Math Problem Statement

The graph of f(x) and a line tangent to the curve at x = 4 are shown. If g(x) = √f(x), evaluate g'(4).

Solution

To solve this problem, we need to find the derivative g(4)g'(4), where g(x)=f(x)g(x) = \sqrt{f(x)} and f(x)f(x) is the function represented by the red curve on the graph. The problem states that there is a tangent line to f(x)f(x) at x=4x = 4, shown as the blue line.

Solution Steps

  1. Identify the relationship between g(x)g(x) and f(x)f(x): Since g(x)=f(x)g(x) = \sqrt{f(x)}, we can use the chain rule to find g(x)g'(x) in terms of f(x)f(x) and f(x)f'(x): g(x)=ddx(f(x))=12f(x)f(x)g'(x) = \frac{d}{dx} \left( \sqrt{f(x)} \right) = \frac{1}{2\sqrt{f(x)}} \cdot f'(x)

  2. Evaluate f(4)f(4): From the graph, find the value of f(4)f(4), which is the yy-coordinate of f(x)f(x) when x=4x = 4. According to the graph, f(4)4f(4) \approx 4.

  3. Evaluate f(4)f'(4): The derivative f(4)f'(4) represents the slope of the tangent line to f(x)f(x) at x=4x = 4. Based on the graph, we can estimate the slope of this tangent line (blue line). By observing the slope, it appears to go from approximately y=4y = 4 at x=4x = 4 to y=0y = 0 at x=6x = 6, giving us: f(4)0464=42=2f'(4) \approx \frac{0 - 4}{6 - 4} = \frac{-4}{2} = -2

  4. Calculate g(4)g'(4): Substitute f(4)=4f(4) = 4 and f(4)=2f'(4) = -2 into the expression for g(x)g'(x): g(4)=124(2)=122(2)=14(2)=12g'(4) = \frac{1}{2\sqrt{4}} \cdot (-2) = \frac{1}{2 \cdot 2} \cdot (-2) = \frac{1}{4} \cdot (-2) = -\frac{1}{2}

Final Answer

g(4)=12g'(4) = -\frac{1}{2}

Would you like further explanation on any part of this solution?

Here are some related questions to deepen understanding:

  1. What is the chain rule, and how is it applied in this example?
  2. How can we estimate the slope of a tangent line more precisely?
  3. Why is it necessary to evaluate both f(4)f(4) and f(4)f'(4) to find g(4)g'(4)?
  4. What other methods can be used to approximate the derivative at a point on a graph?
  5. How would g(x)g(x) change if g(x)=f(x)2g(x) = f(x)^2 instead?

Tip: When working with graphs, use points on the tangent line to calculate the slope accurately by choosing coordinates that are easy to identify.

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Math Problem Analysis

Mathematical Concepts

Calculus
Derivative
Chain Rule
Tangent Line

Formulas

g'(x) = (1 / (2√f(x))) * f'(x)

Theorems

Chain Rule

Suitable Grade Level

Grades 11-12