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# Secant lines: challenging problem 1

Sal solves a challenging problem involving slopes of secant lines to a curve. Created by Sal Khan.

## Want to join the conversation?

• What is the difference between the tangent and secant lines?
• A secant line intersects the curve locally in exactly two points. A tangent line intersects the curve locally in exactly one point.
• Once we establish that the slope of the secant line over [a, 0] is greater than a line with a slope of 1, how do we extrapolate that to mean that f(-a) is less than that (around , part I)? Is it implied that f(-a) atually means "the slope of the line tangent to f(-a)?" Or am I forgetting something that says if the slope is greater, then the value of the function is always greater?
• The actual numeric value of f(-a) is less than 1 (as Sal points out between - ). He later shows that the value of the slope (average rate of change) between points (-a,f(-a)) and (0,1)) is greater than 1. (That's the 1-f(-a)/-a, part).

So he is ultimately comparing the two values, not two slopes. It was confusing to me too, at first. :)
• What is the slope 1 line for?
• It's to compare the secant line to see whether the slope is greater than or less than to 1.
• I don't know what f means, when I started doing calculus I saw stuff like f(x) which I thought sal would explain, but now I need to know to understand this video.
• You should have had that in a previous course. So, you may need to review Algebra II or Pre-calculus.
f(x) is a symbol that means "a function of the variable x". You then need to define what that function is. NOTE: this notation does NOT mean f times x.
For example,
f(x) = 3x²+x-4
Means you have defined the function f(x) to mean 3x²+x-4
Once we have that definition, we can replace the x with any valid mathematical entity. For example
f(7) means to replace x in the definition with the number 7.
f(7) = 3(7)²+ 7 - 4 = 150
But we are not limited to numbers, we can plug any valid mathematical entity we like
For example, f(9x+π) would be to replace all of the x's in the definition with "9x+π":
f(9x+π) = 3(9x+π)² + (9x+π) - 4
Finally, we can use letters other than f for the function notation, usually if we need to use more than one function in a problem. Common letters to use in defining a function are f, g, and h -- though we can use most any letter we like.
• "A secant line intersects the curve locally in exactly two points. A tangent line intersects the curve locally in exactly one point. "

• The names for most of the trig functions have to do with how they may be constructed with respect to the unit circle. Both the tangent and the cotangent can be defined as lengths of segments of a line that is tangent to the unit circle. Secant and cosecant are lengths of segments of lines secant to the unit circle. The words sine, secant, and tangent come from Latin. 'Sinus' means bending, 'secans' means cutting, and 'tangens' means touching.
• If this topic is differential calculus, what is integral calculus?
• Differential Calculus is all about calculating change, or how much a function wiggles or how fast it wiggles. We study all these rules for calculating this change such as the product rule, chain rule, power rule, etc.
Integral Calculus is a little more abstract and a little harder to understand - it's main focus is to find the area under a curve. As Keith said, integrating is really just the reverse process of differential calculus. But there are even more methods than just reversing the process, there are series methods - and even other types of integration. Also the theory seems to me to be a little more abstract with greatest lower bounds and least upper bounds.
• Sal mostly proved everything visually but how do I prove them mathematically?
• maybe you can consult your textbook or just wiki it, and i believe this ( what Sal did ) is to help you understand more directly because you may sometimes find the mathematical proof a real mess to understand
• Why is it ( x, f(x)) and not (x , y )?
(1 vote)
• Because we defined y = f(x).
So we can replace y in (x , y) with f(x). It becomes (x , f(x))
• See into the video. Referring to the denominator, why is it 0-(-a) and not (-a)-0. Thanks ;)
(1 vote)
• You could do that, but the answer would be the same.
For example, if the 2 points were (9,2) and (7,5),
5-2/7-9 would equal 3/-2, which equals -(3/2.) If you switched the order, it would be 2-5/9-7, which is -3/2. The answers are the same. It does not matter which you put first, as long as when you put the y value of it first, then that point's x value has to be in front too.