Math Problem Statement

Find f(a), f(a + h), and the difference quotient [f(a + h) - f(a)] / h, where f(x) = 8 - 7x.

Solution

Let's go through the problem step by step.

We are given the function: f(x)=87xf(x) = 8 - 7x We need to find:

  1. f(a)f(a)
  2. f(a+h)f(a + h)
  3. The difference quotient: f(a+h)f(a)h\frac{f(a + h) - f(a)}{h}

1. Finding f(a)f(a):

Substitute x=ax = a into the function f(x)=87xf(x) = 8 - 7x: f(a)=87af(a) = 8 - 7a

2. Finding f(a+h)f(a + h):

Substitute x=a+hx = a + h into the function: f(a+h)=87(a+h)=87a7hf(a + h) = 8 - 7(a + h) = 8 - 7a - 7h

3. Finding the difference quotient:

f(a+h)f(a)h\frac{f(a + h) - f(a)}{h} Substitute the expressions for f(a+h)f(a + h) and f(a)f(a): (87a7h)(87a)h\frac{(8 - 7a - 7h) - (8 - 7a)}{h} Simplify the numerator: =87a7h8+7ah=7hh= \frac{8 - 7a - 7h - 8 + 7a}{h} = \frac{-7h}{h} Cancel out hh (since h0h \neq 0): =7= -7

Final answers:

  1. f(a)=87af(a) = 8 - 7a
  2. f(a+h)=87a7hf(a + h) = 8 - 7a - 7h
  3. f(a+h)f(a)h=7\frac{f(a + h) - f(a)}{h} = -7

Would you like further details or clarifications on any part of this solution?

Here are 5 related questions for practice:

  1. What is the derivative of the function f(x)=87xf(x) = 8 - 7x?
  2. Find the difference quotient for the function f(x)=x24xf(x) = x^2 - 4x.
  3. What is the physical interpretation of the difference quotient in real-world problems?
  4. How does the difference quotient relate to the slope of a secant line?
  5. How would the difference quotient change if we used a quadratic function?

Tip: The difference quotient is the foundation of the derivative. Understanding how it works will help you with concepts in calculus, especially when studying limits and instantaneous rates of change.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Functions
Difference Quotient

Formulas

f(x) = 8 - 7x
Difference Quotient = [f(a + h) - f(a)] / h

Theorems

Linear Function Analysis

Suitable Grade Level

Grade 10-12 (Pre-Calculus)