This video has provided valuable information regarding the use case of derivatives, which was not previously taught in our years of computation. I am grateful for this newfound understanding.
Think of the curve made up of hundreds of thousands straight lines. The slope foubd is at 2 points so close together that it can be seen as a single point. Think of it being infinitely small.
So to find the instantaneous velocity we find the derivative of the given equation(curve) and plug in the given point into the derivative of said equation?
Yes, assuming the given equation is the equation for position at any time. Finding the derivative and then plugging in the value for time is usually the most convenient approach.
So basically an instantaneous velocity is just a smaller sample of average velocity? In that you’re just taking a portion( a very small point in time) but you can never be entirely accurate because you would be trying to find and infinitely small distance over an infinitely small time?
From my understanding so far: The Derivative is the formula that gives you the slope of the tangent line that intersects the graph at a single point, "x." This formula is found by simplifying the difference quotient of the function as h approaches 0 (dealing with limits). Once you've found this formula from analyzing the difference quotient, you can insert the x-value of the single point into the formula, which will yield the slope of the tangent line that touches the one point on the graph. That slope represents the instantaneous velocity of a single point. Since you now have the x and y coordinates, as well as the newly obtained slope of the tangent line, or simply the slope at that particular point, you can use the point-slope equation to find the equation of that tangent line. (If you needed to for homework) Is that about right?
Yes, that is correct. In addition: Simplifying the difference quotient can be rather cumbersome, but there are various techniques for quickly and easily finding the derivative for various types of functions (the Power Rule, the Chain Rule, etc.)
Many of the videos are indexed here: www.lucideducation.com/?p=VideoIndex.php I hope to get a more complete index put up at derekowens.com sometime soon, hopefully this summer.
Brilliant explanation of the difference between average velocity and instantaneous velocity.. I like how you showed that the slope of the secant line to the position function graph is the average velocity and the slope of the tangent line is the instantaneous velocity.
your method allows us to calculate instantaneous velocity without knowing the path graph formula by using two points on the tangent line, or am i seeing something that isn't here? Are we even allowed to Arbitrarily draw a tangent line?
Amazingly well explained. Now I understand both, derivatives and instantaneous velocity! (I was looking for explanation of instantaneous velocity since average is pretty straightforward)
You might need to start with videos that explain other concepts leading up to this. Or watch multiple videos of the same concept from different people. You may find bits and pieces from each video that help the over all concept come together in your mind :)
