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The displacement (in meters) of a particle moving in a straight line is given by the equation of motion $ s = 1/t^2 $, where $ t $ is measured in seconds. Find the velocity of the particle at times $ t = a $, $ t = 1 $, $ t = 2 $, and $ t = 3 $.

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04:40

Daniel Jaimes

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 7

Derivatives and Rates of Change

Limits

Derivatives

Missouri State University

Harvey Mudd College

Boston College

Lectures

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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The displacement (in meter…

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A particle moves in a stra…

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suppose the position of a particle at time T is defined by the function S of T, which is equal to one over t squared. And here we want to find the velocity at time T, which is equal to a 1, 2 and three to do this, we find the derivative of S at a point T, which is equal to a. Now by definition of the derivative at the point we have s prime of a. This is equal to limit as T approaches A. Of S F T minus S. Of a fish all over t minus A. So from here we have limits. S T approaches A. Of S F T, which is one over T squared. This minus S of a. Which is one over a squared all over t minus A. Now combining the numerator, we have limit S. T approaches A. Of we have a common denominator of a square T squared and then we have a squared minus d squared. This times the reciprocal of t minus A, which is one over t minus A. And then from here we get limit. SD approaches a. We can factor out a squared minus d squared into a minus t, times a plus t. This all over a square times t squared and then times the reciprocal of t minus A, which is one over t minus E. And then know that the a minus T. We can rewrite this into negative of t minus A. And so we can cancel this along with the T -8 in the Denominator. And so we have limit as T approaches a of the negative of a plus T over a square times T squared. And so evaluating at T. Which is equal to a. We have negative of a plus A All over a square times a square. Distrust negative to a All over 8 to the fourth power, or this is just negative 2/8 to the third power. And so this is the velocity of the particle at time T. Which is a. Now we will use this to find the velocity At Times 1, 2 and three. So if the velocity which is just S prime of E. is -2 over a cube, this is the velocity at time equals A. Then when T. Is that's a one, we have s prime of one, this is just negative two over 1 to the third power or negative two. And when he is too we have S. Prime of two, which is just negative too, Over 2 to the 3rd power, That's negative to over eight or -1/4. And when these three we have s prime of three, That's equal to negative two over 3 to the 3rd power. Or that's just negative two over 27. So these are the velocities at specific times T, Which is 1, 2 and three

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