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# Secant lines: challenging problem 2

Sal interprets an expression as the slope of a secant line between a specific point on a graph and any other point on that graph. Created by Sal Khan.

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• The second question: When is (f(x) - 6) / x at its greatest value?
It looks very much like we're supposed to replace x and f(x) with the coordinates from ONE of the points plotted, yet Sal's answer is about the slope between two points.
Why is that? •   We could answer the question by plugging in the values of x and f(x) for each of the six labeled points and see which one produces the greatest value. However, Sal is making use of an insight to answer the question more easily. The formula we're evaluating happens to be exactly the formula we would need to calculate the slope from point A to any of the other points. The numerator is the difference in y-values, and the denominator is equivalent to the difference in x-values (which would be x-0, which is of course equal to x). Because the formula has exactly the same form as the formula for the slope from point A, we can answer the question by looking for the point where the slope from point A is greatest, because the greatest slope will produce the greatest value for the slope formula. This saves us the trouble of calculating the formula for each of the points, because we can easily see which point will produce the greatest slope.
• Why couldn't A be the answer to the second part of the question, as A is 0? •  At point A, the numerator becomes 6-6=0, but the denominator also becomes 0. So instead of having the greatest value at A. the value becomes undefined. You can't have 0/0 and we must look for a different point.
• Do we not get to do some of this stuff ourselves? Why do we have practice sessions for the last two videos but not this one, which is harder than the other two combined. • Hello! I have small question. :)
At , Sal says that the tangent line to point B is horizontal. How can we know this? The line might have a non-zero slope so small as to be invisible at this scale.
This is actually something that jumps out at me every time (unlabeled) graphs and (discrete) tables of values are used to "prove" anything, especially with limits, and it always really confuses me.
-Yehonathan • You are shown f(x), and given 6 points to choose from and asked where does f(x)·f'(x)=0.
The question is telling you that one of the 6 labeled points will make this true.
By process of elimination, it must be B. Also, when you look at B, it appears to be at the vertex of a curve, where the derivative is 0, which confirms the process of elimination.

Now if the question asked "What is the derivative of f at B" then you could say "it appears as if f' could be 0 at B but unless I am given the formula of f, I can't say for certain". This would be fine - but that is not the case here.

The whole point here is to get an intuitive feel for visually assessing graphs of functions and their derivatives.
• Just to clarify, f(x) gives you the actual value at any point and f'(x) gives you the value of the slope of the tangent line for that point • In the second question, couldn't you eliminate all the points further back than d because there are already higher x coord points and y coord points? • Is there some point between D and E where the slope of the tangent line of that point is 0? It seems to be that would be the case for all functions that have a "wave" in them. • Yes, as the slope changes from being positive to being negative, the slope reaches 0 for one point. The same is true if the slope of a function is moving from negative to positive and the interval in question is continuous. An interval in which a transition in the signs of the slope is made without the slope reaching zero would be discontinuous (for instance, a graph with a sharp peak or dip), because the slope would be approaching infinity instead of 0. Later, you will learn about "critical points", which are points where the derivative = 0. The critical points represent minimum or maximum points in the graph.
(1 vote)
• at ish, if you choose a for part b, is the answer just undefined? • At point E, when (x,f(x)) = (8,5), f(x)-6/x=(2,~2.8). The secant line here has a positive slope. Why is this (point E) not the answer? • The secant line from A to E has a negative slope, not positive. in fact, all of the points have a negative slope from A to it, meaning that we're looking for the least negative slope. Since from point A to point D, the line is least slanted, it is the least negative, and thus, the greatest. The slope of the line from point A to point E is actually the second greatest because the line is second least slanted. The slope of the line from A to E is (f(x)-6)/x, or (5-6)/8, or -1/8.

I hope this helps!

P.S. I'm curious as to how you got the point (2, ~2.8), though. It doesn't seem relevant at all. How did you get it? 