derivatives velocity

**jahichuanna** · May 9th 2009, 07:30 AM

I need to check my answers on part a, and see what's wrong on b. Also, what units do i use?

An abnormal projectile is shot in the air, so that after t seconds, its height in feet is

a) find the velocity at times 0 and 5
b) When is it highest? When does it land?

my answers:
I have the derivative as $-7tan(\frac{x}{2}+1.67)$
a) @t=0, 70.33 (units?)
@t=5, -11.6 (units?)

b) I wasn't totally sure how to do it, so I set the derivative equal to 0. I thought that that should give me the max height, but I came out with -3.34, which is obviously wrong. Did i make a math error or should I not have set it equal to 0? please help!

**running-gag** · May 9th 2009, 08:06 AM

Hi

Is the angle unit specified (degree, radian) ?
You have assumed it is radian. In this case are you sure about +1.67 ? I mean if it was +1.57, we could say that it is $\frac{\pi}{2}$

In any case due to ln domain $)0,+\infty($ we must have $-\cos\left(\frac{x}{2}+1.67\right) > 0 \Rightarrow -\frac{\pi}{2} < \frac{x}{2}+1.67 < \frac{3\pi}{2}$

Setting the derivative to 0 is OK to find the maximum height.

**Scott H** · May 9th 2009, 08:11 AM

You are correct that $s'(t)=-7\,\tan\left(\frac{x}{2}+1.67\right)$ .

To find the maximum height, we find the point at which $s'(t)=0$ :

$\begin{aligned}<br /> -7\,\tan\left(\frac{x}{2}+1.67\right)&=0\\<br /> \tan\left(\frac{x}{2}+1.67\right)&=0.<br /> \end{aligned}<br />$

Hint: $\tan x$ measures the slope of an angle. When does $\tan x = 0$ ? For what value of $x$ is $\ln \left(-\cos\left(\frac{x}{2}+1.67\right)\right)$ defined?

**jahichuanna** · May 9th 2009, 08:23 AM

thanks to both of you, i got 2.94, which makes sense. and jsyk, assumed angle measure is radians

So how do I find out when it lands? that I have no idea how to do...

**Scott H** · May 9th 2009, 08:54 AM

To find where it lands, we solve

$14\ln\left(-\cos\left(\frac{x}{2}+1.67\right)\right)+34=0.$

Before we take $\arccos$ , we must remember that there will be two solutions to consider.

**jahichuanna** · May 9th 2009, 09:03 AM

Originally Posted by Scott H

To find where it lands, we solve

$14\ln\left(-\cos\left(\frac{x}{2}+1.67\right)\right)+34=0.$

Before we take $\arccos$ , we must remember that there will be two solutions to consider.

O yeah...I was thinking that i had to do something with the derivative to find where it lands...ok...now i feel smart. thanks!

**davidsons** · May 9th 2009, 12:17 PM

for part a, (assuming this is a calculator active question), i would pop s(t) into the Y1 of a TI calculator and use nDerive(Y1,x,0) and nDerive(Y1,x,5). I get the same answers you do: 70.33 and -11.61. since velocity is the rate of distance over time, the units are feet per second.

Hope this helps

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