What do we mean by 'average gradient'?

[Cambridge101]

I am coming across a lot of labels like find the 'average gradient', but what do they mean by average in this context, it seems like a crap way to say it. For me, the idea behind finding the gradient of a chord on a curve when starting calculus and differentiation is to get close, or better to approximate the gradient of the curve. Because if you try to understand it by thinking about the average as it is defined in stats, (re distributing an amount until each group is equal). It does not make sense in terms of 'average gradient'.

Why does finding the gradient on the chord give us 'average gradient' - how do you guys think about it?

 

[BigBeachBanana]

For non-linear functions, the gradients are different at every point. However, if we're interested in the rate of change over an interval [a,b], we take the average of the gradients or the slope of the secant line connecting the 2 endpoints of the interval to give us an idea of how the gradients behave.

 

[Cambridge101]

For non-linear functions, the gradients are different at every point. However, if we're interested in the rate of change over an interval [a,b], we take the average of the gradients within the interval to give us an idea of how the gradients behave.

Ye I obviously already know that hahaha - all you have done is repeat what a basic google search says. I am asking why it is an average. Surely it is more an approximation. Explain to me how you define average then?

Plz

 

[Harry_the_cat]

If you think about a non-linear graph with time on the x-axis and distance on the y-axis, the gradient (representing instantaneous speed) at each time point will be different. However you could work out an average speed for a given time interval. Just like if you go for a drive covering 60km in one hour. Your average speed is 60km/h but your actual speed at any instant most probably would change a lot.

 

[BigBeachBanana]

Ye I obviously already know that hahaha - all you have done is repeat what a basic google search says. I am asking why it is an average. Surely it is more an approximation. Explain to me how you define average then?

Plz

First of all, if you want help, don't be condescending. Second, you asked a question without stating what you already know. I gave a standard explanation. Next time if you don't want people telling you what you already know, communicate it upfront because we can't read your mind.

To answer your question: Why it is an average gradient? Let's look at a concrete example, [imath]y=x^3 \implies y'=3x^2[/imath] over [imath](-20,-15.5)[/imath].



The term average gradient is referring to the statistical average of all the gradients in the interval. Select sample points within the interval (ideally equally spaced; otherwise, your sample would be skewed). Compute their gradients and average them. You would get approximately the same result as using the Avg. Rate of Change Formula. Of course, the Avg. RoC Formula is less computative expensive than collecting samples while giving an excellent approximate of the behaviour of the gradients within the interval.
Test it out yourself.

 

[Cambridge101]

First of all, if you want help, don't be condescending. Second, you asked a question without stating what you already know. I gave a standard explanation. Next time if you don't want people telling you what you already know, communicate it upfront because we can't read your mind.

To answer your question: Why it is an average gradient? Let's look at a concrete example, [imath]y=x^3 \implies y'=3x^2[/imath] over [imath](-20,-15.5)[/imath].

View attachment 31486

The term average gradient is referring to the statistical average of all the gradients in the interval. Select sample points within the interval (ideally equally spaced; otherwise, your sample would be skewed). Compute their gradients and average them. You would get approximately the same result as using the Avg. Rate of Change Formula. Of course, the Avg. RoC Formula is less computative expensive than collecting samples while giving an excellent approximate of the behaviour of the gradients within the interval.
Test it out yourself.

I see. But how does drawing a general chord from -15 to -20 and finding the gradient of that line do what is being done in this table. How does it work in a way that takes all the gradients in between, compute them and average them. This is what I am interested about.

Cheers

 

[BigBeachBanana]

I see. But how does drawing a general chord from -15 to -20 and finding the gradient of that line do what is being done in this table. How does it work in a way that takes all the gradients in between, compute them and average them. This is what I am interested about.

Cheers

Think of it in the context of distance and time travelled.
The formula estimates the distance and time travelled with one big "jump" from point a to point b. By taking the sample of the points in the interval, you take many smaller hops that add up to that one big jump. Regardless of which approach, you took the same amount of time to travel the same distance, therefore, your average velocity should be roughly the same.