Chapter 8: Derivative Applications 

1.        Given the point 4, 1; , which point on f(x) = sqrt(x); is the closest to it? 

2.        Given f(x) = sqrt(`+`(`-`(`*`(`^`(x, 2))), 36)); determine all values that satisfy the MVT on the interval  [-3, 5]; . 

3.       Given f(x, y) = ln(`+`(`*`(2, `*`(x)), `*`(2, `*`(y)))); determine all values that satisfy the MVT on the interval from point  P(`+`(`*`(`/`(1, 2), `*`(exp(`/`(1, 2)))), `-`(`/`(1, 2))), `/`(1, 2));  to point 

4.       A conveyor belt is dumping corn into a large empty silo, forming a circular based cone shaped - heap on the floor. The base radius is twice the size of the height of the pile. If the corn is filling in the space at a rate of 2 `*`(`^`(m, 3)); / minute, find the rate at which the height is changing when the height is 3 m deep. 

5.      Housing developers hoping to attract buyers will often build in amenities for community members to enjoy. In a particular community, fishing is a key attraction, so the housing developers agree to build a small lake and stock it with trout. The population of trout over time can be modeled by p(t) = `+`(`/`(`*`(1200), `*`(`+`(1, `*`(29, `*`(exp(`+`(`-`(`*`(.36, `*`(t))))))))))); ,   where time is measured in months.  

6.     Solve exp(`+`(`*`(.5, `*`(x)))) = `+`(x, 3); using Newton's Method.