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

1. Product rule
2. Quotient rule
3. Chain rule
1. applies to function compositions: d/dx [f (g(x))] = f ′ (g(x)) g′ (x)
2. some specific applications of the chain rule:
1. Generalized power rule: d/dx [g(x)] n = n[g(x)] n-1g′ (x)
2. Derivatives involving exponentials: d/dx e g(x) = g′ (x) e g(x)
3. Derivatives involving natural logs: d/dx ln(g(x)) = g′ (x) / g(x)
4. Implicit differentiation
1. 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

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

Exponentials and Logarithms

1. exponentials with base e
1. bx may always be written as ekx for an appropriate value of k
2. f(x) = Cekx has property that rate of change is proportional to size (follows from the facts d/dx ex = ex and chain rule)
3. is invertible
2. the natural log function
1. ln ex = x
2. e ln x = x
3. ln 1 = 0
2. d/dx b x = (ln b) b x
3. d/dx ln x = x -1
3. Be able to solve equations involving exponentials, logarithms (like those found in Problems 13-32, p. 256)
4. Be able to determine an appropriate exponential function of the form Cekt 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

1. Exponential growth/decay
1. corresponds to geometric growth/decay
2. equivalent statements
1. The function P grows/decays exponentially.
2. P(t) = Cekt
3. P has a rate of change proportional to its size.
4. P′(t) = kP(t)
2. Compound interest
1. Compound interest formula (“discrete” compounding): A(t) = P(1 + r/n) nt
1. P is initial amount invested (principle) in dollars
2. n is number of compoundings per year
3. r is annual interest rate (as a decimal, not as a percentage)
4. t is number of years
2. Continuous compounding
1. relationship to discrete compounding: the result of the ratio r/n going to zero
2. results in exponential formula: A(t) = Pert
3. often closely approximates discrete compounding (does better and better as ratio r/n shrinks)
3. Elasticity of demand
1. use demand function in altered form: q = f(p) (quantity sold is output for input price)
2. definition: E(p) = -pf ′ (p) / f(p)
1. Note the use of a relative rate of change here.
3. 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