Tangents And Normals

To solve the problem, we need to evaluate for the function , where

1. Define the Function :
Given , we have .

2. Compute the Derivative :
Differentiate using the chain rule:
.

3. Evaluate :
Using the result from step 2:
.

4. Analyze the Options:
Option 1: , which is the definition of , not related to .
Option 2: , which does not equal .
Option 3: , which matches .
Thus, .

Final Answer:
The correct option is .

We are given the equation . We need to find the rate of change of when is on the curve, and the x-coordinate is decreasing at the rate of 8 units per second.

Step 1: Differentiate the given equation with respect to time . The equation is: Differentiate implicitly with respect to : Using the product rule, we get:

Step 2: Substitute the known values at the point . At and , we know (since the x-coordinate is decreasing). Substituting these values into the equation: Simplifying: Solving for :

Step 3: Conclusion. Thus, the y-coordinate of is increasing at the rate of 6 units per second. \bigskip