AP®︎/College Calculus AB
- 2011 Calculus AB free response #1a
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Mean Value Theorem and differentiability. Created by Sal Khan.
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- Why is the reason "not differentiable " rather than "not contionous" Is there really a difference on the AP Exam(2 votes)
- While a function is necessarily not differentiable at a point of discontinuity, a continuous function can also be non-differentiable at cusps.
For example, f(x) = |x| is continuous, but it is not differentiable at x=0.
Thus, a discontinuity is one, but not the only, reason a function would not be differentiable at some point.(7 votes)
- So does the Mean Value Theorem not apply to any function that is at some point not differentiable, or only if that point is x=0?(2 votes)
- At2:27, why does Sal multiply the integral by 1/7? Should it not have been -2/7 instead?(1 vote)
- Because the average value for integrals states: f(c)=1/(b-a)•∫[a,b] f(x)dx, all one has to do is plug in numbers. Letting a= -4 and b= 3 when you place these numbers into f(c) you will get f(c)= 1/(7)•∫[-4,3] f(x)dx. At this point the math is straight forward. Now note that on the inside of the definite integral (integrand) are the slopes of f(x) everywhere. So when integrating the derivative function you come back to the original function while also having the capability of getting the area underneath the curve. Since the average value, f(c), is equal to one over the two endpoints of f(x) (the original function) subtracted from each other, multiplied by the definite integral (the area of f(x)), generates the average rate of change of the entire graph of f(x). Hopefully I didn't confuse you with all the extra material behind reasoning.(3 votes)
- When calculating the average rate of change, why does Sal use average value? I know these two are not the same so why did he used the average value equation (1/b-a)(integral from a to b of f(x)dx) to find the rate of change?(1 vote)
- Okay, so the question gives us this statement, "There is no point c, -4<c<3, for which f '(c) is equal to that average rate of change." This basically means: 'The MVT doesn't apply.' And we know that this is because f is not continuous or differentiable at x = 0. So, the statement is true.
The question asks us to tell WHY the (true) statement does NOT contradict the MVT?
So does I read that question as saying why "this statement" does not DISprove the MVT. Which means say why it AGREES w/ what the MVT says (not agrees as in applies).
...Am I right?
This isn't so much as a mathematical/calculus problem as it is a question about reading the AP question. Sorry if I broke the rules. Thanks!(1 vote)
- Explain why statement does not contradict the Theorem
Explain why statement does not disprove the Theorem
Theorem requires that f(x) is continuous on [a,b] and differentiable on (a,b)
Statement does not meet requirement. this f(x) is not differentiable on (a,b)(1 vote)
- I'm not sure why the statement does not contradict the Mean Value Theorem. Sal had me at the actual mathy portion of the problem but lost me from that point forward.(1 vote)
- At5:15, Sal says that there would be a point where the derivative equaled the average slope (i.e. the Mean Value Theorem would apply). However, wouldn't the MVT still not apply since the derivative of f doesn't exist when x = -3 (because the slope of that point is undefined)?(2 votes)
- it depends on how technical you want to get. intuitively, the slope is continuous at x=-3 because it approaches infinity smoothly and leaves infinity smoothly, so the mvt would apply. different theories of thinking might say that the slope there is undefined so that the mvt doesnt apply.
