- Einstein says the kinetic energy of an object moving very fast is E = mc2( 1/√(1-v2/c2) - 1 ) where m is the object’s (rest) mass, v is its speed, and c is the speed of light.
- This is how much energy has to be put into the object to accelerate it from standing still to speed v
- What is the limit of this as v approaches c?
- Consequence: “Star Trek” spaceships must accelerate discontinuously
- If they didn’t, then in accelerating from stopped to
faster than light, there would be some time when speed equalled c
- (Intermediate Value Theorem)
- … and we just showed that it is impossible to move at the speed of light
- Example of a proof by contradiction
- If they didn’t, then in accelerating from stopped to
faster than light, there would be some time when speed equalled c
- Section 3.2
- Example 1?
- A couple of simplification steps compressed out of book:
- (x+h)(x-1) - x(x+h-1)
- = (x2 - x + xh - h) - (x2 + xh - x)
- A couple of simplification steps compressed out of book:
- Derivatives that don’t exist?
- Key point: f′(x) = limh→0( f(x+h) - f(x) ) / h
- Aren’t we back where we started?
- Started w/ average rate of change
- Wanted instantaneous rate aka slope of curve at a point as interval gets smaller and smaller
- Limits as way of being precise about “smaller and smaller”
- Limit applied to rate of change is derivative
- Example 1?
- Examples
- f(x) = 1
- limh→0( f(x+h) - f(x) ) / h
- = limh→0( 1 - 1 ) / h
- = limh→0 0 / h
- = limh→0 0
- = 0
- f(x) = 1
- More examples
- Start differentiation rules as time permits