Math 1351-011: Monotonic Functions, Derivatives, and Limits - Prof. Victoria Ellen Howle, Study notes of Mathematics

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Announcements

• Tuesday 10/30 is the last day to drop
• Homework 9 due this Friday 11/2/

Monotone function theorem: Let f be differentiable on the open interval (a, b).

• If f ′(x) > 0 on (a, b), then f is strictly increasing on (a, b).
• If f ′(x) < 0 on (a, b), then f is strictly decreasing on (a, b).

• If the graph of a function f lies above all of its tangents on an interval I, then it is said to be concave up on I.
• If the graph of a function f lies below all of its tangents on an interval I, then it is said to be concave down on I.
• The graph of a function f is concave up on any open interval I where f ′′(x) > 0, and concave down where f ′′(x) < 0.
• A point P (c, f (c)) on a curve is called an inflection point if the graph is concave up on one side of P and concave down on the other side.

2nd Derivative Test for Relative Extrema

Let f be a function such that f ′(c) = 0 and the second derivative exists on an open interval containing c.

• If f ′′(c) > 0, there is a relative minimum at x = c. (f is concave up on an interval around c)
• If f ′′(c) < 0, there is a relative maximum at x = c. (f is concave down on an interval around c)
• If f ′′(c) = 0, the 2nd derivative test fails. (The point could be either a max, a min, or neither.)

Formal definition of limits to infinity

The limit statement limx→+∞ f (x) = L means that for any number  > 0, there exists a number N 1 such that

|f (x) − L| <  whenever x > N 1

for x in the domain of f.

Similarly, limx→−∞ f (x) = M means that for any number  > 0, there exists a number N 2 such that

|f (x) − M | <  whenever x < N 2

for x in the domain of f.

Limit Rules

If limx→+∞ f (x) and limx→+∞ g(x) exist, then for constants a and b:

Power rule:

x→^ lim+∞[f^ (x)]n^ = [^ x→lim+∞ f^ (x)]n Linearity rule:

x→^ lim+∞[af^ (x) +^ bg(x)] =^ a^ x→lim+∞ f^ (x) +^ b^ x→lim+∞ g(x)

Product rule:

x→^ lim+∞[f^ (x)g(x)] = [^ x→lim+∞ f^ (x)][^ x→lim+∞ g(x)]

Quotient rule:

x→^ lim+∞^ f g^ ((xx) ) = limlimx→+∞^ f^ (x) x→+∞ g(x)^

if (^) x→lim+∞ g(x) 6 = 0

Similarly for limx→−∞ f (x) and limx→−∞ g(x) if they exist.

Infinite Limits

x^ lim→c f^ (x) = +∞ means that f increases without bound as x approaches c (from either side).

xlim→c g(x) =^ −∞ means that g decreases without bound as x approaches c (from either side).

Formal definition of infinite limits

x^ lim→c f^ (x) = +∞ if for any number N > 0 (no matter how large), it is possible to find a number δ > 0 such that f (x) > N whenever 0 < |x − c| < δ.

Similarly,

xlim→c g(x) =^ −∞ if for any number N > 0, it is possible to find a number δ > 0 such that f (x) < −N whenever 0 < |x − c| < δ.

Note that ∞ is not a number. A limit that goes to infinity does not exist (in the sense that it does not approach a number). But this specifies that way in which the limit does not exist.

Vertical tangents and cusps

Suppose the function f is continuous at the point P (c, f (c)). Then the graph of f has

• a vertical tangent at P if limx→c−^ f ′(x) and limx→c+^ f ′(x) are either both +∞ or both −∞.
• a cusp at P if limx→c−^ f ′(x) and limx→c+^ f ′(x) are both infinite with opposite signs (one +∞ and the other −∞).

Summary of Graphing Strategy

• Simplify the equation as much as possible
• Find 1st derivatives and critical numbers
• Determine intervals of increase and decrease
• Apply 2nd derivative test
• Determine concavity and points of inflection
• Apply 1st derivative test
• Find asymptotes, vertical tangents, cusps
• Plot points
• Sketch the curve

(See detailed summary on last page of section 4.3 of Strauss. Page 226 of 5th edition. )