To solve the problem, let's analyze each part carefully using the information given on the graph.

Given:

• $h(x) = \frac{f(x)}{g(x)}$
• We are asked to find the derivatives $h'(2)$, $h'(3)$, and $h'(4)$ using the graph.

The derivative $h'(x)$ of a quotient function $h(x) = \frac{f(x)}{g(x)}$ is given by the quotient rule: $h'(x) = \frac{f'(x) g(x) - f(x) g'(x)}{[g(x)]^2}$ where $f'(x)$ and $g'(x)$ are the derivatives of $f(x)$ and $g(x)$, respectively.

Let's examine each part:

Part (a): Find $h'(2)$

• Solution: From the graph, we determine $f(2)$, $g(2)$, $f'(2)$, and $g'(2)$.
  • $f(2) = 2$
  • $g(2) = 1$
  • $f'(2) \approx -1$ (since $f(x)$ has a slope of $-1$ at $x = 2$)
  • $g'(2) = 0$ (since $g(x)$ is flat at $x = 2$)
  • Using the quotient rule: $h'(2) = \frac{f'(2) g(2) - f(2) g'(2)}{[g(2)]^2} = \frac{(-1)(1) - (2)(0)}{1^2} = -1$
  • Answer: $h'(2) = -1$ (This matches the correct answer already marked.)

Part (b): Find $h'(3)$

• Solution: From the graph, we find $f(3)$, $g(3)$, $f'(3)$, and $g'(3)$.
  • $f(3) = 0$
  • $g(3) = 0$ (since both functions intersect the x-axis at $x = 3$)
  • Because both $f(3) = 0$ and $g(3) = 0$, the function $h(x) = \frac{f(x)}{g(x)}$ is undefined at $x = 3$, so $h'(3)$ does not exist.
  • Answer: $h'(3) = \text{DNE}$ (This also matches the correct answer.)

Part (c): Find $h'(4)$

• Solution: Let's determine $f(4)$, $g(4)$, $f'(4)$, and $g'(4)$.
  • $f(4) = 1$
  • $g(4) = 2$
  • $f'(4) = 1$ (since $f(x)$ has a slope of $1$ at $x = 4$)
  • $g'(4) = 0$ (since $g(x)$ is flat at $x = 4$)
  • Applying the quotient rule: $h'(4) = \frac{f'(4) g(4) - f(4) g'(4)}{[g(4)]^2} = \frac{(1)(2) - (1)(0)}{2^2} = \frac{2}{4} = \frac{1}{2}$
  • Answer: $h'(4) = \frac{1}{2}$

Let me know if you'd like further explanation on any part of the solution or if you have additional questions!

Related Questions:

1. How is the quotient rule applied to find derivatives of function quotients?
2. What happens if both the numerator and denominator equal zero at a given point?
3. How do we interpret the slopes of piecewise functions from a graph?
4. How would $h(x)$ change if $g(x)$ were to become negative?
5. What is the significance of a point where $h(x)$ is undefined in a derivative context?

Tip: When applying the quotient rule, always check for cases where the denominator is zero, as this could indicate an undefined point or require careful handling to avoid division by zero.