Questions: a) The difference quotient is (f(x+h)-f(x))/h=□. (Simplify your answer.) b) Complete the following table. x h (f(x+h)-f(x))/h 3 2 □ 3 1 □ 3 0.1 □ 3 0.01 □ (Round to three decimal places as needed.)

a) The difference quotient is (f(x+h)-f(x))/h=□.
(Simplify your answer.)
b) Complete the following table.
x h (f(x+h)-f(x))/h
3 2 □
3 1 □
3 0.1 □
3 0.01 □
(Round to three decimal places as needed.)

Transcript text: a) The difference quotient is $\frac{f(x+h)-f(x)}{h}=\square$. (Simplify your answer.) b) Complete the following table. \begin{tabular}{|c|c|c|} \hline$x$ & $h$ & $\frac{f(x+h)-f(x)}{h}$ \\ \hline 3 & 2 & $\square$ \\ \hline 3 & 1 & $\square$ \\ \hline 3 & 0.1 & $\square$ \\ \hline 3 & 0.01 & $\square$ \\ \hline \end{tabular} (Round to three decimal places as needed.)

Solution

Solution Steps

Step 1: Simplify the Difference Quotient

The given function is $f(x) = f(x)$. The difference quotient is defined as $\dfrac{f(x+h)-f(x)}{h}$. After substituting and simplifying, we get: $\dfrac{f(x+h)-f(x)}{h} = \frac{- f{\left(x \right)} + f{\left(h + x \right)}}{h}$.

Step 2: Evaluate for Specific Values and Fill in the Table

| x | h | Difference Quotient | |-|-|-| | 3 | 2 | -0.5_f(3) + 0.5_f(5) | | 3 | 1 | -f(3) + f(4) | | 3 | 0.1 | -10_f(3) + 10_f(3.1) | | 3 | 0.01 | -100_f(3) + 100_f(3.01) |

Final Answer:

The simplified form of the difference quotient for the function $f(x) = f(x)$ is $\frac{- f{\left(x \right)} + f{\left(h + x \right)}}{h}$. The table above shows the evaluated values of this expression for the given values of x and h.

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