Think about at what time does the particle change direction and then, I can figure out its position by taking the integral So the way that I'm gonna tackle it is first I need to For t between zero and six the particle changesĭirection exactly once. Parenthases on the sine and then I have plus one and then there ya go. Whoops, let me make sure I do this right. The absolute value of two sine of e to the, I'll use x again just cause its easier to Now we'll hit math again, and then we'll go to number, Which is going to beĪpproximately equal to. So its the integralįrom zero to six of the absolute value of two sine And then we can type this, actually let me just write it out, since you would wanna do this if you're actually taking the test. So this is just going to be the integral from zero to six of the absolute value of v of t, dt. So if you wanna figure out distance, what you wanna do is take the integral of the absolute value of velocity, you could think about it as you're taking the integral of the speed function. Is the total path length so it would be that, plus this. Something that starts here, it goes over there and then it comes back And to remember theĭifference between distance and displacement, if I have We just figure that out? No, this is displacement. So part C Find the total distance traveled by the particle from time tĮquals zero to t equals six. To three decimal places unless they tell you otherwise, And I get And when you're taking an AP Calculus exam its important to round And I am going to get that,Īnd then I divide by six. Here with the respect to x just cause it was more convenient. Parentheses for the sine, plus one and then I'm integrating 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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