a good video on the calculus intro which a 5th grader can understand.
- 1/x when x = 0 : Newton created Limit in order to solve instantaneous speed of planets at change of t -> 0
- We invented imaginary number because negative numbers cannot have square roots in Real Number Set.
MC does not cover these two topics (textbook Ch10-12)
analytic geometry,
systems of equations, sequences, and series
table and graph
- Limit does not care what happens to the point when x=a.
- Relate case (a) to continuity
- This limit concept helps us to study the points that are not defined, such as denominator = 0, square root < 0; log X where X =<0.
- As denominator approaches 0, numerator approaches a nonzero number
- Left side limit Not equal to right side limit:
- absolute function at a=0
- Greatest integer functions
At a point (a, f(a)):
Limit (two definitions as x -> a OR as h ->0) [x-> infinite OR 1/x ->0]
- Slope of the tangent line (Point-slope form)
- Velocity (instantaneous velocity)
Derivative: f'(a)=m=v(a)
Instantaneous Rate of change: lim [delta y / delta x ]
f is differentiable <=> limit exists. [by definition]
- at point a,
- on an open interval (a,b)
f is not differentiable <=> limit DNE
2.2 Limits
2.5 Continuity
2.6 Limits at Infinity
2nd part: precise definition
2.7: Derivative and Rate of Change
At a point (a, f(a)):
Limit (two definitions as x -> a OR as h ->0) [x-> infinite OR 1/x ->0]
- Slope of the tangent line (Point-slope form)
- Velocity (instantaneous velocity)
Derivative: f'(a)=m=v(a)
Instantaneous Rate of change: lim [delta y / delta x ]
2.8
f is differentiable <=> limit exists. [by definition]
- at point a,
- on an open interval (a,b)
f is not differentiable <=> limit DNE
3.1
2.2 Limits
2.3 Limits Laws — Prove after learning 2.4
(P170)
(P136)
(P134)
2.5: Prove all the theorems 4-10
(2.5 P126)
(2.5)P127
2.7: derivatives
s=f(t) = 4.9t^2
2.8. derivative functions
Prove Theorem 4
3.1 Prove the Power Rule. sum rule, constant rule
The normal line is perpendicular to the tangent line at the point.
2.3 Limits Laws — Prove after learning 2.4
2.5: Prove all the theorems 4-10
2.6. Limits at Infinity
Hint: Limit laws 6,11 and Example 2
2.8 Derivative functions
3.1 Prove the Power Rule. sum rule, constant rule
Review Binomial:
- Note:
- Review: the Binomial Theorem
Prove: if the exponential function f(x) = a^x is differentiable at x=0, then it is diffentiable everywhere and
Note: What does the above equation 4 say about the rate of change of any exponential function (the slope)?
Note: what is the geometric significance of this fact? [draw the graph]
Proof: youtube