So we won’t actually be using this in class until April, but might as well get you thinking this way: Assuming θ is in radians. sinθ = tanθ = θ as shown in the image above.
Another image about sinθ = tanθ = θ
Chapter 2 begins.
v bar is average velocity. just plain v represents instantaneous velocity.
Secant lines (green) slowly turn into a tangent line (red) as the ∆t gets smaller and smaller and smaller until it finally turns into dt. So the slope of secant lines (∆x / ∆t) represents average velocities. The slope of tangent lines (dx/dt) represents instantaneous velocity at the point at which the line “kisses” the curve.
Showing secant (v bar) vs. tangent (v)
overkill probably, but making the very important point again. You’re welcome.
The “trio” derivative (slope graph of x vs. t, in other words dx/dt) and second derivative (slope of the slope graph of x vs. t . . . or . . . dv/dt . . . or d²x/dt²)
A student shows an example of the trio.
Derivative (slope) of y = x³ is y = 3x² Power rule shown graphically.
Zero slopes and point of inflection (turning into a local minimum or maximum)
A student determined that the slope graph had the function dy/dx = (x-4)²-2 .
The problems from the class discussion. deaths per week , vs. deaths per week per week. Okay . . . you had to be there.
two equations for v bar.
Average acceleration vs. instantaneous acceleration.
1st Orange is the derivative of the 2nd Orange. Ain’t Physics cool.
Part 1 of 4. Symbol proof of the power rule for a parabolic function.
Part 2 of 4: You will have to know this for the test.
Part 3 of 4
Part 4 of 4
A couple of derivative rules.
Tthe two scenarios where acceleration is negative. Don’t use the word deceleration. It’s not even in the Physics dictionary.
the two cases of negative acceleration on one graph.
This si from the homework. I think it is the last problem on 2.1 I need to see THIS KIND of work on your papers.
Power rule for second derivatives.
x double dot; x triple dot
V vs. t with slope of secant being average acceleration (a bar) and slope of tangent being instantaneous acceleration (a).