<p>I'm having trouble with the following two homework problems. I would appreciate any help anyone can give. </p>
<p>"Use the position function s(t) = -16t^2 + 1000 which gives the height (in feet) of an object that has fallen for t seconds, from a height of 1000 feet. The velocity at time t = a seconds is given by:</p>
<p>lim s(a) - s(t)/a-t
t->a </p>
<p>If a construction worker drops a wrench from a height of 1000 feet, how fast will the wrench be falling after 5 seconds?" </p>
<p>Isn't the direct substitution method perfectly valid here? That's what I tried, and I keep getting -80, while the answer is -160. What am I doing wrong??</p>
<p>The other problem I'm having trouble with is: "Use the function -4.9t^2 + 150 which gives the height in (in meters) of an object that has fallen from a height of 150 meters. The velocity at time t = a is given by:
the same limit as the first problem, but find the velocity when t = 3. "</p>
<p>On this problem I'm getting -14.7 where the answer in the book is -29.4. It seems my answer is also 1/2 the correct one. What am I getting wrong?? Is the book wrong?? Please help, I'm a little frustrated. Thanks a lot everyone!</p>
<p>First of all, do you have to use the formal defnition of the derivative (for showing work, etc.)? If not, all you have to do is find the derivative (using the handy dandy shortcut) of these position/time functions and plug in the time to get your velocity.
1) derivative of -16t^2 + 1000 is -32t. When t = 5, the velocity is
-32 x 5 = -160 m/s
2) derivative of -4.9t^2 + 150 is -9.8t. When t = 3, the velocity is
-9.8 x 3 = -29.4 m/s
I hope that helps. I can show you the long way...but maybe you can see it more clearly now and you could figure it out?</p>
<p>RiesAray89, Thanks so much for your help! Now I was able to solve for the 2 limits you mentioned and got the correct answers (I assume the "handy shortcut" for derivative you mentioned was [f(x+h)-f(x)]/h). Is there another shortcut you were thinking of? That's the only one I know right now)</p>
<p>I guess the only real question I have is (and this is kind of scary I know, but I'm new to this), but, why do you have to find the derivative to compute the limit? Why isn't direct substitution OK, considering it's just 2 rational functions and there is no division by zero? I'm guessing it's because the velocity of the object changes instantaneously, but I mean mathematically, what is the reasoning?</p>
<p>I'm not required to use delta-epsilon to show work... We actually haven't learned delta-epsilon yet. If you could show me what it looks like in delta-epsilon, I'd appreciate it though, as I always like to be a little ahead of my homework :)</p>
<p>Thanks again!
Andre</p>
<p>ObsessedAndre:
If you compute [s(t) - s(0)] / [ t - 0], you are computing the <em>average</em> velocity of the object over all t seconds, not its instantaneous velocity at the end of the t'th second. That's why your 'direct substitution' answer is always coming up short. If an object is constantly accelarating, its average velocity will always be less than its velocity at time t.</p>
<p>Thanks optimizerdad, now it makes sense.</p>
<p>No problem Andre. The [f(x+h) - f(x)]/h was actually the long way...The short way is to use (d/dx)x^n = nx^n-1. I'm guessing you haven't learned this and some of the other shortcuts yet. That will come up very soon. Unfortunately, I'm probably in the same boat as you (AP Calc AB??), so we haven't even learned delta-epsilon yet! Hopefully, some of the more advanced calc-ers on this website can help you on that one :)</p>