Graph the curves. a. Where do the graphs appear to have...

Graph the curves. a. Where do the graphs appear to have vertical tangents? b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction to Exercises 37 and 38 . $$y=\left\{\begin{array}{ll}-\sqrt{|x|}, & x \leq 0 \\\sqrt{x}, & x>0\end{array}\right.$$

Graph the curves.
a. Where do the graphs appear to have vertical tangents?
b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction to Exercises 37 and 38 .
$$y=\left\{\begin{array}{ll}-\sqrt{|x|}, & x \leq 0 \\\sqrt{x}, & x>0\end{array}\right.$$

Key Concepts

Limit Calculations for Derivatives

Limit calculations are used to evaluate the derivative at points of interest, particularly where the derivative may be undefined in the standard sense. By applying the definition of the derivative as a limit, one can confirm whether the slope approaches infinity, which is indicative of a vertical tangent. This approach is essential for rigorous verification of the geometric intuition gained from the graph.

Vertical Tangents

A vertical tangent occurs when the slope of the curve becomes infinite or undefined at a point. This typically happens when the derivative of the function approaches ±?, indicating that the tangent line at that point is vertical. Understanding vertical tangents is crucial for analyzing the behavior of curves, especially at points where the function's rate of change becomes abrupt.

Piecewise Functions

Piecewise functions are defined by different expressions over different intervals of the independent variable. Analyzing such functions involves carefully studying each segment and the behavior at the junctions, where issues like discontinuities or non-standard differentiability may arise. This is important for accurately graphing the function and understanding phenomena like vertical tangents that may occur at the boundaries between the pieces.

Transcript

00:01 I was shifting away from just the average rate of change, we're going to start looking at the tangent line.

00:07 So before we had an example, or for example, we had a function x squared, some parabola, and we were looking at the average rate of change by drawing a secant line from this point to this point, and then calculating the slope.

00:20 Well, now no longer, we actually want to look at the tangent line and the slope of the tangent line.

00:27 This is directly leading us into what the derivative is because the derivative takes these siquant lines and brings them closer and closer together until what we have is a tangent line to the graph.

00:43 So if we draw tangent lines at the different points, we want to estimate their slopes.

00:50 So estimating their slopes, we end up seeing that, for example, if we want to estimate the slope of the tangent line here, we see it would be zero.

00:59 The best way to estimate the slope of a tangent line, in my opinion, would be to look closely at the graph and draw this tangent line.

01:08 The closer we zoom into a function, the more we realize it is a line.

01:12 It's called local linearity.

01:15 So if we look at this now, we see that close to this point, this is at 0 .6, i chose this arbitrarily.

01:23 We see that it rises about one and runs one.

01:26 So the slope is about one, but it's going to be slightly more than one, slightly steeper slope than one.

01:34 So that's how we interpret this, and then we could even look closer at the value of one and see that at this point, we have a slope of two.

01:42 That is the given slope that we have, which makes sense as we learn more about the derivative...

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