Questions: Let r(x)=q(x) · p(x). Determine r(0). Write your answer as an integer or reduced fraction.

$Let r(x)=q(x) · p(x). Determine r(0). Write your answer as an integer or reduced fraction.$

Transcript text: Let $r(x)=q(x) \cdot p(x)$. Determine $r(0)$. Write your answer as an integer or reduced fraction.

Solution

Step 1: Identify the functions p(x) and q(x) at x = 0

From the graph, we can see the values of p(x) and q(x) at x = 0:

Step 2: Determine the derivatives p'(x) and q'(x) at x = 0

To find the derivatives, we need to determine the slopes of the lines at x = 0:

For $ p(x) $, the slope between the points (-1, -1) and (1, 1) is $ \frac{1 - (-1)}{1 - (-1)} = \frac{2}{2} = 1 $. So, $ p'(0) = 1 $.
For $ q(x) $, the slope between the points (-1, -2) and (1, 0) is $ \frac{0 - (-2)}{1 - (-1)} = \frac{2}{2} = 1 $. So, $ q'(0) = 1 $.

Step 3: Apply the product rule to find r'(x) at x = 0

The function $ r(x) = q(x) \cdot p(x) $. Using the product rule, $ r'(x) = q'(x) \cdot p(x) + q(x) \cdot p'(x) $: