Math 132A/B
Calculus for Management,
Life and Social Sciences
Spring, 2006
Study Guide for Exam 2
Exam 2 will focus on material from Chapters 25, ending with
Section 5.1. Of course, we have continued in these chapters
to call upon knowledge that came earlier.
Differentiation Rules
 Product rule
 Quotient rule
 Chain rule
 applies to function compositions:
d/dx f(g(x)) = f '(g(x))g '(x)
 some specific applications of the chain rule:
 Generalized power rule:
d/dx [g(x)]^{ n} = n[g(x)]^{ n1}g '(x)
 Derivatives involving exponentials:
d/dx e^{g(x)} = g '(x) e^{g(x)}
 Derivatives involving natural logs:
d/dx ln(g(x)) = g '(x)/g(x)
 Implicit differentiation
use when dependent variable is not explicitly
written as a function of the independent variable
(like when an equation involving y as
dependent variable and x as independent
is not explicitly solved for y; or
like in typical relatedrate problems when
the independent variable time doesn't
even appear in the equation)
Applications of derivatives

Curve sketching
 describing increasing/decreasing behavior, relative extrema
 Main fact: If f '(x) is positive (negative) for
a < x < b, then f increases (decreases)
on that interval
 finding extrema
 find places where f ' is equal to zero
 test to see that f ' changes sign as x values
move by these numbers
 describing concavity, inflection points
 Main fact: If f ''(x) is positive (negative) for
a < x < b, then f is concave up (down)
on that interval
 finding inflection points
 find places where f '' is equal to zero
 test to see that f '' changes sign as x values
move by these numbers
 Be able to take information like that you gather
in determining extrema, inflection points and sketch
a reasonable graph.

Related rates
 set things up (may already be done for you)
 define appropriate variables
 shouldn't give a variable name to constant values
 relevant quantities are those for which a rate of
change is either provided or desired
 be specific about what the variable represents
and its units of measure
 determine an equation relating your variables
may need an auxilliary equation(s) to eliminate
variables which are not relevant (see above)
 differentiate (usually with respect to time;
implicit differentiation almost always necessary)
 plug in known values and solve for desired rate
 express answer with appropriate units

Optimization
 set things up
 define appropriate variables
 determine which variable is to be optimized (dependent variable)
 determine function for dependent variable
 must involve just one independent variable
(if more than one, find an auxilliary equation(s)
which may be used to eliminate the extra variable(s))
 domain of the function you've written?
 some specific scenarios/quantities which we have encountered:
economic order quantity, economic lot size, revenue,
demand (often linear in our problems),
carrying costs, profit
 differentiate your function
 find critical numbers (places where derivative is zero)
 check value of the function at critical numbers and endpoints
of the domain (unless you can provide reasons why you don't need
to do the latter) to determine which one optimizes (maximizes
or minimizes as appropriate for the problem) the dependent
variable
 give answer(s) with appropriate units
Exponentials and Logarithms
 exponentials with base e
 b^{x} may always be written as
e^{kx} for an appropriate value of
k
 f(x) = Ce^{kx} has property that rate
of change is proportional to size (follows from the
facts d/dx e^{x} = e^{x}
and chain rule)
 is invertible
 the natural log function
 is the inverse of e^{x} (All we
know about this function follows from this)
 ln e^{x} = x
 e^{ln x} = x
 ln 1 = 0
 d/dx b^{ x} = (ln b) b^{ x}
 d/dx ln x = x^{ 1}
 Be able to solve equations involving exponentials,
logarithms (like those found in Problems 1332, p. 256)
 Be able to determine an appropriate exponential function
of the form Ce^{kt} to fit scenarios
in which the rate of change is proportional to the size
of the function (a la Section 5.1)
In addition
Be able to use the quadratic formula adeptly.
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This page maintained by:
Thomas L. Scofield
Department of Mathematics and Statistics,
Calvin College
Last Modified:
Thursday, 16Feb2006 21:43:09 EST