Video Transcript
The biomass of a bacterial culture in milligrams as a function of time in minutes is given by 𝑓 of 𝑡 is equal to 71𝑡 cubed plus 63. What is the instantaneous rate of growth of the culture when 𝑡 equals two minutes?
The question gives us a polynomial function which tells us the biomass of a bacterial culture in milligrams after 𝑡 minutes. We are asked to calculate the instantaneous rate of growth of this bacterial culture at the time when 𝑡 is equal to two minutes.
There are many different ways of measuring growth. For example, we could measure the population size. However, in this question, we are told that we are measuring the biomass of a bacterial culture in milligrams to represent the growth. This means that when we are asked to find the rate of growth, we are actually trying to find the rate of change in the biomass of our bacterial culture. And we know how to find the rate of change of a function 𝑓 at 𝑎. This rate of change denoted 𝑓 prime of 𝑎 is equal to the limit as ℎ approaches zero of 𝑓 evaluated at 𝑎 plus ℎ minus 𝑓 evaluated at 𝑎 all divided by ℎ if this limit exists.
The question tells us that we want to find the rate of change when 𝑡 is equal to two minutes. And since our biomass function is measured in minutes, we’ll set 𝑎 equal to two. So to find the rate of growth of our culture when 𝑡 is equal to two, we want to find the derivative function 𝑓 prime evaluated at 𝑡 is equal to two. And this is equal to the limit as ℎ approaches zero of 𝑓 evaluated at two plus ℎ minus 𝑓 evaluated at two all divided by ℎ.
We will begin by evaluating 𝑓 of two plus ℎ. This is equal to 71 multiplied by two plus ℎ cubed plus 63. Using the binomial expansion of 𝑎 plus 𝑏 cubed, two plus ℎ cubed is equal to eight plus 12ℎ plus six ℎ squared plus ℎ cubed. We can then multiply this by 71 and add 63. Our expression simplifies to 631 plus 852ℎ plus 426ℎ squared plus 71ℎ cubed. Evaluating 𝑓 of two, we have 71 multiplied by two cubed plus 63. This is equal to 631. We can now substitute these into our expression for 𝑓 prime of two. We have the limit as ℎ approaches zero of 631 plus 852ℎ plus 426ℎ squared plus 71ℎ cubed minus 631 all divided by ℎ. 631 minus 631 is zero. We can then divide through by ℎ.
We have the limit as ℎ approaches zero of 852 plus 426ℎ plus 71ℎ squared. As this is a polynomial in ℎ, we can attempt to use direct substitution. Substituting ℎ equals zero gives us 852 plus 426 multiplied by zero plus 71 multiplied by zero squared. This is equal to 852. The rate of change is equal to 852. Since the output of our function 𝑓 of 𝑡 is measured in milligrams and 𝑡 is measured in minutes, we can add units to our answer. These will be milligrams per minute. We have therefore shown that if the biomass of a bacterial culture measured in milligrams as a function of time in minutes is given by 𝑓 of 𝑡 equal to 71𝑡 cubed plus 63, then the instantaneous rate of growth of this culture at the time where 𝑡 is equal to two minutes is 852 milligrams per minute.
