a. Let f(x) be a function satisfying |f(x)| ≤x^2 for -1 ≤x ≤1 Show that f is differentiable at x

step 1 So section 3 .2 looking at problem number 58. Okay, this comes in two parts. So part number A, s

A. Let f(x) be a function satisfying |f(x)| ≤x^2 for -1 ≤x...

Key Concepts

Limit Definition of the Derivative

The derivative of a function at a point is defined as the limit of the difference quotient as the increment approaches zero. Specifically, f'(0) is given by the limit as h approaches 0 of [f(h) - f(0)]/h. This concept is crucial in testing whether a function is differentiable at a point.

Squeeze Theorem

The Squeeze Theorem is a method used to determine the limit of a function when it is bounded between two other functions that converge to the same limit at a particular point. This theorem is particularly useful when dealing with functions that are difficult to evaluate directly, as it allows one to 'squeeze' the target function between bounds that simplify the computation of the limit.

Behavior of Functions Near a Point

Understanding how functions behave in the vicinity of a point, especially in terms of their growth or oscillatory characteristics, is essential in assessing continuity and differentiability. For instance, a function that is bounded by a quadratic function near zero will have its oscillations dampened sufficiently, ensuring that the derivative exists at that point.

Piecewise Functions and Continuity/Differentiability

Piecewise functions require careful consideration at the points where their definitions change. To establish differentiability at such points, it is necessary to check that the limit of the difference quotient exists and is the same from both directions. This ensures that the function transitions smoothly between the different pieces.

Transcript

00:02 So section 3 .2 looking at problem number 58.

00:06 Okay, this comes in two parts.

00:09 So part number a, so where they're telling us that f of x is a function such that absolute value of f of x is less than or equal to x squared.

00:37 So that means bounded by this parabola on the interval from negative 1 to 1.

00:44 Okay.

00:45 So what does that mean for us? okay, so let's first, i'll draw this parabola here.

00:51 So x squared is going to be at the origin.

00:56 So it's going to be here's x squared, then minus x squared is going to look like this.

01:04 Okay, so here you have x squared and you have minus x squared.

01:09 So i've given a function, and so let's just say that this is one and negative one.

01:15 I've got a function that is bounded by that.

01:18 So it means that the function i have, i don't know what it's doing over here, but it cannot be above the parabola.

01:27 So it might be doing several things.

01:28 I'm not sure what it's doing here out in this area, but i know that on this interval, it has to be bounded.

01:34 So it could be doing things like this.

01:37 It could have spikes.

01:38 I don't know what's happening out there.

01:40 But i know what has to happen eventually.

01:41 No matter what it does, it has to go through this point.

01:45 There's no other way around it.

01:47 So it has to go through this particular point right here.

01:51 Okay.

01:52 So i know it has to go through the origin.

01:54 Okay.

01:55 If i know that f of x is x squared, i know that f prime of x, and we can work that out, but i'm not going to do now is 2x.

02:07 So f prime of x evaluated at x equals 0 is 2 times 0, which is 0.

02:15 So it tells me that m, the slope is zero when i go through those two parabolas.

02:21 Okay, so it tells me that the rate of change, the slope, the pitch, all of those things, has to be zero when i go through there.

02:28 So this red function, whatever f of x, if i look at this function as it goes through here, sorry about that, if i go through here, there's only one choice that it goes through at x equals zero and it has to go through with a, slope of zero.

02:49 If the slope or anything but then i would find myself, my curve would be rising above the parabola at that point.

02:56 But if these two curves go in exactly with the slope of zero, and i've got a curve that goes with the same spot, doesn't rise on either side of it...

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