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If a ball is thrown into the air with a velocity of 40 $\mathrm{ft} / \mathrm{s}$ , its height (in feet) after $t$ seconds is given by $y=40 t-16 t^{2}$ . Find the velocity when $t=2$ .

$\left.v\right|_{t=2}=-24 \mathrm{ft} / \mathrm{s}$

Calculus 1 / AB

Chapter 3

Derivatives

Section 1

Derivatives and Rates of Change

Harvey Mudd College

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Idaho State University

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we're told a ball is thrown in the air with velocity afford a few per second, and its height can be given by the equation. Why is even 40 team minus 16 t squared? You want to find a velocity when time is acquitted, too. Now the thing will need to recall from physics is that velocity is equal to the derivative or the rate of change of our displacement. So we normally right, this is like Delta lie or don't t. But in this case, what we're really finding is the derivative of this function active. So if we were to call this function, f a t were wanting to find what is f prime of two. So what we would do for that is the limit as tr purchased two o f t minus Uh uh too All over. T minus two. Now we need to figure out wood after two is so let's see. Effort to is going to be so it B a D minus 16 times two squared, which is or which is going to be 64 and then subtracting those should give us 16. So we're gonna plug 16 in perfect too. So we're gonna have the limit as t approaches to of so effort ease the puncture were given, and I'm just going to rewrite it like this. Where e. T squared is first. That may have minus 16 all over tea and minus two. Now, if we were to just plug this in directly, we would get 0/0, and I don't know what that means. So what we can try to do is defied these two functions to see if they cancel. And I'm going to do this with synthetic division off on the side here. So it'd be too on the outside because that's the root of T minus two. Read a negative 16 40 negative 16. So dropping the 16 down to that give us negative 32 which would give eight and then two times eight is 16 0 So that means this would be negative 16 x plus, not expert t t plus eight. Which means we can rewrite this us the limit as t approaches to, uh, negative 16 t plus eight. And now we can go ahead and plug to win directly into this, which would give us so negative 16 times two special. Let me just go ahead and write it out first. That's gonna be negative. 32 plus eight or negative. 24. And the units with this would be feet per second. So this would be our velocity after two seconds.

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