(a) Find the value of a making f(x) continuous at x=1:...

(a) Find the value of $a$ making $f(x)$ continuous at $x=1:$ f(x)=\left\{\begin{array}{ll}
a x & 0 \leq x \leq 1 \\x^{2}+3 & 1<x \leq 2\end{array}\right.
(b) With the value of $a$ you found in part (a), does $f(x)$ have a derivative at every point in $0<x<2 ?$ Explain.

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Key Concepts

Continuity at a Point

Continuity at a point means that as one approaches that point from both the left and right, the values of the function converge to the function’s value at that point. For piecewise-defined functions, this requires matching the function values from different sub-domains at the boundary point.

Differentiability

Differentiability refers to the existence of a derivative at a point, meaning the function has a well-defined tangent or rate of change there. For piecewise functions, it is necessary that not only the function values but also the slopes (or one-sided derivatives) from either side of the point coincide.

Piecewise-Defined Functions

Piecewise-defined functions are functions that have different expressions over different intervals in their domain. Analyzing such functions requires ensuring that the segments match up appropriately at the boundaries in order to maintain properties like continuity and differentiability.

One-Sided Limits

One-sided limits evaluate the behavior of a function as the variable approaches a specific point exclusively from one direction, either from the left or the right. They are crucial in the study of piecewise functions to verify continuous behavior and matching derivatives at transition points.

Transcript

00:01 I always enjoy these piecewise functions where you have to figure out, you know, what the a value is to do this.

00:13 And so they give you a few bounds.

00:15 They say that this has to be between zero and one.

00:21 And then on the right side, it's one to two.

00:28 All right.

00:28 So we only care about these values.

00:30 But in part a, they're just asking you what value for a can we have in order to make.

00:36 This continues.

00:37 Well, in order to be continuous at x equals 1, we need to plug in 1 for all of these xes.

00:45 So, a times 1, and we have 1 squared plus 3.

00:51 And in order for that to be continuous, they need to be equal on either side.

00:55 Well, it's pretty easy once you realize that that's what they're asking you to do, because a times 1 is a.

01:01 1 squared is 1 plus 3 is 4.

01:05 So therefore, a must be equal to 4.

01:07 Now, the second part to this is a little bit trickier because if it's differentiable, then the derivative on the left side has to equal the derivative on the right side.

01:20 So what's the derivative of ax? well, if you follow the rules from this section, the derivative is just a.

01:26 And then the derivative of the right side of this function is 2x, or the derivative of a constant zero...

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