Chapter 2: Derivatives
xyz = 3 ; pretend each is a function of t, and differentiate (implicitly).
General procedure:
Draw a picture, describing the situation; label things with variables.
Which variables, rates of change do you know, or want to know?
Find an equation relating the variables whose rates of change you know or want to know.
Differentiate!
Plug in the values that you know.
Chapter 3: Applications of Derivatives
d is an (absolute) minimum for a function f(x) if f(d) £ f(x) for every other x
max or min = extremum
Extreme Value Theorem: If f is a continuous function defined on a closed interval
[a,b], then f actually has a max and a min.
Goal: figure out where they are!
c is a relative max (or min) if f(c) is ³ f(x) (or £ f(x)) for every x near c. Rel max or min = rel extremum.
An absolute extremum is either a rel extremum or an endpoint of the interval.
c is a critical point if f^{¢}(c) = 0 or does not exist.
A rel extremum is a critical point.
So absolute extrema occur either at critical points or at the endpoints.
So to find the abs max or min of a function f on an interval [a,b] :
(1) Take derivative, find the critical points.
(2) Evaluate f at each critical point and endpoint.
(3) Biggest value is maximum value, smallest is minimum value.
Mean Value Theorem: if f is containous on [a,b] and differentiable on (a,b), then there is at least one c in (a,b) so that
Consequences:
Rolle's Theorem: f(a) = f(b) = 0; between two roots there is a critical point.
So: If a function has no critical points, it has at most one root!
A function with f^{¢}(x)=0 is constant.
Functions with the same derivative (on an interval) differ by a constant.
f is increasing on an interval if x > y implies f(x) > f(y)
f is decreasing on an interval if x > y implies f(x) < f(y)
If f^{¢}(x) > 0 on an interval, then f is increasing
If f^{¢}(x) < 0 on an interval, then f is decreasing
Near a local max, f is increasing, then decreasing; f^{¢}(x) > 0 to the left of the critical point, and f^{¢}(x) < 0 to the right.
Near a local min, the opposite is true; f^{¢}(x) < 0 to the left of the critical point, and f^{¢}(x) > 0 to the right.
If the derivative does not change sign as you cross a critical point, then the critical point is not a rel extremum.
Basic use: plot where a function is increasing/decreasing: plot critical points; in between them, sign of derivative does not change.
f is concave up on an interval if f^{¢¢}(x) > 0 on the interval
Means: f^{¢} is increasing; f is bending up.
f is concave down on an interval if f^{¢¢}(x) < 0 on the interval
Means: f^{¢} is decreasing; f is bending down.
A point where the concavity changes is called a point of inflection
Graphing:
Find where f^{¢}(x) and f^{¢¢}(x) are 0 or DNE
Plot on the same line.
In between points, derivative and second derivative don't change sign, so graph looks like one of:
Then string together the pieces!
Second derivative test: If c is a critical point and
f^{¢¢}(c) > 0, then c is a rel min (smiling!)
f^{¢¢}(c) < 0, then c is a rel max (frowning!)
lim_{x®¥}f(x) = L means f(x) is close of L when x is really large.
lim_{x® -¥}f(x) = M means f(x) is close of M when x is really large and negative.
Basic fact: lim_{x®¥} [1/x] = lim_{x® -¥} [1/x] = 0
More complicated functions: divide by the highest power of x in the denomenator.
f(x),g(x) polynomials, degree of f = n, degree of g = m
lim_{x®±¥} [f(x)/g(x)] = 0 if n < m
lim_{x®±¥} [f(x)/g(x)] =
(coeff of highest power in f)/(coeff of highest power in g) if n = m
lim_{x®±¥} [f(x)/g(x)] = ±¥ if n > m
The line y = a is a horizontal asymptote for a function f if lim_{x®¥}f(x) or lim_{x® -¥}f(x) is equal to a.
I.e., the graph of f gets really close to y = a as x®¥ or a®-¥
The line x = b is a vertical asymptote for f if f®±¥ as x®b from the right or left.
If numerator and denomenator of a rational function have no common roots, then vertical asymptotes = roots of denom.
f®¥ or f®-¥ : can use f incr or decr on either side of b to decide (so long as you already know it is blowing up!)
Draw a picture; label things.
What do you need to maximize/minimize? Write down a formula for the quantity.
Use other information to eliminate variables, so your quantity depends on only one variable.
Determine the largest/smallest that the variable can reasonably be (i.e., find your interval)
Turn on the max/min machine!