Find f^'(c) if it exists. f(x)={ 4 x, x<1 2...

Key Concepts

Differentiability

Differentiability means that a function has a defined derivative at a point, which in the context of piecewise functions requires that both the left-hand and right-hand derivatives exist and are equal at the point of interest. This property indicates that there is a unique tangent line to the function at that point.

One-Sided Derivatives

One-sided derivatives involve computing the derivative as a limit from one side—either from the left or the right of a specific point. This concept is crucial when dealing with points where a function's definition changes, ensuring that the derivative is well-defined from both directions at that point.

Piecewise Functions

Piecewise functions are those that are defined by separate expressions on different intervals of the domain. They require careful examination at the boundary points where the function definitions change, as behavior such as continuity and differentiability may not automatically carry over across these boundaries.

Continuity

Continuity is a fundamental concept that requires a function to have no breaks, jumps, or holes at a point. A function is continuous at a point if the left-hand limit, the right-hand limit, and the value of the function at that point all agree.

Transcript

00:01 Defined function for x less than one f of x equals 4x for x greater than or equal to one f of x equals 2x squared plus 2 we want to find f prime of 1 if it exists now f prime of x the derivative is found by taking the limit of the difference quotient as h approaches zero now i want you to pay careful attention to this piece right here, the f of x plus h.

00:35 Now when we take this limit as h approaches zero, that means h is approaching zero from the positive side, the right side of zero, but h also is approaching zero from the left side, the negative side of zero.

00:52 So this limit has to exist as h approaches zero.

00:58 That means from the left side and the right side.

01:01 Now here's the situation.

01:04 We want to find f prime of 1, okay, when x is 1.

01:10 The situation that we have is that the function is defined in two separate pieces right at 1.

01:20 So, when x is greater than or equal to 1, this is f of x.

01:27 But when x is less than 1, this is f of x.

01:31 Well, when we need to take the limit as h approach 0 of f of x plus h, if h is approaching zero from the right side of zero to positive side, that means it's a positive number.

01:46 So x plus a positive number, of course, x is 1 because we want to find f prime of 1.

01:52 So don't forget x is 1.

01:53 Let's put a little thing right there.

01:56 X is 1 in this particular problem.

01:59 All right because we want to find f prime of one so we need to find f of one plus h now we're taking the limit of this quotient as h approaches zero if h is approaching zero zero from the right side the positive side that means h is a small positive number so then f of x plus h which is really f of one plus h one plus h h being a small positive number is going to be greater than one if h is approaching zero from the right side the positive side then 1 plus h is greater than 1 which means this is the function that we would have to use when we substitute in x plus h or 1 plus h in other words if we're finding f of x plus h or f of 1 plus h if h is approaching 0 from the right positive side then we need to use use this expression for f of x when we find f of one plus h one plus h needs to be plugged in to x in this part of the definition however if we're then taking the limit as h approaches zero remember the limit has to exist as age approaches zero from the right side and the left side if h is approaching zero from the left side that means it's a small uh negative number so then 1 plus h, 1 plus a small negative number is going to be less than 1.

03:36 1 plus a small negative numbers like 1 minus a tiny number.

03:41 It's going to be less than 1.

03:42 So if 1 plus h is less than 1 because h is a small negative number, then 1 plus h being less than 1, we would have to use this part of the definition of f of x.

04:01 So we got a situation.

04:03 H is approaching zero from the positive side and the left side, but 1 plus h sometimes it's going to be greater than one, sometimes it's going to be less than one, depending on which side h is approaching zero from.

04:19 So to make a long story short, we have to find the right -hand limit and the left -hand limit of this difference quotient and then see if agree.

04:32 So we need to find the limit as h approaches zero from the right side of f of 1 plus h members it's 1 plus h because we're trying to find f prime of 1 so f of 1 plus h minus f of 1 all divided by h and then we have to do the limit of this different quotient as h approaches zero from the left side, the negative side, because h is approaching zero, but it approaches from the positive side and approaches from the negative side.

05:41 And if this limit is going to exist as h approaches zero, then the right side limit and the left side limits have to exist and be equal.

05:51 So we have to find the left side limit, the limit as h approaches zero from the left side, the negative side of f of 1 plus h minus f of 1.

06:37 So the limit as h approaches 0 from left side to negative side of f of 1 plus h minus f of 1.

06:45 Okay, remember these are ones, the x's are ones because we're trying to find f prime of 1.

06:49 So f of 1 plus h minus f of 1, all divided by each.

07:03 Now here is where we have to pay attention to which part of the definition of f prime of 1.

07:09 Of x we're going to use.

07:21 Okay, if h is approaching zero from the positive side, that means it's a small positive number, then one plus h, one plus a small positive number is going to be greater than one.

07:34 So that means we're working with one plus h, basically our x value, greater than or equal to one.

07:42 So we have to use this definition for the right side limit.

07:59 So we need to take the limit, as h approaches 0 from the right side of f of 1 plus h.

08:07 Let's draw a little thing showing that we're using this function for the right hand limit...

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