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If a ball is thrown into the air with a velocity of $ 40 ft/s $, its height in feet $ t $ seconds later is given by $ y = 40t - 16t^2 $.
(a) Find the average velocity for the time period beginning when $ t = 2 $ and lasting (i) 0.5 seconds (ii) 0.1 seconds (iii) 0.05 seconds (iv) 0.01 seconds(b) Estimate the instantaneous velocity when $ t = 2 $.
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05:24
Daniel Jaimes
01:10
Carson Merrill
05:04
Cindy Rodgers
Calculus 1 / AB
Chapter 2
Limits and Derivatives
Section 1
The Tangent and Velocity Problems
Limits
Derivatives
Juan D.
August 29, 2021
If a ball is thrown into the air with an initial velocity of 42 ft/s,
Hank S.
September 23, 2020
Paucity is the presence of something only in small or insufficient quantities or amounts; scarcity. Hope that helps Frank.
Frank V.
I'm confused by the term paucity.
Sam L.
Hey Jessica, Velocity is defined as a vector measurement of the rate and direction of motion. Hope that helps.
Alexis B.
Can someone explain what the Velocity is?
Holly S.
Average velocity can be defined as the displacement divided by the time. Hope that helps Jessica.
Jessica G.
What is a average velocity?
Angel S.
June 8, 2020
I agree with Samantha. This is not helpful at all.
Samantha G.
March 17, 2019
seeing someone else use excel is not helpful in teaching how to complete a problem like this. As someone who struggles with understanding math computation, I need to see the problem worked out and broken into steps.
Missouri State University
Oregon State University
University of Michigan - Ann Arbor
University of Nottingham
Lectures
04:40
In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.
In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.
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If a ball is thrown into t…
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The height $s$ in feet of …
Alright, here's a fun physics problem, we've got a ball being thrown upward with the starting velocity of 40ft per second and we have the equation that describes the height of the ball with time being 40 t minus 16 T squared and our goal is to find the average velocity. And we're going to do this um basically um for different um different lengths of time and we're gonna kind of compare and see what our average velocity is for the different time periods. Okay, so in general average velocity is just basically the displacement over time. So it's basically going to be our final white time minus y at two seconds. Because we're interested to find um We're going to use approximate the velocities the average velocity, but our starting point is always two seconds And then that's going to be over T -2. Alright, so we can substitute in our equation 40 T minus 16 T squared and then we got to figure out what Why is that too? So that's 80 -16 times four, so that ends up being 16. So we're going to subtract 16 All over T -2. Okay, so that's gonna be what we used to plug in their values um So our first t is um .5 seconds later, so it'll be at two sorry it will be a half a second later. So it We'll go from 2-2.5. So this will be basically the end the end time that we're using for a little interview interval to find the average velocity. Alright, so if we plug in 2.5 we'll use our calculator, plug it into the equation the top right? Um we end up with 32 as our feet per second average velocity If we go to 2.1, so notice it's only quite one time spread from the two original time And that will give us -25.6 If we get really close to 2,2.05, so a smaller interval with we get minus 24.8 and if we get really close only .01 seconds after the two and do that interval, we might get -24.16. So you can see we're honing in on 24. So if we're doing our best guess At the velocity at two seconds, I would say it's about 24 ft per second. That would be our best guess for our instantaneous velocity based on seeing the average velocity where the two points become really close to what would be the tangent line. So anyway, hopefully that helped having a wonderful day. See you next time
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