Math 132A
Calculus for Management,
Life and Social Sciences
Spring, 2010
Study Guide for Exam 2
Exam 2 will focus on material from Chapters 3-5, ending with
Section 5.3. 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)]^{ n-1}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 related-rate problems when
the independent variable time doesn't
even appear in the equation)
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
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 13-32, 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)
Applications of exponentials
- Exponential growth/decay
- corresponds to geometric growth/decay
- equivalent statements
- The function P grows/decays exponentially.
- P(t) = Ce^{kt}
- P has a rate of change proportional to its size.
- P′(t) = kP(t)
- Compound interest
- Compound interest formula (“discrete” compounding):
A(t) = P(1 + r/n)^{ nt}
- P is initial amount invested (principle) in dollars
- n is number of compoundings per year
- r is annual interest rate (as a decimal, not as
a percentage)
- t is number of years
- Continuous compounding
- relationship to discrete compounding:
the result of the ratio r/n going to zero
- results in exponential formula:
A(t) = Pe^{rt}
- often closely approximates discrete compounding (does
better and better as ratio r/n shrinks)
- Elasticity of demand
- use demand function in altered form: q = f(p)
(quantity sold is output for input price)
- definition: E(p) = -pf ′ (p) / f(p)
Note the use of a relative rate of change here.
- demand is elastic (revenue goes up as price goes down and
vice versa) when E(p) > 1, inelastic (revenue and
price increase/decrease together) if E(p) < 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:
Wednesday, 07-Apr-2010 23:19:14 EDT