f(x+ h) - f(x) Find and simplify the difference quotient- ,h 0 for the given function. h 5) f(x) = 8x - 3

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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solve question 5 and 6

**Educational Content Transcription**

---

### Graph the Function

**4)** \( f(x) = \begin{cases} 
x + 4 & \text{if } x \leq 1 \\ 
-1 & \text{if } x > 1 
\end{cases} \)

#### Graph
- **Axes:** The graph includes a y-axis labeled from -5 to 5 and an x-axis labeled similarly.
- **Graph Details:** 
  - The function \( f(x) \) behaves differently based on the value of \( x \).
  - For \( x \leq 1 \), the function is represented by the linear equation \( x + 4 \).
  - For \( x > 1 \), the function is a horizontal line \( y = -1 \).

**5)** Find and simplify the difference quotient \(\frac{f(x+h) - f(x)}{h}\), \(h \neq 0\), for the given function.

\( f(x) = 8x - 3 \)

---

### Solve

**6)** A faucet is used to add water to a large bottle that already contained some water. After it has been filling for 5 seconds, the gauge on the bottle indicates that it contains 12 ounces of water. After it has been filling for 12 seconds, the gauge indicates the bottle contains 26 ounces of water. Let \( y \) be the amount of water in the bottle \( x \) seconds after the faucet was turned on. Write a linear equation that models the amount of water in the bottle in terms of \( x \).

---

(Page A-2)
Transcribed Image Text:**Educational Content Transcription** --- ### Graph the Function **4)** \( f(x) = \begin{cases} x + 4 & \text{if } x \leq 1 \\ -1 & \text{if } x > 1 \end{cases} \) #### Graph - **Axes:** The graph includes a y-axis labeled from -5 to 5 and an x-axis labeled similarly. - **Graph Details:** - The function \( f(x) \) behaves differently based on the value of \( x \). - For \( x \leq 1 \), the function is represented by the linear equation \( x + 4 \). - For \( x > 1 \), the function is a horizontal line \( y = -1 \). **5)** Find and simplify the difference quotient \(\frac{f(x+h) - f(x)}{h}\), \(h \neq 0\), for the given function. \( f(x) = 8x - 3 \) --- ### Solve **6)** A faucet is used to add water to a large bottle that already contained some water. After it has been filling for 5 seconds, the gauge on the bottle indicates that it contains 12 ounces of water. After it has been filling for 12 seconds, the gauge indicates the bottle contains 26 ounces of water. Let \( y \) be the amount of water in the bottle \( x \) seconds after the faucet was turned on. Write a linear equation that models the amount of water in the bottle in terms of \( x \). --- (Page A-2)
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