X divided by four power, so that's my sine and then let me close the Variable x instead of t cause its easier to type in. So let me do, second e to the And I'll just use the And we are going to go from zero to six of, this is going to be two sine of e to the. So on our calculator we would hit math and then we would wanna do number 9 which is definite Where did I get this from? Well they tell us, what our velocity asĪ function of time is, its that right over there. Going to be equal to six and so, this is going to beĮqual to the integral from zero to six of two sine of e to the t over four power plus one, dt and then all of that divided by six. Zero to t equals six and then we're going to divide that by the amount of time that goes by. And so to figure out the average velocity, we could figure out our displacement, which is going to be equal to Our position function but they do give us our velocity function. In position going to be? Well they don't give us Our change in position, which we could view as our displacement, divided by our change in time. So our average velocity, that's just going to be Find the average velocity of the particle for the time period from zero is less than or equal to t is less Now we need to add distances traveled during all of those periods of time to each other - and we do it by integration. Therefore distance traveled in that interval of time equals to v(t) * dt. Since those intervals are infinitely small, we can assume that in each of those intervals velocity is constant, and it equals v(t). Now that we know that v(t) is derivative of x(t), we can also say that x(t) is integral of v(t) - since integral is opposite of derivative.Īnother way to understand it is to split time into very tiny little intervals - call them dt. Therefore, its derivative, will be velocity - v(t) - because it tells us how fast the position changes, which is consistent with our understanding of velocity. So let's define x(t) to be function of position, with respect to time. Derivative of a function with respect to given variable tells us how fast value of given function changes as the variable grows. Well in mathematics there is a way to describe that - derivative. We need to ask ourselves question - what is velocity? One answer I could give you is that velocity is how fast object moves, or how fast it changes its position.
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