The 6 Pillars
When we cannot obtain an exact answer to a problem, we approximate it with limits
close is good enough
traditional slope formula
change in y/change in x or y2-y1/x2-x1 or delta y/delta x
slope of a line passing through (5,6) and (-3,9)
-3/8
5x(x2-5x+2)
By observing the change in variables over short periods of time, we can determine the overall trend of a function
one step at a time
the rate of change of a function at a certain point, "instantaneous rate of change"
derivative
give two examples of real life application of derivatives
ex. instantaneous velocity, marginal cost and revenue, population growth rate, temperature variation, etc.
simplifiy (x-2)2
(x-2)(x-2) = x2-4x+4
We use derivatives to express average rates of change
track the change
linear approximation
after setting up this equation, what is the next step see slide A
remove the fraction on top of the fraction by multiplying by (1/h)
find the least common denominator: 8/3h+6 and 2/5
5(3h+6) or 15h+30
A person estimates a value that isn't able to be calculated exactly. What pillar are they using?
close is good enough
as you zoom in on a smooth curve, it looks straighter, what is the equation used to approximate this
microscope equation,
f(x) = f(a) + f’(a) (x-a)
find and correct the mistake in this step, see slide B
the negative wasn't distributed to the 'h', the numerator of the fraction should be 5-5-h
find the LCD and subtract the numerator: (1/3+h)-(1/3)
-h/3(3+h) or -h/9+3h
A person looks at the change over a short period of time
one step at a time
The Derivative: Best Definition
f'(a)=limh->0 f(a+h)-f(a)/h
with the information given, what is the derivative, see slide C
-1/25
simplify: (x+3)3
(x+3)(x+3)(x+3) = x3+9x2+27x+27