Increasing and decreasing functions:

Function is increasing for:

x < - 1.3

x > 0

 

 

 

Function is decreasing for:

- 1.3 < x < 0

 

 

 

Increasing functions in the interval of x1 and x2:

 

x1 < x2

f (x1) < f (x2)  (by using the function)

f (x) > 0  (by using first derivative)

 

 

Decreasing functions in the interval of x1 and x2:

 

x1 < x2

f (x1) > f (x2)  (by using the function)

f (x) < 0  (by using first derivative)

 

 

Critical number:

 

c is a critical number if

f(c) = 0 or

f(c) does not exist (undefined).

 

Critical point:

 

Critical Point:  (c, f (c))

 

 

First derivative test:

 

Let f(c) = 0 and c: critical number

Local maximum at c if f(x) changes sign from positive to negative at x = c.

Local minimum at c if f (x) changes sign from negative to positive at x = c.

Neither local maximum nor a local minimum if f (x) does not change sign at x = c.

 

 

Second derivative test:

 

Local maximum at c if f(c) = 0 and f" (x) < 0

Local minimum at c if f(c) = 0 and f" (x) > 0

 

 

Three ways of finding local maximum and local minimum:

 

  local maximum local minimum

the original function

f(c) > f (x) for all x values near c
(on both sides of c).
f(c) < f (x) for all x values near c
(on both sides of c).
First derivative test

f(c) = 0 and f(x) changes sign from positive to negative at x = c

 

Neither local maximum nor a local minimum if f (x) does not change sign at x = c

f(c) = 0 and f(x) changes sign from negative to positive at x = c

 

Neither local maximum nor a local minimum if f (x) does not change sign at x = c

Second derivative test

f(c) = 0 and f" (x) < 0

f(c) = 0 and f" (x) = 0 (no information if it max or min)

f(c) = 0 and f" (x) > 0

f(c) = 0 and f" (x) = 0 (no information if it max or min)

 

 

Cusp:

 

If f(x) is undefined, there is a cusp at that point.

 

 

Concavity

 

Concave upward:

The graph lies above all its tangents.  f" (x) > 0

 

Concave downward:

The graph lies below all its tangents.  f" (x) < 0

 

 

Point of inflection at c

 

f (c) = 0 and f (x) changes sign at c from positive to negative.  The the graph changes from concave upward to concave downward.

 

f (c) = 0 and f (x) changes sign at c from negative to positive.  The the graph changes from concave downward to concave upward.

 

If f (c) = 0, but f (x) does not change sign at c, there is no point of inflection.

 

 

Vertical asymptotes:

 

 

Example:

 

 

Horizontal asymptotes:

 

 

Example:

 

 

Therefore, the graph approaches the asymptote from above at +∞ and -∞

 

 

Oblique asymptotes:

 

They occur with rational functions in which the degree of the numerator is exactly one more than degree of denominator.

 

Example:

 

Method for Curve Sketching:

 

Determine if there are any discontinuities.

 

Determine the domain.

 

Investigate the values on either side of the discontinuity.

 

Determine if there are any vertical asymptotes.

 

Find the direction of the curve at left and right side of the asymptotes.

 

Determine x-intercepts and y-intercepts.

 

Determine the critical points.  f (x) = 0 at the critical points.

 

Test the critical points whether they are local maxima, local minima, or neither.

 

Determine the horizontal asymptotes if there are any.

 

Determine the behaviour of the function for large positive and negative values of x.

 

Find the points of inflections for setting f (0) = 0

 

Determine the oblique asymptotes if there exist.

 

Draw the sketch.