Alculus in Context #2 Math 113 Chapter 2 and section 3.1 How does our understanding of derivatives allow us to understand the relationship between weight and caloric requirements? Biologists often investigate relationships between properties of organisms and their size. One of the first researchers who investigated the issue of 10 0 dependency between metabolic rate and animal mass was Max Kleiber, who in 1932 showed that 10 - body weight scales with metabolic weight pursuant to a power law with exponent /4. Latest research shows this law even extends to plants! cold blooded orgasms Source: www.scientropy.wordpress.com unicellular organisms To the right you can see a log-log plot showing Kleiber's Law. (The log-log plot transforms values of input and output so that the graph is linear even though the original equation is not.) Kleiber's law states that the daily calorie 1 keathy - 1.162 watts mass (z. log scale) 104 requirement, C(w), of a mammal with weight w pounds is: C(w) = 42w0.75 The domain (acceptable input for w) is w 2 0 pounds. (Although the max weight of a whale is roughly 360,000 pounds, so the domain does NOT go to infinity!) 1. Calculate the derivative of Kleiber's Law, C'(w). Show your work in the space below. Simplify, but you may leave this expression in terms with negative exponents. Car ) = 42 W 0 = 42W To C' car = ARX Wit = 63 w / 4 - 0-25 W - 313 W- 025 2. Fill in the table: C' (w) C(w) X 4 40 400 4000 Page 179 ctrlSummarize what happens to the daily caloric requirement, C(w), as weight increases by anawering the following, using calculus to support your answers.: a, Since C'(w) positiveegative (choose one), this tells me that daily caloric requirement is (Fill in the blank.) b. Since C'(w) increasing/decreasing (choose one), this tells me that the instantaneous rate of change in daily caloric requirement is (Fill in the blank.) Since C'(w) increasing/decreasing (choose one), this tells me that the function C(w) concave up/down (choose one) 4. Find the second derivative of Kleiber's Law, C"(w). Show your work in the space below. Simplify, but you may leave this expression in terms with negative exponents. Please neatly print your name (First Last): Page 180