Math Problem Statement

Let G = 4f - g, where the graphs of f and g are shown. Find G'(2).

Solution

We are given that G(x)=4f(x)g(x)G(x) = 4f(x) - g(x), and we need to find G(2)G'(2), which is the derivative of G(x)G(x) evaluated at x=2x = 2.

Using basic differentiation rules, we know:

G(x)=4f(x)g(x)G'(x) = 4f'(x) - g'(x)

Thus, to find G(2)G'(2), we need to compute 4f(2)g(2)4f'(2) - g'(2).

Steps:

  1. From the graph, we will estimate the slopes (derivatives) f(x)f'(x) and g(x)g'(x) at x=2x = 2.
  2. We will multiply f(2)f'(2) by 4, then subtract g(2)g'(2) from it to get G(2)G'(2).

Looking at the graph:

  • For f(2)f'(2): The graph of f(x)f(x) appears to be a straight line at x=2x = 2, and the slope is the change in yy over the change in xx. By estimation from the graph, we see that for f(x)f(x), the slope appears to be negative. A rough estimate gives f(2)1f'(2) \approx -1.

  • For g(2)g'(2): Similarly, for g(x)g(x), the slope at x=2x = 2 also looks negative but steeper than f(x)f(x). A rough estimate gives g(2)2g'(2) \approx -2.

Now substitute these estimates into the equation for G(2)G'(2):

G(2)=4(1)(2)=4+2=2G'(2) = 4(-1) - (-2) = -4 + 2 = -2

So, G(2)2G'(2) \approx -2.

Would you like more details on how to estimate slopes from graphs?

Here are some related questions:

  1. How is the derivative of a function defined geometrically?
  2. How do you differentiate a function with multiple terms like G(x)=4f(x)g(x)G(x) = 4f(x) - g(x)?
  3. What is the significance of the slope of a function at a particular point?
  4. How do you compute the derivative when dealing with piecewise or non-linear graphs?
  5. Can we use numerical methods to estimate derivatives from graphical data?

Tip: When estimating slopes from graphs, focus on identifying two clear points on the line and use the formula for the slope: slope=ΔyΔx\text{slope} = \frac{\Delta y}{\Delta x}.

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Math Problem Analysis

Mathematical Concepts

Differentiation
Graphs
Slope estimation

Formulas

G'(x) = 4f'(x) - g'(x)
Slope formula: (Δy / Δx)

Theorems

Derivative definition
Linear approximation

Suitable Grade Level

Grades 11-12