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11. Find and simplify the difference quotient for f(x)=4x-5 a. 4x+h b. 4h c. 4 d. 4h-5

Pergunta

11. Find and simplify the difference quotient for f(x)=4x-5
a. 4x+h
b. 4h
c. 4
d. 4h-5

11. Find and simplify the difference quotient for f(x)=4x-5 a. 4x+h b. 4h c. 4 d. 4h-5

Solução

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MayraEspecialista · Tutor por 3 anos

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To find and simplify the difference quotient for the function \( f(x) = 4x - 5 \), we follow these steps:<br /><br />1. **Difference Quotient Formula**: The difference quotient is given by:<br /> \[<br /> \frac{f(x+h) - f(x)}{h}<br /> \]<br /><br />2. **Calculate \( f(x+h) \)**:<br /> \[<br /> f(x+h) = 4(x+h) - 5<br /> \]<br /> Simplify this expression:<br /> \[<br /> f(x+h) = 4x + 4h - 5<br /> \]<br /><br />3. **Substitute \( f(x) \) and \( f(x+h) \) into the Difference Quotient**:<br /> \[<br /> \frac{f(x+h) - f(x)}{h} = \frac{(4x + 4h - 5) - (4x - 5)}{h}<br /> \]<br /><br />4. **Simplify the Numerator**:<br /> \[<br /> \frac{(4x + 4h - 5) - (4x - 5)}{h} = \frac{4x + 4h - 5 - 4x + 5}{h}<br /> \]<br /> Combine like terms:<br /> \[<br /> \frac{4h}{h}<br /> \]<br /><br />5. **Simplify the Fraction**:<br /> \[<br /> \frac{4h}{h} = 4<br /> \]<br /><br />Therefore, the simplified difference quotient for \( f(x) = 4x - 5 \) is \( 4 \).<br /><br />The correct answer is:<br />c. 4
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