)^3
\]
2. **Expansão de \( (x - \Delta x)^2 \)**:
\[
(x - \Delta x)^2 = x^2 - 2x\Delta x + (\Delta x)^2
\]
Portanto,
\[
-5(x - \Delta x)^2 = -5x^2 + 10x\Delta x - 5(\Delta x)^2
\]
3. **Expansão de \( c(x - \Delta x) \)**:
\[
c(x - \Delta x) = cx - c\Delta x
\]
Agora, juntando tudo, temos:
\[
F(x - \Delta x) = \left(2x^3 - 6x^2\Delta x + 6x(\Delta x)^2 - 2(\Delta x)^3\right) + \left(-5x^2 + 10x\Delta x - 5(\Delta x)^2\right) + \left(cx - c\Delta x\right) - a
\]
Agrupando os termos semelhantes:
\[
F(x - \Delta x) = (2x^3 - 5x^2 + cx - a) + (-6x^2 + 10x - c)\Delta x + (6x - 5)(\Delta x)^2 - 2(\Delta x)^3
\]
Agora, substituímos \( F(x) \):
\[
F(x - \Delta x) = F(x) + (-6x^2 + 10x - c)\Delta x + (6x - 5)(\Delta x)^2 - 2(\Delta x)^3
\]
Agora, vamos calcular \( F(x - \Delta x) - F(x) \):
\[
F(x - \Delta x) - F(x) = (-6x^2 + 10x - c)\Delta x + (6x - 5)(\Delta x)^2 - 2(\Delta x)^3
\]
Dividindo por \( \Delta x \):
\[
\frac{F(x - \Delta x) - F(x)}{\Delta x} = -6x^2 + 10x - c + (6x - 5)\Delta x - 2(\Delta x)^2
\]
Agora, tomamos o limite quando \( \Delta x \rightarrow 0 \):
\[
\lim_{\Delta x \rightarrow 0} \frac{F(x - \Delta x) - F(x)}{\Delta x} = -6x^2 + 10x - c
\]
Portanto, o resultado final é:
\[
\lim_{\Delta x \rightarrow 0} \frac{F(x - \Delta x) - F(x)}{\Delta x} = -6x^2 + 10x - c
\]
Quick Answer
O resultado final é \( -6x^2 + 10x - c \).
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