Math Problem Statement

solve

Let G = 7f - g, where the graphs of f and g are shown in the figure. Find G'(6).

Solution

We are tasked with finding G(6)G'(6), given that G(x)=7f(x)g(x)G(x) = 7f(x) - g(x), where the graphs of f(x)f(x) and g(x)g(x) are provided. We need to evaluate the derivative of GG at x=6x = 6.

Steps:

  1. Find G(x)G'(x): G(x)=7f(x)g(x)G(x) = 7f(x) - g(x) Taking the derivative of both sides: G(x)=7f(x)g(x)G'(x) = 7f'(x) - g'(x) So, we need to compute f(6)f'(6) and g(6)g'(6) from the graphs.

  2. Determine f(6)f'(6) and g(6)g'(6): The derivative at a specific point is the slope of the tangent line at that point.

    • For f(x)f(x) at x=6x = 6:
      From the graph, f(x)f(x) appears to be a straight line with a positive slope after x=4x = 4. We can calculate the slope between points (4,4)(4, 4) and (8,8)(8, 8), giving: f(6)=8484=1f'(6) = \frac{8 - 4}{8 - 4} = 1

    • For g(x)g(x) at x=6x = 6:
      From the graph, g(x)g(x) appears to have a negative slope between x=4x = 4 and x=8x = 8. The slope between points (4,8)(4, 8) and (8,0)(8, 0) is: g(6)=0884=84=2g'(6) = \frac{0 - 8}{8 - 4} = \frac{-8}{4} = -2

  3. Evaluate G(6)G'(6): Using the values for f(6)f'(6) and g(6)g'(6): G(6)=7(1)(2)=7+2=9G'(6) = 7(1) - (-2) = 7 + 2 = 9

Thus, G(6)=9G'(6) = 9.


Would you like further details on the steps, or do you have any questions? Here are 5 related questions you might find useful:

  1. How do you calculate the derivative from a graph in general?
  2. What is the geometric interpretation of the derivative at a point?
  3. How would the solution change if the functions were non-linear?
  4. What is the physical meaning of multiplying a derivative by a constant, like 7 in this problem?
  5. Can you explain the relationship between the slopes of different functions and their derivatives?

Tip: Always check the slope of the graph carefully when calculating derivatives from visual data.

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Math Problem Analysis

Mathematical Concepts

Differentiation
Graph interpretation
Slopes of linear functions

Formulas

G(x) = 7f(x) - g(x)
G'(x) = 7f'(x) - g'(x)
Slope formula: (y2 - y1) / (x2 - x1)

Theorems

Derivative as slope of the tangent line

Suitable Grade Level

Grades 10-12