Rates Of Change MATH-221 Assignment

5.1. Acceleration as the rate at which velocity changes. As you are driving in your car your velocity does not stay constant, it changes with time. Suppose v(t) is your velocity at time t (measured in miles per hour). You could try to figure out how fast your velocity is changing by measuring it at one moment in time (you get v(t)), then measuring it a little later (you get v(t).

You conclude that your velocity increased by v = v(t + t) +v(t) during a time interval of length t, and hence
{average rate at which your velocity changed} = v = v(t + t) -v(t) /t .

This rate of change is called your average acceleration (over the time interval from t to t + t). Your instantaneous acceleration at time t is the limit of your average acceleration as you make the time interval shorter and shorter:

{acceleration at time t} = a = lim t0 v(t + t) - v(t) /t .

the average and instantaneous accelerations are measured in “miles per hour per hour,” i.e. in (mi/h)/h = mi/h2.

Or, if you had measured distances in meters and time in seconds then velocities would be measured in meters per second, and acceleration in meters per second per second, which is the same as meters per second2 , i.e. “meters per squared second.”

5.2. Reaction rates. Think of a chemical reaction in which two substances A and B react to form AB2 according to the reaction A + 2B+AB2.

If the reaction is taking place in a closed reactor, then the “amounts” of A and B will be decreasing, while the amount of AB2 will increase. Chemists write [A] for the amount of “A” in the chemical reactor (measured in moles). Clearly [A] changes with time so it defines a function. We’re mathematicians so we will write “[A](t)” for the number of moles of A present at time t.

To describe how fast the amount of A is changing we consider the derivative of [A] with respect to time, i.e. [A]0 (t) = lim t0 [A](t + t) + [A](t) t .

This quantity is the rate of change of [A]. The notation “[A]0 (t)” is really only used by calculus professors. If you open a paper on chemistry you will fifind that the derivative is written in Leibniz notation: d[A] /dt .

More on this in

How fast does the reaction take place? If you add more A or more B to the reactor then you would expect

Are you struggling to keep up with the demands of your academic journey? Don't worry, we've got your back! Exam Question Bank is your trusted partner in achieving academic excellence for all kind of technical and non-technical subjects.

Our comprehensive range of academic services is designed to cater to students at every level. Whether you're a high school student, a college undergraduate, or pursuing advanced studies, we have the expertise and resources to support you.

To connect with expert and ask your query click here Exam Question Bank