Complete the following statement. If dy/dx is small, then smal...

Question

Complete the following statement. If dy/dx is small, then small changes in x will result in relatively ______ changes in the value of y.

Accepted Answer

Hello there. Today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we're going to need to use in order to solve this problem. A biologist is studying the growth rate of a bacterial culture. The population size of the bacteria is a function of time T in hours. If the derivative of P with respect to T DP by DT is small, what can be inferred about the population of the bacteria when small changes are observed in T? Awesome. So it appears for this particular problem, we're trying to figure out what can be inferred about the population of this bacterial culture when small changes are observed in tea. Considering the derivative of P with respect to T. So now that we know that we're ultimately trying to figure out what can be inferred about this population of this bacterial culture based on the information that's provided to us by the prom itself. So with that said, now that we know what we're solving for, let's read off our multiple choice answers to see what our final answer might be. A is the population size is changing rapidly with respect to time. B is the population size is growing exponentially with respect to time. See as the population size is changing slowly, and the small changes in time result only in small changes in the population. And D is the population size remains constant over time. So, in order to solve for this problem, we need to think back to our background knowledge of, you know, derivatives and the respect to population growth. So we're trying to think of rate of change or growth change, so rate of change equations and what we know about them from our textbook, as we should recall and note what we've learned. And as we should recall, based on what we've learned in the past, we need to note that when small changes are made to T. Which means that small intervals of time pass. It's important to note that the resulting changes in the population size P will also be small. So let's make a note of this. So if small, which I'll put an S for small, small changes are made to T, the time. So if small changes are made to made in time or small intervals of time, like, you know, occur. So as small intervals of time passes, the resulting changes in the population, which the population is denoted by capital P. So due to the, so it's really important to note that when small changes are made to T, which means that if small intervals of time passes, the resulting changes in this population size P will also be small, which I'll put another little S for small. So therefore, without a doubt, based on this knowledge, our final answer has to be multiple choice answer letter C. So the population size is changing slowly, and small changes in time result only in small changes in the population. And that's it. This is a fairly easy problem to solve as long as you have a strong understanding of the conceptual mathematics that are associated with a problem like this, and that's it. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Complete the following statement. If dy/dx is small, then small changes in x will result in relatively ______ changes in the value of y.

Key Concepts

Derivative

Differential

Continuity

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