CALC Homework:
Battaly notes. If you want more
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AMSU
How to Do Word Problems:
- Read the problem.
- Read the problem again, identifying what's given and what you
need to find.
- Organize the information: draw a diagram, construct a table,
etc.
- Identify the unknown variables, including the appropriate rates,
such as dy/dt or dA/dt, etc.
- Write an equation to relate the given and the to find.
- Solve the equation. For related rates, this involves
implicit differentiation.
- Check the solution: Is "to find" found?
Does solution make sense?
Do numbers fit?
Integrating
Rational Expressions:
- Is the denominator a monomial?
If yes, divide and integrate.
If no, proceed to #2.
- If the quotient is rewritten as a product, is the exponent (-1)?
If yes, think ln
and look for du/u
If no, think un
and look for un du
To Find
Absolute Extrema on a Closed Interval [a,b]:
- 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
To Find
Relative Extrema of a continuous function using intervals and the First Derivative Test:
- 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)).
To Find
Relative Extrema of a continuous function using Concavity and the Second Derivative Test:
- 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)
To Find the Limit at
Infinity for a Rational Function, f(x) = g(x)/h(x):
- 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.
Using
the Limit Definition to Find Area:
- Sketch the function f and indicate the region on the interval [a,b].
- Is f continuous and nonnegative on the interval? If yes, then:
5. Substitute ci for x in the function
and proceed with the sums, using Summation Formulas, Theorum 4.2.
6. Evaluate the limits at infinity.
Curve Sketching:
DRIve
A Car
NEXt
TRIP
D:
Domain, R:
Range, I:
Intercepts, A:
Asymptotes,
CN: Critical
Numbers, EXTR:
Extrema, IP:
Inflection Points
Finding
the Area of the Region Bounded by 2 or More Curves:
- Sketch the curves:
- Find the points of intersection of the curves.
- 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 = k1 or y = k2 is given this
may be one of the limits.
Volumes of Revolution - Disk Method
- 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: R2,
or R2 - r2 ?
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 R2 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 R2 - r2 as
the integrand.
Webmaster: G. Battaly, merlin@pipeline.com
Last updated: September 25, 2001