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Take it to the LIMIT!
Numerical Representation and Formal Definition
Algebraic Methods for Evaluating Limits
Limit Theorems and Properties
Continuity and Discontinuity
Limits Involving Infinity
Intermediate Value TheoremVolumes of Solids with Known Cross Sections
Deriving Me CRAZY!!
Two Definitions of Derivatives
Parametric Derivatives and Second Derivatives
Uniqueness Theorem, Mean Value Theorem, Rolle's Theorem
Critical points, Points of Inflection and Optimization
Ain't It Great to Integrate!
Formal Def of Indefinite Integrals
Integral Rules For All Functions
Displacement, Velocity and Acceleration
Riemann Sums and Trapezoidal Approximations
Mean and Average Value Theorems for Integration
Integration by Parts
Areas Between Curves
Volumes of Solids of Rotation
Volumes of Solids with Known Cross Sections
Solving Differential Equations by Separating Variables: Exponential Growth and Decay
Slope FieldsThe Logistic Function: Including Solving Using Partial Fractions
Integral Properties and Techniques
Integrating Higher Power Trigonometric Functions
What's Your Vector Power Series?
Power Series for Basic Functions
Writing Series from Known Series
Geometric Series and Sequences
Taylor and McLaurin Series
Interval of Convergence Using Ratio Technique
Distance, Displacement and Acceleration
DISTANCE, DISPLACEMENT AND
Where do we begin? Let's nail down some essential terms first:
: total units traveled regardless of direction.
: distance from starting point (affected by direction traveled)
: Speed and direction.
: You can think of this as the absolute value of velocity, it involves no direction.
Hopefully by this point you have successfully navigated yourself through wiki-world and are now particularly learned in the ways of deriving and integrating. To be thorough, however, let's clarify that integrating is the process of finding an original function based on the function modeling the rate of change. In plain English, integrating is the opposite of deriving.
So let's say you've got a basic function:
where y represents distance and x represents time. Many learned mathematicians would refer to this sort of function as a
function. Now what do you do with this function?
First, you can derive it. This gives you:
This function now describes the rate of change of your distance, otherwise known as speed or
. Why does deriving your displacement function give you velocity, exactly? Recall that the slope of a function is your change in y over change in x, and the derivative of a function describes slope at a point. Because slope in a displacement function is represented by distance/time (speed), and the derivative represents the slope at each point of the original function, naturally the derivative becomes a graph representing velocity.
The same concept can be further applied to our new velocity function. Deriving it, we get:
Using the same logic, this function represents the rate of change of velocity. When finding the slope on a velocity graph, we come up with (distance/time)/time, or distance/time
(acceleration). Hence, we have found our acceleration graph!
OKAY! So we have learned that...
1. Velocity is the derivative of displacement
2. Acceleration is the derivative of velocity
3. Displacement is the integral of velocity
4. Velocity is the integral of displacement
But don't relax just yet!!! It is important to remember that an INTEGRAL also represents the area under a curve!! When we think about this in terms of units, it makes wonderful sense. Let's take the integral of the velocity graph, for example. The y-axis represents distance/time, and the x-axis represents time. To find the area, you multiply some value of x by some value of y, or (distance/time)(time). When we do this, time cancels out, and we are left with simply
, the integral of velocity!! The same concept applies to your acceleration graph, when you multiply (distance/time
)(time) and you are left with distance/time, or velocity.
Now let's get into some specifics. Suppose some person shows you a velocity graph of some particle and says, "find the total distance traveled by this particle, and you will never have to take a math class again!" You would really want to be able to find that distance correctly, wouldn't you? So let's do it.
Now, looking at this graph and knowing what we know about integrals, it would be silly in this particular instance to simply integrate from one point (say, 0) to another (say, 5). Why? Because here the graph contains negative integrals, where the y-coordinates are less than zero. If you were to take a sum of the integral without accounting for negatives, you would simply have the distance of the particle from the starting point as opposed to the total distance traveled. With me so far? Because of this, when you want to find the total distance traveled it is important to either take the integral of the absolute value of the function or integrate in sections, adding the positive areas with the absolute values of the negative areas. This will give you the correct distance traveled! It would most likely be a smart move to check out a graph of the function first to find exactly what you're dealing with regarding negative integrals and such before proceeding to evaluate.
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