Math
105
Thursday, April 27
Review from last class:
What is the calculus?
A systematic method for
answering these questions of velocity,
tangents, change in position and area that uses the algebraic
representation of a relation between y and x .
Examples: if y = .... then the slope of the tangent to
the curve described by this relationship at the point (a,b) is....
If the position s = ... t.... then the instantaneous
velocity of the
object moving to correspond to this relation at time t is
.................
If the velocity of a moving object is given by v= ....
t..... , then
the object has a net change during the interval from t = a to t = b of
.....
the area of the region in the plane bounded by the Xis,
the lines X = a and X= b and the graph of y= ...x... is....
Newton: Tangent lines,
velocity, and the derivative.
Using systematic estimates to find i) the slope of the
tangent line and ii) the instantaneous velocity.
The derivative: a limiting number interpreted as
i) the slope of the tangent line or
ii) the instantaneous velocity
A calculus for derivatives: core functions- example powers of
x.
Rules: sum . constant multiples, etc. [Ways
that we put together core functions]
Activity: Finding the derivative of a
function by estimation and by rules
Application: Finding maximum - old
Macdonald: Fence Problem- find
dimension of largest rectangular area enclosed by a fence of length 100
meters using the side of a barn as part of the boundary of the
rectangle.
A = f (x).
What does the maximium area have to do with the slope of the
tangents to the graph of A?
When is tangent horizontal? calculus - formula for slope of
the tangent as a function of x: ...
Notation:
y = f(x). dy/dx = f'(x) Leibniz.Lagrange
Reversing the process-
given the derivative- find the original function(s): "Integration".
Interpretation: Given slopes of tangent lines- fit a curve to
them. [A dot to dot problem.]
Given velocity, find a position motion that corresponds. [A
detective problem]
Determining position and areas. Change in position - area of
rectangle.
Putting concepts together with computations.
To find area under curve y = v(x). Find s where s' = v. Area
corresponds to net change in position.
Fundamental Theorem of
Calculus: Area = s(b) - s(a).
Notation: v = F
'(x); int v dx
= F(x) + C; int_a^b v dx = F(b) - F(a)
"Seeing
the Infinite"
