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(a) Graph the function $ f(x) = \sin x - \frac{1}{1000} \sin (1000x) $ in the viewing rectangle $ [-2 \pi, 2 \pi] $ by $ [-4, 4] $. What slope does the graph appear to have at the origin?

(b) Zoom in to the viewing window $ [-0.4, 0.4] $ by $ [-0.25, 0.25] $ and estimate the value of $ f'(0) $. Does this agree with your answer from part (a)?

(c) Now zoom in to the viewing window $ [-0.008, 0.008] $ by $ [-0.005, 0.005] $. Do you wish to revise your estimate for $ f'(0) $?

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03:13

Daniel Jaimes

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 7

Derivatives and Rates of Change

Limits

Derivatives

Missouri State University

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Baylor University

Idaho State University

Lectures

04:40

In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

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So here we are going to graph the function sign x minus Sign of 1000 X. All over 1000. And then observe it in different viewing rectangles. And Try to observe what the slope at zero is. So I've taken the first bounds -2 pi to pie And -4- four. Those are X and Y bounds respectively. And so it looks something like this here. I'm using Dismas but you can use your graphing calculator, your own graphing calculator or any other graphing utility you have. And so it looks like a zero. We have a pretty straight line that goes from negative one negative 1 to 1 comma one. So we can approximate the slope to be about one. So we're going to say the slope is about one. Now we have to change our bounds to these values. So I can do that in Desmond's here by using this graph settings button and changing the X axis bounds the ones we need and are Y axis bones similarly. And so now can see that it's a little different or you can see but it's not your typical sine function, there is a little bit of a squiggle going on and so line very close to the origin is kind of hard to see from here. Well assuming just a tiny bit and it looks like it goes flat just right where it hits the origin. So let's go ahead and approximate This answer. This slope as zero which of course does not agree with our answer from party. And then we're going to zoom in one last time here To negative 0.008 And positive 0.008 and the Y axis we have 0.005. And as you can see where we kind of zoomed in on the squiggle now and you can see that Very clearly hit zero at or the slope it looks like it has an inflection point at the origin. So we can confirm our estimate That the slope is approximately zero my observation at the origin.

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