This week in AP Calc we discussed using velocity functions to find the displacement of a curve. We revisited the fact that the derivative of a position function is a velocity function:
s(t) --> position, s'(t) = v(t) --> velocity
And, the second derivative of a position function as well as the derivative of a velocity function is an acceleration function:
s"(t) = v'(t) = a(t) --> acceleration
Essentially, the displacement of a curve is the rate multiplied by the time. Displacement in this case is very similar to distance and therefore can be found using the distance formula.
So, for example, say you are given the function: ds/dt = t^2 - 8/(t+1)^2. You know the ds/dt represents s'(t) or the derivative of the position function which we know is the velocity function or v(t). If you are asked to find how far a particle with this velocity function traveled in the first second (at (0,1) ), this is how you would find the displacement:
s(t) --> position, s'(t) = v(t) --> velocity
And, the second derivative of a position function as well as the derivative of a velocity function is an acceleration function:
s"(t) = v'(t) = a(t) --> acceleration
Essentially, the displacement of a curve is the rate multiplied by the time. Displacement in this case is very similar to distance and therefore can be found using the distance formula.
So, for example, say you are given the function: ds/dt = t^2 - 8/(t+1)^2. You know the ds/dt represents s'(t) or the derivative of the position function which we know is the velocity function or v(t). If you are asked to find how far a particle with this velocity function traveled in the first second (at (0,1) ), this is how you would find the displacement:
We also learned how to find the new position of a particle using both initial position and displacement. First, you would need to use the anti-derivative of the velocity function (which is the position function since you're looking for position). When you solve the integral for the given time duration which will be (t,v(t)), you will be left with the displacement of the function. To find the initial position of the function you must find the anti-derivative and solve for what the position function was at s(0). After finding the initial position, you would add it to the displacement and that would give you the new position of the particle.
For the last example above, we already found the displacement: -11/3. The anti-derivative of t^2 - 8/(t+1)^2 is 1/3t^3 + 8/(t+1). S(0) of the anti-derivative function would equal 8, therefore the initial position of the function is 8. Thus, the new position of the function would be equal to 8 + -11/3 or 13/3.
Now, to get the total distance of a function you would need to take the absolute value of the position function. Finding total distance can be modeled by:
Total Distance = Velocity * Time
I got caught frequently forgetting to take the absolute value of my answer which is really important. Total distance traveled has to be positive whereas net distance can be negative depending on where and which direction the particle moved. Another thing I messed up a few times when practicing finding total distance was matching the labels. At one point all my answers were correct, but were written in terms of miles per hour times seconds which doesn't really make any sense. So, I had to remind myself to set up proportions so that there would only be one time label like miles per second because then I could get the correct amount of miles I am looking for in order to answer the question.
For the last example above, we already found the displacement: -11/3. The anti-derivative of t^2 - 8/(t+1)^2 is 1/3t^3 + 8/(t+1). S(0) of the anti-derivative function would equal 8, therefore the initial position of the function is 8. Thus, the new position of the function would be equal to 8 + -11/3 or 13/3.
Now, to get the total distance of a function you would need to take the absolute value of the position function. Finding total distance can be modeled by:
Total Distance = Velocity * Time
I got caught frequently forgetting to take the absolute value of my answer which is really important. Total distance traveled has to be positive whereas net distance can be negative depending on where and which direction the particle moved. Another thing I messed up a few times when practicing finding total distance was matching the labels. At one point all my answers were correct, but were written in terms of miles per hour times seconds which doesn't really make any sense. So, I had to remind myself to set up proportions so that there would only be one time label like miles per second because then I could get the correct amount of miles I am looking for in order to answer the question.