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Nick starts jogging and runs faster and faster for 3 minutes, then he walks for 5 minutes. He stops at an intersection for 2 minutes, runs fairly quickly for 5 minutes, then walks for 4 minutes.$$\begin{array}{l}{\text { (a) Sketch a possible graph of the distance } s \text { Nick has covered }} \\ {\text t \text { minutes. }} \\ {\text { (b) Sketch a graph of } d s / d t}\end{array}$$

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Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 8

The Derivative as a Function

Limits

Derivatives

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

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.

03:41

Nick starts jogging and ru…

02:20

A jogger begins her daily …

01:08

If Katie walked at 3 miles…

01:13

Last night Phil went joggi…

04:39

Two cars are approaching a…

08:27

The accompanying graph sho…

08:05

From $t=0$ to $t=5.00 \ma…

Okay. So question is giving his next start jogging and run faster for three minutes. Okay, so you run faster for three minutes and he walks for five minutes. Another point is he walks for five minutes. He stops at an intersection for two minutes. Run fair quickly for five minutes again. He ren faire quickly for five minutes and then walks for four minutes. Okay, so this is the data given and we're told toe plot or graph. Distance s verses. Time T. Okay, So let's have our access first. De excess and distance success as so For three minutes, he walks faster and faster, so for three minutes, you'll accelerate after three minutes. So this is time. Three. After three minutes, he walks for five minutes, so it will be a constant speed so we can ever straight lines in spirits constant. Then it stops at an intersection for two minutes. So this is three plus five. That is eight. Andi stops for two minutes, so that will be original alliances. There is no incriminating distance, so it stops for two minutes. So eight plus two will give us then and then he start walking fairly quick. Okay, so again, it's all weekend. It's given running fairly quickly. Okay, so that can be the president came by off a poor train line. Okay, A slope with laden high. And it is done for four minutes, so we can add four minutes. So 10 plus four is 14. So this is the graph or s word says T Okay. And your point we have to remember is since in real life we don't just directly switch from a particular speed toe Another we haven't inertia. So there will be a curvy er. It will be a small transition, and similarly here we'll have a smooth transition. Now we have to plot the diagram or graph for D s by D t. Okay, so we'll have your same time. TXs What says the s pay DT. Okay, so since at time three still time. Three minutes is for increasing his speed on that, assuming it's a constant speed increments. So we will have ah, straight line till time three again after time. Three minutes. He started walking, but this initiation off walking will not start suddenly. Okay, so he starts walking at constant rates, so we'll take it as a horizontal line. And after five minutes, that is it around eight minutes, that is three plus 58 minutes. He stops running or walking. So at that time velocity, This D s by D. T is nothing but velocity is zero. And this change in its velocity from constant walking to zero is not an instant but a gradual process. So we'll have ah, slope like this and the velocity will come to zero and it will remain zero for next two minutes. So that is that time, then. Okay, Now, after 10 minutes again, he starts walking fairly quickly so that walking is four four minutes. So we can again have in cream mint in its velocity. And it's a quick walking so well are a velocity to it. And this working is again not instantaneous. Well, gradually to increase to quick walking. This will go till 14 minutes. Now, the important fact here we have to understand is there is no discontinuity due to sudden stopping or walking. Since in real life we cannot stop instantaneously or start walking incent. Initially, we need we have an inertia. And that is what is represented by this girl. Thank you

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