Number and Algebra
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Calculus
Derive the velocity function v(t)v(t)v(t) for the position s(t)=e−t+sin(−t)t.s(t)=e^{-t}+\frac{\sin(-t)}{t} \text{.}s(t)=e−t+tsin(−t).
Let s(t)=e−2t(t+1)s(t)=e^{-2t}(t+1)s(t)=e−2t(t+1) for t≥0t\ge 0t≥0. Find all times when the acceleration is zero.
A particle moves on a line with position s(t)=5sint+ts(t)=5\sin t + ts(t)=5sint+t for t≥0t\ge 0t≥0. Find the first time the particle is at rest.
An object moves along a line with position s(t)=t3−6t2+9ts(t) = t^3 - 6t^2 + 9ts(t)=t3−6t2+9t for t≥0t \ge 0t≥0. Find the first time ttt when the object is at rest.
Find the acceleration function a(t)a(t)a(t) by differentiating your result for v(t)=−e−t+−t cost+sintt2.v(t)=-e^{-t}+\frac{-t\, cos t+\sin t}{t^2} \text{.}v(t)=−e−t+t2−tcost+sint.
An object has velocity v(t)=t2−4t+3v(t) = t^2 - 4t + 3v(t)=t2−4t+3 for all real ttt. Determine all times ttt when the object is speeding up.
An object has velocity v(t)=sint−12v(t)=\sin t - \tfrac{1}{2}v(t)=sint−21 for t∈[0,4π]t\in[0,4\pi]t∈[0,4π]. Find all times in [0,4π][0,4\pi][0,4π] when the object is decelerating (i.e., when its speed is decreasing).
Show that v(t)=−e−t+−tcost+sintt2v(t)=-e^{-t}+\frac{-t\cos t+\sin t}{t^2}v(t)=−e−t+t2−tcost+sint has exactly one root for t>0t>0t>0.
Find the first positive time t>0t>0t>0 at which the object is at rest, i.e. v(t)=−e−t+−tcost+sintt2=0ext.v(t)=-e^{-t}+\frac{-t\cos t+\sin t}{t^2}=0 ext{.}v(t)=−e−t+t2−tcost+sint=0ext.
Determine the intervals on which the object is slowing down for t>0t>0t>0.
Determine the intervals on which the object is speeding up for t>0t>0t>0.
Find the total distance traveled by the object from t=0.1t=0.1t=0.1 to t=πt=\pit=π given s(t)=e−t+sin(−t)t.s(t)=e^{-t}+\frac{\sin(-t)}{t}\text{.}s(t)=e−t+tsin(−t).
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Question Type 1: Given the displacement or velocity, finding the velocity or acceleration respectively
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Question Type 3: Going from acceleration and velocity, to velocity and displacement