Math Problem Statement
Find the difference quotient \(\frac{f(x+h) - f(x)}{h}\), where h ≠ 0, for the function \( f(x) = -5x + 8 \).
Solution
Let's solve the given problem step-by-step.
Problem:
We need to find the difference quotient
for the function .
Step 1: Calculate
Since , substitute in place of :
= -5x - 5h + 8$$ --- ### Step 2: Find the Difference $$f(x + h) - f(x)$$ Now, subtract $$f(x)$$ from $$f(x + h)$$: $$f(x + h) - f(x) = (-5x - 5h + 8) - (-5x + 8)$$ Simplify: $$f(x + h) - f(x) = -5x - 5h + 8 + 5x - 8 = -5h$$ --- ### Step 3: Form the Difference Quotient Now, plug the result of $$f(x + h) - f(x) = -5h$$ into the difference quotient: $$\frac{f(x+h) - f(x)}{h} = \frac{-5h}{h}$$ Since $$h \neq 0$$, we can cancel $$h$$: $$\frac{-5h}{h} = -5$$ --- ### Final Answer: $$\frac{f(x+h) - f(x)}{h} = -5$$ This means the slope of the linear function $$f(x) = -5x + 8$$ is $$-5$$, which is consistent with the derivative of this linear function. --- Do you need more clarification or have any further questions? ### Five Related Questions: 1. How does the difference quotient relate to the derivative of a function? 2. What would change if the function were $$f(x) = -5x + 8 + x^2$$? 3. How does the slope interpretation change for non-linear functions? 4. Why do we cancel $$h$$ in the difference quotient only when $$h \neq 0$$? 5. How is the difference quotient useful in calculus beyond derivatives? ### Tip: For linear functions $$f(x) = mx + b$$, the difference quotient always gives the slope $$m$$, which is also the derivative.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Difference Quotient
Linear Functions
Algebra
Formulas
Difference Quotient: \(\frac{f(x+h) - f(x)}{h}\)
Linear Function: \(f(x) = mx + b\)
Theorems
Linear function slope theorem
Suitable Grade Level
Grades 9-12