If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

## AP®︎ Calculus AB (2017 edition)

### Course: AP®︎ Calculus AB (2017 edition)>Unit 2

Lesson 3: Secant lines & average rate of change with arbitrary points

# Secant line with arbitrary point

Sal finds the slope of the secant line on the graph of ln(x) between the points (e,1) and (x,lnx). Created by Sal Khan.

## Want to join the conversation?

• Could someone tell me the difference of tangent lines and secant lines?
• A tangent line has a slope equal to the instantaneous rate of change of the function at one point. It touches the graph at that point.

A secant line has a slope equal to the average rate of change of the function between two points. It touches the graph at those two points.
• I don't get it, what am I missing. How is e^1 = 0 :S?

ln(0) is undefined, it is impossible?
• Yes, log (0) is undefined for all bases.
In the video Sal misspoke. He meant to say that e⁰ = 1
• At what does e represent, is it a number, if so what is the reason or use of e, and did I miss a video introducing e.
• e = lim h→0 (1+h)^(1/h)
This can also be written as the equivalent:
e = lim h→∞ [1 + (1/h) ]^(h)
e is a number, a constant, and is irrational and transcendental. The first few digits of e are 2.718281828459....

e is used extensively in calculus as well as applied, real-world mathematics. It comes up even more frequently than π. It is used as the base of the natural logarithm as well as the base of most exponential functions. The reason for choosing e will be become very clear when you advance in math, but for now just know that it is the easiest number to work with for log bases and for exponential functions.
• This is my understanding so far of what we've learned in differential calculus: The secant line, being the average slope between two points on the curve, is a rough approximation of the true slope, while the tangent line is the exact value of the slope of the curve at that point. Finding the limit of the secant line should give us the closest approximation of the tangent line. Is this correct?
• Almost correct. The secant line is not the average slope between two points, it is a line drawn between two points, and it has a slope equal to the average slope of the graph between those points.
• So... no matter what the function is, we just need to take `Δ = (y2 - y1) / (x2 - x1)` to get the slope of secant line? What if it is a periodic functions such as `f(x) = cos(x)` or `f(x) = sin(x) + 2`, or a complex function which is only possible to graph?
• The slope of a line between any two points (x1, y1) and (x2, y2) is always m = (y2 - y1) / (x2 - x1). So if you're finding the slope of a secant line, it does not matter what kind of function it is, so long as you have coordinates to work with. f(x) = cos(x) and g(x) = sin(x) + 2 work just fine as well!
• hello
I'm from Tunisia in my school we have learned derivatives but we haven't learn "logs" , what i should do ?
• The question states "Write an expression in x". What does that mean?
• The first one I saw was the following: "A curve has an equation y=cosx and passes through the points P=(0,1) and Q=(x,cosx). Write an expression in x that gives the slope of the secant joining P and Q."

With the secant line being the line joining these two points, (0,1), and (x, cosx), you simply write a slope equation, change in Y/Change in X. In this case, it would be (cosx - 1) / (x - 0). Make sure that you use parentheses so that the exercise knows which operations to perform first. Hope that helps answer your question. Furthermore, I had quite a bit of trouble wrapping my head around another problem which is as follows:

Sometimes this also requires an additional step, like the similar question about the slope of the secant joining P=(π, 0) and Q=(h, Sin(h) / h ). In this case you have the slope equation as follows: (Sin(h) / h - 0) as the change in Y.. all of that over change in x, which is h - π. So your equation would be (Sin(h) / h - 0) / (h - π). But since you have a fraction in the numerator, it would be (Sin(h) / h) * (1 / (h - π)), which would give you (Sin(h)) / (h (h - π)). Hope that helps as well.
• is the concept of instaneous slope physically existent apparent and consistent? or was it invented by humans in order to get a direct answer of a scale for a specific point because it was a high demand?
(1 vote)
• Like all of math, instantaneous slope is an abstraction; but, it is an abstraction designed to be very useful in describing reality.
• could you make it (1-ln)/(e-x) and still get the correct answer for this problem?
• While Sid is correct that ln needs input I am assuming that you meant to put ln(x).

If you did mean this then the answer is yes you could do that, because all you did was multiply (ln(x)-1)/(x-e) by -1/-1 which is a valid operation as you are essentially multiplying by 1 which has no effect on the equation except showing it in a new form.

Hope this helped!