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A particle moves along a straight line with equation of motion $ s = f(t) $, where $ s $ is measured in meters and $ t $ in seconds. Find the velocity and the speed when $ t = 4 $.
$ f(t) = 80t - 6t^2 $
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03:18
Daniel Jaimes
Calculus 1 / AB
Chapter 2
Limits and Derivatives
Section 7
Derivatives and Rates of Change
Limits
Derivatives
Oregon State University
Harvey Mudd College
Boston College
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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A particle moves along a s…
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The velocity of a particle…
suppose the position of a particle at time T is defined by the function F f T Which is equal to 80 T -60 square. And in this problem we need to find the velocity and speed at time T equals four. Now to do this we know that velocity, this is equal to f prime of T. And speed is just the absolute value of velocity. So that's the absolute value of f prime of T. Now define the Derivative given T is equal to four. We will use the definition of derivative at a point. So in here we have velocity. The this is equal to F. Prime of four. That's a limit. ST approaches four of f of t minus F before this all over t minus four. So from here we get a limit SD approaches four of fft which is a TT minus 60 squared and then Fo four which is just Evaluating F at four. That will be 80 times four -6 times four squared fish all over, t minus four. So expanding the numerator, we have limit S. T approaches for We have 80 T -60 squared -80 times four. That's 3. 20 Plus six times 16. That's 96. This all over t minus four. Simplifying the numerator, we have limits SD approaches four of you have negative six t squared plus a D t minus 224. This all over t minus four. A factoring out a -2 in the numerator we have limit as T approaches four of negative two times three t squared minus we have 40 T plus You have 112. This all over t minus four. Factoring the numerator completely. We have Limit as T approaches four of negative two times we have T -4 times three t -28. This all over t minus four. And then from here we can cancel out the T -4 and we get limit as T approaches four of -2 times three T -28 and so evaluating a T equals four, we have -2 times three times 4 -28. This is just equal to 32 and so this is the velocity of the function at time T which is equal to four seconds. The units must be meters per second now because this is already positive, then the this is also the speed of the particle. Therefore the velocity and speed of the particle is 32 m/s. This is when T is equal to four seconds.
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