another example of this is 1/x. depending on how strict you are about your maths, the integral from negative infinity to infinity of 1/x dx (the area between the two branches and the x-axis) is either undefined or zero. intuitively, it might make sense that the area would be zero, since 1/x is an odd function and the area under the two branches should cancel each other out, but different methodology might say that the area from negative infinity to zero is infinity and the area from 0 to infinity is infinity, so that negative infinity minus infinity is undefined. of course, it's a lot easier to prove it's undefined than to logically -prove- intuition, so schools usually just teach that the integral is undefined.(0 votes)
Part d, find the average rate of change of f on the interval from x is greater than or equal, negative four is less than or equal to x, which is less than or equal to three. So really the interval that they've depicted right over here. And then they say there's no point c between in that interval for which f prime of c is equal to the average rate of change. Explain why the statement does not contradict the Mean Value theorem. Fascinating. Alright. Let's do this first part first. The average rate of change of f on the interval. Sounds like a very fancy thing, but the average rate of change on the interval is really just a slope of the line that connects the end points of the interval. So these, this right over here, those are the end points, and let's figure out the slope of that line. So, going we, going from this point to that point, our change in x, we do this in a color that you're likely to see. Our change in x over here, our change in x, change in x, is equal to 7. And you can get that by taking 3 minus negative 4 or you could literally just count, 1, 2, 3, 4, 5, 6, 7. That's our change in x. And our change in y, when we, when we run over 7, when we go 7 to the right, our change in y, our change in y is equal to negative 2. We went from negative 1 to negative 3 is equal to negative 2. So, slope, slope, which is change in y over change in x, rise over run. Is equal to 7 over negative 2, or well actually let me, the other way around. The change in y is negative 2 over 7. So negative two 7s and that's all it is. Now you could, you could kind of. Think of it in, in, kind of a more fancy sense, but you're gonna get the exact same answer if you said, oh well look, you know, the average, the average rate of change of f on the interval. Well, the rate of change of f is f prime, f prime of x. That is the rate of change of f at any point x. And so if you want to find the average value of this over the interval, you would integrate it from our starting point, from negative 4 to 3 dx, and then you would divide it by your change in x. So then you would divide it by, you would have it over 1 over 7. Over 1 over 7. But then this, this part right over here is just going to be the same thing as f of 3 minus f of 4. And then this over here you have a 7 in the denominator. So this is really just your change in x. This is your change in x which is, which was by definition or how we, how we actually set up this average right over here. And this over here is really just your change in y. So it really is just a slope between the end points. So, we did the first part. Our average rate of change of f on the interval is negative two 7s. And then let's think about the second part. You see there is no point c in that interval. For which f prime of c is equal to the average rate of change. Explain why this statement does not contradict the Mean Value Theorem. So the Mean Value Theorem, just as a little bit of a review, as a little bit of review, it says that if we have some type of an interval. If you have an interval. So, let me draw some axes right over here. If you have an interval. Let me draw an interval like this. And over that interval you have a differentiable function. You have a differentiable function, so maybe my function looks like this. It says at least, there is at least one point c on that interval where the derivative at that point c is equal to the average rate of change. So the way I've drawn it right over here, the average rate of change of this function, I'll do it in magenta, is this right over here. And the Mean Value Theorem says, is it, if this is deferentiable, there's at least one point c in this interval. Where I have the same slope. Where the tangent line has the same slope. Where the derivative is the same as the slope, as that average slope. And you can see here, is probably right over here you have one of those points and actually we probably have multiple of them. We probably have another point right over here that's like that, and then another point there that's like that. And if you think about it, it's kind of. It's kind of intuitive that at some point, you know, here we're, we obviously have a larger slope and over we have a smaller slope and since our, since it's differentialable, our derivative is continuous. So at some point, at some point the slope has to get to exactly what the average slope is. Now, let's think about our little conundrum with this question right over here. Why is there not a point c for which f prime of c is equal to the average rate of change? You can even verify that for yourself because from negative 4 to 0 our slope is positive. We have a positive slope here, and then the slope just jumps down to negative 2. It just jumps down to negative 2 which is a much more negative slope than this. So, it never goes to negative 2 7s, and the reason is that. This is not a differentiable function at x is equal to 0. It is not differentiable at x equals 0. Our slope jumps here and because it's not differentiable the Mean Value theorem doesn't apply. Now you can imagine if this was differentiable, if this did have a continuous derivative, then you would find a point. So if this looks something like this instead, I'll continue it over here. If it looks something like this instead where it was, it had a continuous derivative then there would be a point where the slope was the same as the average slope. Maybe it would have been right over there. So it's really because, it's really because f is not differentiable over the entire interval. It does not have a continuous derivative. The derivative jumps from a, from a positive value approaching 0 here and it jumps just straight down to negative 2 right over here. It doesn't go continuously through all the values in between.