advertisement

Math 124 Abegg Handout (3.4) Chain Rule Application Problems Note: This homework assignment must be turned in on a separate sheet of paper. Final answers are given to help you know if you are doing the problems correctly. Full solutions will be posted on Canvas. If the quantity is given by a composition ( ), then the chain rule shows that the rate of change of with respect to is calculated by multiplying the rate of change of with respect to with the rate of change of with respect to . You will need to be comfortable with the following notations of the chain rule. General derivative: Derivative evaluated at a specific point: = ⋅ = ! ( ) ⋅ ! !!! = ⋅ !!!(!) !!! = ! ⋅ ! 1. Bacterial growth produces heat. The biomass (in grams) of a particular sample of bacteria grows according to the equation = 0.4 + 0.1 !.! , where is in minutes. The temperature (in °C) of grams of biomass is given by = 20 + 2. For each of the following, find the value and interpret what it represents with regard to the situation. a. 15 b. ! (15) = c. ( 15 ) d. !!!" !!!" Ans: a. 3.78 grams; b. 0.29 grams/min; c. 22.7°C; d. 0.1°C/min 2. The population of a city is given by = 8 + !!.!"! thousand people, where is the number of years since 2015. The number of garbage trucks needed by the city can be modeled by the equation = 4 + + 5! , where is the population in thousands. Find the value of each of the following in 2025. Include units in your answers. a. () b. () c. d. e. Ans: a. 14,060 people; b. 32 garbage trucks; c. 303 people/year; d. 2.2 trucks/thousand people; e. 0.68 trucks/year 3. A beehive with bees produces ℎ = 20(1 − 0.95! ) grams of honey each day. The price !"" !" of a gram of honey, in dollars, is given by ℎ = !!!. Find and interpret !" !!!" . Ans: –1.4 cents per additional bee # Rabbits 4. The graph shows the number of rabbits, , living in a certain garden weeks from now (in reality the graph should be discrete points since you cannot have fractional rabbits, but let's just consider it a continuous function for this problem). The number of carrots in the garden, , depends on the number of rabbits: = 300 !"!! !"!!! . Calculate the rate at which the number of carrots in the garden is changing with time when: a. = 1 week b. = 5 weeks c. = 5 rabbits Handout (3.4) – Page 1 of 1 (weeks) Ans: a. –6.7; b. 0; c. 30 (units: carrots/week)