Best way to explain if the approximation of area is an overestimation or under?

[duhherro]

Best way to explain if the approximation of area is an overestimation or under of Left/right rectangles and linear approximations ?

Thanks in advance!

[lnaa19]

My way: Over actual, under actual.
Lol.

[Moko]

For linear approximations; you would say that "Because the tangent at x=a lies below/over the value of b on the curve, then the actual value of b has been underestimated/ overestimated" (respectively)

[duhherro]

For linear approximations; you would say that "Because the tangent at x=a lies below/over the value of b on the curve, then the actual value of b has been underestimated/ overestimated" (respectively)

could you go a bit in depth of what a and b are ? So b is the value of the point they want you to approximate, but when you do it, it is actually just on the tangent at x = a ?

[Shenz0r]

I like to think about this graphically.

Say we're drawing a tangent to a log graph. Do you see that the tangent will be above the graph?

No, it's not because the function is increasing. is also an increasing function, but you find that the tangent cuts underneath the graph. Why is this?

Well, for an over-approximation, the gradient of the function is DECREASING where the tangent intersects with the function. So it's getting less and less steep, more flatter and flatter, allowing the tangent to lie on top of it.

For an under-approximation, the gradient of the function is INCREASING where the tangent intersects with the function. The function itself will be getting more and more steep, allowing the tangent to lie underneath it.

That's how I feel will be the best way to explain it.

[Moko]

could you go a bit in depth of what a and b are ? So b is the value of the point they want you to approximate, but when you do it, it is actually just on the tangent at x = a ?

What I mean is that when you use the approximation formula, you are using a tangent at a certain x value (eg. x=2) to approximate another x value on the curve, eg x=2.2....

If the curve gets steeper and steeper as x increases (y increases as x increases) then your tangent at x=2, when extended, will fall BELOW the curve for any value of x above 2. So this means that at x=2.2, your approximation (based on the tangent you obtained for the curve at x=2) will be LOWER than the actual value (the one that you would obtain if you sub in x=2.2 into your curve). 

Do this: draw an e^x curve.   Locate on the curve x=2 for example. Find the tangent of e^x at x=2. draw the tangent which passes through x=2. You will see that if you were to estimate the value of 2.2 using the tangent line (not the curve) then it would be less than the actual value. To verify this, sub x=2.2 into the tangent equation (estimation), then sub it into e^x (actual). See what you get..is the estimated value over the actual value? Should be, otherwise repeat.