Motion and Antiderivatives None of what you’ll find in these notes is new, but it’s important to pull it all together to make sure we can think about motion not just in terms of derivatives, but also in terms of antiderivatives. Problem: A particle moves along the y-axis and has position y = 3 when t = 0 . A graph of the particle’s velocity as a function of t is shown below.
a. Determine the particle’s speed at t = 6 . Is the speed increasing, decreasing or constant at that time?
b. At what time(s) does the particle reach the origin?
c. Find the absolute lowest and absolute highest points reached by the particle. (Absolute max/min, eh? What test is this going to involve?)
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Problem: (Calculator) A particle moves along a straight line. For 0 ≤ t ≤ 3 , the velocity of the particle is 5/3 1 given by v ( t ) = 2 + ( t 3 − 4t ) − t 2 , and the position of the particle is given by s ( t ) . It is given that 5 s (0) = 3 . a. Find all values of t in the interval 0 ≤ t ≤ 3 for which the speed of the particle is 1.
b. Write an expression involving an integral that gives the position s ( t ) . Use this expression to find the position of the particle at t = 2 .
c. Find all times t at which the particle changes direction. Justify your answer.
d. Is the speed of the particle increasing or decreasing at time t = 1 ? Give a reason for your answer.
Problem: The velocity, in miles per minute, of an object moving along a horizontal axis is modeled by the piecewise-linear function whose graph is shown below.
a. Find the acceleration at t = 1.5 minutes. Indicate units of measure.
b. Using correct units, find the value and explain the meaning of
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∫ v (t ) dt . 10
0
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t 0 6 18 24 30 v (t ) 5 -4 -8 -1 6 Problem: The velocity of a particle moving along the x-axis is modeled by a differentiable function v, where x is measured in feet, and t is measured in seconds. Selected values of v ( t ) are given in the table above. The particle’s position is x = 4 feet when t = 0 . a. Estimate the acceleration of the particle at t = 12 seconds. Show the computations that lead to your answer. Indicate units of measure.
b. Using correct units, explain the meaning of
∫ v (t ) dt 24
6
in the context of the problem. Use a
trapezoidal sum with two subintervals indicated by the data in the table to approximate
∫ v (t ) dt . 24
6
c. For 0 ≤ t ≤ 30 , must the particle change direction in any of the subintervals indicated by the data in the table? If so, identify the subintervals and explain your reasoning. If not, explain why not.
d. Suppose that acceleration of the particle is negative for 0 < t < 6 seconds. Explain why the particle’s position at t = 6 must be less than x = 35 feet.
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Problem: For 0 ≤ t ≤ 6 , a particle is moving along the y-axis. The particle’s position, y ( t ) , is not
( )
explicitly given. The velocity of the particle is given by v ( t ) = −3cos et 3 − 1 . The acceleration of the
( )
particle is given by a ( t ) = et 3 sin et 3 and y ( 0 ) = 3 .
a. Is the speed of the particle increasing or decreasing at t = 4 ? Give a reason for your answer.
b. Find the average velocity of the particle on the time interval 0 ≤ t ≤ 6 .
c. Find the total distance traveled by the particle on the time interval 0 ≤ t ≤ 6 .
d. Does the particle ever reach the origin on the time interval 0 ≤ t ≤ 6 ?
e. Find the particle’s maximum acceleration on the time interval 0 ≤ t ≤ 6 .
f. Use the line tangent to y ( t ) at t = 0 to estimate y (1) . Is this an over or underestimate?
g. Write an expression involving an integral that gives the position y ( t ) . Use this expression to find y (1) .
A graph of the particle's. velocity as a function of t is shown below. a. Determine the particle's speed at t = 6 . Is the speed increasing, decreasing or constant at that time? b. At what time(s) does the particle reach the origin? c. Find the absolute lowest and absolute highest points reached by the particle. (Absolute max/min,.
Page 1 of 6. Calc AB Notes 02 1 of 6 www.turksmathstuff.com. The Chain Rule. The final rule of taking derivatives that we need to learn (although it won't seem like it's the last rule. since you'll find yourself doing all sorts of weird problems in w
Calc AB Notes 08 3 of 6 www.turksmathstuff.com. This is a hugely useful result. Don't go crazy and over use it, though! You have to get an indeterminate. form (either 0. 0. or ± â. â ), so always check. Also, it's still more expedient to notice
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Page 1 of 6. Calc AB Notes 08 1 of 6 www.turksmathstuff.com. L'Hopital's Rule. Let's start with some problems that involve taking limits. Problem: Evaluate each of the following limits. a. limxâ3. x. 3 â 27. x â 3 b. lim. hâ0. csc(3( x + h))
f ( x) -4 3 6 8. f â²( x) 4 1 5 3. a. Approximate f ( x)dx. 5. 16 â« with a left Riemann sum and intervals indicated in the table. b. Approximate f â²(11). c. Evaluate f â²( x)dx. 5. 16 â« . Page 4 of 9. Calc AB Notes 16.pdf. Calc AB Notes 16.pd
eventually vanish if you keep taking them repeatedly...I'm looking at you polynomials!) ⢠You have to pick dv so that you can actually integrate it or you've defeated the entire purpose... Other people use the acronym L.I.A.T.E. for making the choi
You are expected to attend all classes (on time!) prepared with your assignments,. COVERED textbook, notebook, graphing calculator/app, and writing utensils.
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Page 1 of 1. Calc 1 Exam review notes.notebook. 1. January 16, 2014. Jan 167:49 AM. Semester Examnotes. 1. Optimization from 4.6. 2. Related Rate 4.1. 3. Antideriv. (velocity) 4.8. 4. Tangent line implicit diff. 3.5. 5. Given graph of f ' determine m
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resolved to continue with investment in infrastructure and. has put in place appropriate measures to ensure fiscal. prudence. Michael M. Mundashi, SC. Chairman. Whoops! There was a problem loading this page. Retrying... Bargain - AB Guthrie.pdf. Barg
Page 1 of 1. Acton-Boxboroughâ âElementaryâ âBusâ âPass. This form must be delivered to the school office beforeâ â2:00â âpmâ âtheâ âdayâ âbeforeâ âtheâ âchangeâ âisâ âeffectiveâ, OR bus pass will not
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and stay close to the BO surface. - Optimization method instead of variational equation. - Equation of motion instead of matrix diagonalization. Be careful.
The EQUATION section of Figure 2 shows how the. data distribution and decoding logic works. Equations. starting with A-G are generic seven segment display.
Page 1 of 1. A.G.S. Middle School. 2017 - 2018 School Calendar. December 2017. Monday Tuesday Wednesday Thursday Friday. 1. A Day. 4 5 6 7 8. B Day A Day B Day A Day B Day. 11 12 13 14 15. A Day B Day A Day B Day A Day. 18 19 20 21 22. B Day A Day B