Ok so derivative are an expression of the rate of change of a function. Cool I get that.
– F(x) = 5 : the product of this function is always 5 there is no increase or decrease so there is no change no matter what X is and it makes sense that the derivative would equal 0.
– F(x) = 5x : it is obvious that each time x increases by 1 the product of this function increases by 5. I get it.
– F(x) = x² => F'(x) = 2x : starting from here the numbers stop matching and make me feel like I am missing something. F'(1) = 1. This makes perfect sense. F(x) did in fact increase by 1 when going from F(0) to F(1). Then I try F'(2) = 2×2 = 4. Huh ? But F(x) only increased by 3 between F(1) and F(2) ? Maybe I am looking at the rate of change as compared to F(0) ? after all there is an increase of 4 between F(0) and F(2). Let’s check with 3 then. F'(3) = 6. Wtf ?!
I don’t get it what does it mean when F'(2) = 4 ? When X = 2 then …? and what does it tell me about the original function. Thanks and hope my english isn’t too awfull.
In: 40
You get in your car, buckle up, turn it on, and put the pedal to the metal. The trip-o-meter started at 0 miles but quickly starts going up really fast, and soon everybody else is whizzing past you in a blur.
You may look at your speedometer and think it measures your speed, and it does. But it measures how fast you’re going even if your buddy takes a picture of you. While the picture doesn’t show you moving, per se, it does show you’re whizzing super fast cause your car is a blur. This picture of you, the glance at your speedometer is the derivative. It is an instant in time.
But if your buddy on the street wanted to measure how fast you were going they’d have to use a stopwatch and start it when you whizzed past the tree and then stop it when you flew by the lamp post. He’d then measure how many feet between those 2 points, then divide by the time. His measured speed didn’t match up with your speedometer between the tree and lamp post because you saw the indicator smoothly go from 0 to infinity and beyond. It only hit your buddy’s measured speed once. He would be wrong thinking you went this measured speed the whole time between the tree and lamp post.
His was an average over some range while your was instantaneous snapshot, just as your F(3)-F(2)=5 was different than F'(3)=6. Your speed probably started at 4 when you went past the tree and got to 6 when you past the lamp post, but this averages out to 5 from your buddy’s perspective. The instant you past the lamp post your speedometer showed 6.
Latest Answers