About this video
The video finds a limit as x approaches pi/4 by recognising it as the derivative of cos(x) at pi/4. It uses the alternative definition f'(c) = limit as x approaches c of (f(x) - f(c))/(x - c), then gets -sin(pi/4) = -sqrt(2)/2.
You'll be able to
- recognise a limit as the definition of a derivative at a point.
- identify f(x) and c from a limit written as (f(x) - f(c))/(x - c).
- evaluate the limit by finding f'(c) with derivative rules.
Key words
- derivative at a point
- The slope of the tangent line at one input, written f'(c).
- alternative form of the derivative
- f'(c) = limit as x approaches c of (f(x) - f(c))/(x - c).
- unit circle
- A circle of radius 1 used to define sine and cosine; it gives sin(pi/4) = sqrt(2)/2.
Transcript
- [Instructor] So here we have, we wanna figure out what the limit as x approaches pi over four of all of this business is. So pause this video and see if you can have a go at that. All right, now the key realization here is this looks a lot like the definition of a derivative at a point.
You might say, "Well, I'm familiar with this definition of a derivative at the point A. It's the limit as h approaches 0 of f(a + h) - f(a), all of that over h. This doesn't look a lot like that. There's some similarities, but I have x minus pi over here. Here I just have an h.
But there's another definition or another way of thinking about the derivative at a point C as being equal to the limit as x approaches c of f(x) - f(c), all of that over over x - c. Now, this over here looks an awful lot like this over here. If we said that f(x) is equal to cosine of x, if we said that c is equal to pi over four, so if we said that this is pi over four, then this would be pi over four, then this would be pi over four.
And since f is cosine of x, this would be cosine of pi over four. So what's really happening here, if f(x) is cosine of x, this is really, so we're assuming here that f(x) is cosine of x and we wanna figure out what f prime of pi over four is. That's what c is, is equal to question mark.
Well, if we take the derivative of cosine of x, f'(x) = -sinx. So f' of pi over four is going to be equal to negative sine of pi over four. And you might know from the unit circle definition already that sine of pi over four is square root of two over two. That's a good one to just know.
Square root of two over two. So we have negative square root of two over two, which is choice C, and we're done.
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