Instantaneous Rate - Online Tutor, Practice Problems & Exam Prep

Instantaneous Rate Concept 1

Instantaneous rate is the rate of a reaction at any particular point in time. And we're going to say here that calculated using the slope of the tangent line to the curve of that point is the way we obtain our instantaneous rate. So if we're given basically a plot of Y&X coordinates, we can use that to figure out the slope which will be equal to our instantaneous rate.

Now we're going to say while the rate of reaction decreases with time because the amount of reactant is decreasing in time. Instantaneous rate though remains constant here. If we take a look, we say that if we're given points Y&X, we can figure out the slope. Remember that slope is equal to changing Y over change in X rise over run. This really means Y2 - Y1X2 - X1.

Here we would change. We would keep the change in Y as changes in our concentration and our changes in X as changes in time. Now here we have the graphical representation of a curve and again we've used two points in terms of this which would relate to our tangent line. We'd utilize this formula for slope and use that to figure out our instantaneous rate.

So just keep this in mind as we start investigating more and more questions dealing with calculating the instantaneous rate of an overall reaction or at any particular point.

Instantaneous Rate Example 1

Determine the instantaneous rate for the following reaction. Here we have methanol reacting with hydrochloric acid to produce chloromethane plus water as a liquid. We are given the times for the reaction at different points and from this we're also given the change in concentration of hydrochloric acid. Now, since we're given all these times and all these concentrations, remember our times will represent X and our concentrations will represent Y.

From this information we can figure out our instantaneous rate by determining the slope. So here Y2 - Y1X2 - X1. We're going to say that this is our first time given to us, which is 0. So we'll say that this is X1 and if this is X1 then this has to be Y1. Now X2 we can make the last time taken, which would be 247, and if this is X2 then this would be Y2. Now we would take those and plug them in.

So we have 1.01 - 1.90 and these are in concentration, so molarity divided by 247.0 - 0.0 in seconds. When we plug this in we get -3.60 × 10-3 molarities per minute. So this will represent our instantaneous rate for the reaction. Notice here that the sign is negative because we can see that the concentration of our reactant overtime is indeed decreasing as we expect it to be.

So this is the approach we would take when they're giving us a list of points for X&Y. We determine what the slope is from these coordinates, and with that we can relate it to our instantaneous rate.

Consider the decomposition of dinitrogen pentoxide:2 N₂O₅ (g) → 4 NO₂ (g) + O₂ (g)

What is the instantaneous rate of this reaction at 20 seconds?

• A. 0.0325 M/s
• B. 0.0270 M/s
• C. 0.0380 M/s
• D. 0.015 M/s

Here’s what students ask on this topic:

How can I find the instantaneous rate of change?

To find the instantaneous rate of change of a function at a specific point, you'll need to use the concept of the derivative. The instantaneous rate of change is essentially the slope of the tangent line to the function at that point. Here's how you can find it:

1. Identify the Function: Make sure you have the function f(x) for which you want to find the instantaneous rate of change.
2. Compute the Derivative: Find the derivative of the function, f'(x), using the appropriate rules of differentiation (such as the power rule, product rule, quotient rule, or chain rule).
3. Plug in the Point: Once you have the derivative, plug in the x-value of the point you're interested in into the derivative. This means if you want the rate of change at x = a, you'll calculate f'(a).

The result will give you the instantaneous rate of change of the function at that particular point. This value represents how fast the function's value is changing at that exact point with respect to x. If you're dealing with a real-world problem, make sure to interpret the result in the context of the problem, such as "meters per second" or "dollars per item."

How can the instantaneous rate of change be found?

The instantaneous rate of change of a function at a specific point is essentially the slope of the tangent line to the curve at that point. It represents how fast the function's value is changing at that exact moment. To find it, you typically use calculus, specifically, the concept of the derivative.

Here's how you can calculate it:

1. Start with the function $f\left(x\right)$ that you're interested in.
2. Take the derivative of the function, denoted as $f\text{'}\left(x\right)$ or $\frac{d}{\mathrm{dx}}$. The derivative represents the rate of change of the function with respect to $x$.
3. Plug the specific $x$-value you're interested in into the derivative. The resulting value is the instantaneous rate of change at that point.

For example, if your function is $f\left(x\right)=x^2$, the derivative is $f\text{'}\left(x\right)=2x$. To find the instantaneous rate of change at

When does the instantaneous rate of change equal the average?

The instantaneous rate of change of a function at a particular point is the slope of the tangent line to the function's curve at that point. It represents how fast the function's value is changing at precisely that point and is given by the derivative of the function at that point.

The average rate of change of a function over an interval is the slope of the secant line that connects two points on the function's curve. It represents how fast the function's value is changing on average over that interval.

The instantaneous rate of change equals the average rate of change when the function's curve is linear over the interval in question. In other words, if the function is a straight line between two points, then the slope of the tangent at any point between those two points (instantaneous rate of change) will be the same as the slope of the secant line connecting them (average rate of change). This is because, for a linear function, the rate of change is constant.

How can the instantaneous rate of change be estimated?

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The instantaneous rate of change of a function at a particular point can be estimated using the concept of the derivative. To find this rate, you would typically calculate the derivative of the function, which represents the slope of the tangent line at any given point on the curve.

Here's a basic method to estimate it:

1. Choose two points that are very close to the point of interest on the function. Let's say you're interested in the instantaneous rate of change at $x=a$; you might choose points at $x=a-h$ and $x=a+h$, where $h$ is a very small number.
2. Calculate the average rate of change between these two points. This is done by finding the difference in the y-values of these points $f\left(a+h\right)-f\left(a-h\right)$ and dividing by the difference in the x-values $2h$.
3. As $h$ approaches zero, this average rate of change approaches the instantaneous rate of change. In calculus, this is done by taking the limit as