Questions: Previous Problem f(x)=6 x+4 Part 1 - Difference Quotient If h ≠ 0, then the difference quotient can be simplified into the form A+B h+C h^2. That is: (f(x+h)-f(x))/h=A+B h+C h^2 where the coefficients A, B, and C can be functions of x or just constant terms. Find these coefficients: A= B= C= (Note: It's possible for one or more of these coefficients to be 0.) Part 2 - Derivative Use the simplified expression from Part 1 to then calculate the derivative f′(x) : f′(x)=lim (h → 0) (f(x+h)-f(x))/h f′(x)=

Previous Problem f(x)=6 x+4 Part 1 - Difference Quotient If h ≠ 0, then the difference quotient can be simplified into the form A+B h+C h^2. That is: (f(x+h)-f(x))/h=A+B h+C h^2 where the coefficients A, B, and C can be functions of x or just constant terms. Find these coefficients: A= B= C= (Note: It's possible for one or more of these coefficients to be 0.) Part 2 - Derivative Use the simplified expression from Part 1 to then calculate the derivative f′(x) : f′(x)=lim (h → 0) (f(x+h)-f(x))/h f′(x)=
Transcript text: Previous Problem \[ f(x)=6 x+4 \] Part 1 - Difference Quotient If $h \neq 0$, then the difference quotient can be simplified into the form $A+B h+C h^{2}$. That is: \[ \frac{f(x+h)-f(x)}{h}=A+B h+C h^{2} \] where the coefficients $A, B$, and $C$ can be functions of $x$ or just constant terms. Find these coefficients: \[ A=\square B=\square C=\square \] (Note: It's possible for one or more of these coefficients to be 0 .) Part 2 - Derivative Use the simplified expression from Part 1 to then calculate the derivative $f^{\prime}(x)$ : \[ f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \] \[ f^{\prime}(x)=\square \]
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Solution

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Solution Steps

Solution Approach
  1. For Part 1, we need to find the difference quotient \(\frac{f(x+h)-f(x)}{h}\) for the given function \(f(x) = 6x + 4\). We will then simplify this expression to identify the coefficients \(A\), \(B\), and \(C\).
  2. For Part 2, we will use the simplified expression from Part 1 to calculate the derivative \(f'(x)\) by taking the limit as \(h\) approaches 0.
Step 1: Difference Quotient

To find the difference quotient for the function \( f(x) = 6x + 4 \), we calculate: \[ \frac{f(x+h) - f(x)}{h} = \frac{(6(x+h) + 4) - (6x + 4)}{h} = \frac{6h}{h} = 6 \] This simplifies to \( 6 \), which can be expressed in the form \( A + Bh + Ch^2 \) where \( A = 6 \), \( B = 0 \), and \( C = 0 \).

Step 2: Derivative

Using the simplified expression from Step 1, we calculate the derivative \( f'(x) \) as follows: \[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} = \lim_{h \to 0} 6 = 6 \]

Final Answer

The coefficients are: \[ A = 6, \quad B = 0, \quad C = 0 \] The derivative is: \[ f'(x) = 6 \] Thus, the final answers are: \[ \boxed{A = 6}, \quad \boxed{B = 0}, \quad \boxed{C = 0}, \quad \boxed{f'(x) = 6} \]

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