Problem

The motion of a spring is described by $f(t)=e^{-…

06:32

Problem 45 Hard Difficulty

The motion of a spring is described by $f(t)=e^{-t}$ cos $t .$ Compute the velocity at time $t .$ Graph the velocity function. When is the velocity zero? What is the position of the spring when the velocity is zero?

Answer

$$e^{-7 \pi / 4}(1 / \sqrt{2}) \approx .002896$$

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Calculus 1 / AB

Calculus: Early Transcendental Functions

Chapter 2

Differentiation

Section 7

Derivatives of Exponential and Logarithmic Functions

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Video Transcript

emotional spring is described by the function. F of T is equal to e to the negative team co sign of team and they want us to compute velocity at Time t Well, since this is the motion of the screen, we can find the change in that motion to the velocity scene as using the product roll negative e to the negative t thinking the root of the first factory, leaving the second loan then being Obviously it's the negative z and then we have negative sign of tea by threat of the second factor. And so we end up list. I'd see negative e to the negative T Times CO sanity, uh, sign of team. Okay, pretty full on. Let's see the ask us. When is this velocity zero? We'll remember. This is never zero. So we're on May zero when co sign of plus sine of t is equal to zero. Well, when are they both the same angle? I wanted they both negatives of each other. Well, they definitely call inside when we get squared so obtuse over to our one over Scary to however you want to see it so one of them will need to be positive while one is negative. Well, when is that occurred? I believe it is in quadrant two and in quadrants. Three. But seeing is that right? That might be questions for Yeah, it's quiet for quiet into and qlogic. So when we have t is equal Teoh, let's see, That should be three pi over for him. Eso three pi over four and then any to buy multiple of that. So we're some energy okay and seen, but also, when we're at, let's see, this should be seven pile before because that's when the sign is negative. But coastline is positive and then we'll add to UK pie So that does it actually just find that if I yeah, instead, let's just use a single equation and this is enough to see that. Okay, All the zeros where literally zero. So what's the position of the spring when the velocity is zero? Well, let's see the position we have flt is equal to e to the negative t co sign team. Well, what is co sign any of these values? See, we ah three pi over four you to the negative three pi over horn plus by King good. Actually, all of that should be negative. So we're only gonna ushering this e term as K grows bigger for all our zeros. And then we're going to evaluate at co sign, uh, three pile reform less pai que. But now let's see, when we add two angles for coastline, there's a for me and I let me and confirm since we haven't looked at them in a while Jig, let's see, co sign and I should be someone into the double angle for me a lot. But let's make sure coastline Ah yeah gonna end up with Let's see this factor here and then co sign of the first single times co sign of the second angle and then we're going to you Subtract sign of those angles in the same way a sign of three pile report Sign of by for some images where I never sign our any multiple of by zero. So this time will always go away. And Lynsey, we're going to end up with e to the negative three pi over four last spiking. So remember this magnitude is shrinking in the front. We're all some of the planet by Let's see a negative one over the square root of two and this over here co sign of Pike A. This is just alternating between negative one on one like a and I make sensory. If it's oscillating like a spring, it's going back and forth. Sometimes it's in the negative. If the origin is in the middle, it's the velocity stopping because about we pulled back the other direction and the other direction in the other direction. And what's more, there's this other term that's weighing it down. It's dampening its source so that the further so you can get is lowered every time he oscillated pretty cool.

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