009A Sample Final A, Problem 9

9. A bug is crawling along the $x$-axis at a constant speed of   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dx}{dt}=30} . How fast is the distance between the bug and the point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (3,4)} changing when the bug is at the origin? (Note that if the distance is decreasing, then you should have a negative answer).

Foundations:
Like most geometric word problems, you should start with a picture. This will help you declare variables and write meaningful equation(s). In this case, we will have to use implicit differentiation to arrive at our related rate.

Solution:

Part (a):
We need to find two values a and b such that one is positive, and one is negative. Notice that f(0) = √2, which is greater than zero. We can choose x = -1, to find f(-1) = -2 - 4 + √2, which is less than zero. Since f is clearly continuous, the IVT tells us there exists a c between -1 and 0 such that f(c) = 0.