Our jalapeño pepper plants have been producing peppers like there’s no tomorrow, and Sunrise Farms has starting baking jalapeño poppers to sell at the farmer’s market. During our experiments to find the best jalapeño popper recipe, we’ve discovered that there’s a relationship between the length of a jalapeño pepper and its weight. Roughly speaking, if x is the length (in cm) of a freshly harvested pepper, the weight of the pepper is estimated to be about
w(x)=20 - 1/(x3+1)
(in grams). However, this formula only works for full-grown peppers, which have length x in the interval [4,11].
See, we’ve been studying derivatives in my calculus class, and I was wondering: what is the derivative w’(x)? Also, how could we interpret the meaning of this derivative w’(x)? I think that w’(x) might be helpful to study, because we’ve found that the heaviest peppers make for the best jalapeño poppers!
I found that w'(10) is approximately 0.0003. Since 0.0003 is such a small number, I thought the weight is decreasing when x=10. Is my logic correct?
I also can’t figure out whether w’(x) should be increasing or decreasing for x in the interval [4,11]. Would you please use calculus to solve this last mystery, and explain why?
