Question
Remember that \( f(x)=5 x-3 x^{2} \) Find \( f(x+h)-f(x) \) \( \square \)
Ask by Potter Collins. in the United States
Nov 14,2024
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( f(x+h) - f(x) = 5h - 6xh - 3h^2 \)
Solution
To find \( f(x+h) - f(x) \) for the function \( f(x) = 5x - 3x^2 \), we first need to calculate \( f(x+h) \).
1. **Calculate \( f(x+h) \)**:
\[
f(x+h) = 5(x+h) - 3(x+h)^2
\]
Expanding this:
\[
f(x+h) = 5x + 5h - 3((x+h)^2)
\]
Now, expand \( (x+h)^2 \):
\[
(x+h)^2 = x^2 + 2xh + h^2
\]
Therefore,
\[
f(x+h) = 5x + 5h - 3(x^2 + 2xh + h^2)
\]
Distributing the \(-3\):
\[
f(x+h) = 5x + 5h - 3x^2 - 6xh - 3h^2
\]
Combining like terms:
\[
f(x+h) = (5x - 3x^2) + (5h - 6xh - 3h^2)
\]
Thus,
\[
f(x+h) = f(x) + (5h - 6xh - 3h^2)
\]
2. **Now, calculate \( f(x+h) - f(x) \)**:
\[
f(x+h) - f(x) = (f(x) + (5h - 6xh - 3h^2)) - f(x)
\]
This simplifies to:
\[
f(x+h) - f(x) = 5h - 6xh - 3h^2
\]
Therefore, the final result is:
\[
\boxed{5h - 6xh - 3h^2}
\]
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Simplify this solution Mind Expander
To find \( f(x+h) - f(x) \), we first calculate \( f(x+h) \): \[ f(x+h) = 5(x+h) - 3(x+h)^2 \] Expanding this gives us: \[ f(x+h) = 5x + 5h - 3(x^2 + 2xh + h^2) = 5x + 5h - 3x^2 - 6xh - 3h^2 \] Now, substituting for \( f(x) = 5x - 3x^2 \), we have: \[ f(x) = 5x - 3x^2 \] Now calculating \( f(x+h) - f(x) \): \[ f(x+h) - f(x) = (5x + 5h - 3x^2 - 6xh - 3h^2) - (5x - 3x^2) \] Simplifying the expression: \[ = 5h - 6xh - 3h^2 \] So, the final answer is: \[ f(x+h) - f(x) = 5h - 6xh - 3h^2 \]
