Find relative extrema.
a) Find critical numbers [f '(c) = 0 or f '(c)
undefined]
b) Find f(c) for all c.
Evaulate function at endpoints: find f(a) and f(b)
Compare f values of relative extrema and endpoints and select:
a) the point (x,f(x)) with the largest f value is
the absolute maximum
b) the point (x,f(x)) with the smallest f value is
the absolute minimum
Find critical numbers [f '(c) = 0 or f '(c)
undefined]
Determine intervals for evaluation of f ' and begin the interval
table:
a) Locate the critical numbers along a number line
containing the domain of the function.
b) Determine the intervals, using the critical
numbers as endpoints.
Continue the interval table by: a) Selecting a test value for each interval.
b) Express f '(x) in factored form, and write each
factor in the first column.
c) Find the sign of each factor in each interval
and indicate the sign on the table.
d) For each interval, find the sign of f '(x) by determining the
number of negative factors.
Determine whether f(x), the original function, is increasing (when f '(x) >0) or decreasing
(when f '(x) <0) on each interval.
The critical value for which f(x) is increasing to the left and
decreasing to the right is a relative max. / \
The critical value for which f(x) is decreasing to the left and increasing
to the right is a relative min. \ /
Find the corresponding f or y value for each critical value determined
to be a relative max or min, and write the ordered pair (c,f(c)).
Find critical numbers [f '(c) = 0 or f '(c)
undefined]
Find f ''(x).
Find f ''(c) for all critical numbers.
Determine the relative extrema using the Second Derivative Test:
a) If f ''(c) > 0, then f is concave up and f(c) is a relative min
b) If f ''(c) < 0, then f is concave down and f(c) is a relative
max
c) If f ''(c) = 0, then the test fails. (consider an Inflection
Point - a point where concavity changes)
If degree of numerator < degree of denominator, then limit is 0.
If degree of numerator = degree of denominator, then limit is ratio of
leading coefficients.
If degree of numerator > degree of denominator, then limit does not
exist.
For the first 2 cases, where a limit k exists, then y = k is a
horizontal asymptote. Be sure to consider the limit approaching both
positive infinity and negative infinity.
1. Read the problem carefully, and identify what's given and
what you need to find. 2. Organize the info: draw a diagram, construct
a table, etc. 3. Identify the unknown variables; add to diagram or
table. 4. Write an equation to relate the given and the to find.
5. Reduce the number of variables to 2. 6. Find the derivative
and Critical Numbers. 7. Test the critical numbers for max or min,
using 1st derivative or 2nd derivative test, and state solution. 8.
Check the solution: Is "to find" found? Does solution make sense? Do numbers
fit?
If the curves are close and orientation is difficult to determine,
substitute values between those of the points of intersection to
determine which is above (or to the right of) the other.
Use the sketch to determine which integral to use:
If each curve passes the vertical line test in the
bounded region, use vertical rectangles, the x variable, and the
integral:
If a curve fails the vertical line test but passes
the horizontal line test in the bounded region, use horizontal
rectangles, the variable y, and the integral:
If the bounded area contains more than one distinct
region, write the area as the sum of the areas of each distinct region.
Limits of integration:
Use the coordinates of the points of intersection.
If x = k_{1} or y = k_{2} is given this
may be one of the limits.
Sketch the curves and identify the region, using
the points of intersection.
Locate the axis of revolution on the sketch.
Decide whether to use a horizontal or vertical
rectangle. The rectangle should be perpendicular to the axis of
revolution.
Sketch the rectangle and determine the variable
of integration. a) If the rectangle is horizontal, then integrate with respect to y (use dy).
The integrand must be in terms of y.
b) If the rectangle is vertical, then integrate with respect to x (use dx).
The integrand must be in terms of x.
Determine the integrand: R^{2},
or R^{2} - r^{2} ?
a) If the rectangle
touches the axis of revolution, identify R as the length of the
rectangle. Find R in terms of the appropriate variable (see above),
and use R^{2}as the integrand.
b) If the rectangle does
not touch the axis of revolution, identify R as the distance of the
furthest end of the rectangle from the axis of revolution and r as the
distance of the closest end of the rectangle from the axis of
revolution. Use R^{2} - r^{2} as
the integrand.^{ }
Sketch the curves and identify the region, using
the points of intersection.
Locate the axis of revolution on the sketch.
Decide whether to use a horizontal or vertical
rectangle. Select the orientation that requires the least number of
separate sections.
Decide whether to use the Disc Method or the
Shell Method: a) If the rectange is perpendicular to the
axis of revolution, use the Disc Method.
b) If the rectangle is parallel to the axis of
revolution, use the Shell Method.
Complete Steps 1 to 4 in Volumes of
Revolution, which Method? noted above.
Be sure that your rectangle is parallel to the
axis of revolution.
Determine the variable of integration: a) If the rectangle is horizontal, then integrate with respect to y (use dy).
The integrand must be in terms of y.
b) If the rectangle is vertical, then integrate with respect to x (use dx).
The integrand must be in terms of x.
Determine the integrand: p(x)h(x) or p(y)h(y) ?
a) If the rectangle
is horizontal, identify p(y), the distance of the rectangle from the
axis of revolution, and h(y), the length of the rectangle. Use:
b) If the rectangle
is vertical, identify p(x), the distance of the rectangle from the
axis of revolution, and h(x), the length of the rectangle. Use: ^{ }