Study Guide for Exam 2

Exam 2 will focus on material from Chapters 2-5, ending with Section 5.1. 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
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)

Applications of derivatives

1. Curve sketching
   1. describing increasing/decreasing behavior, relative extrema
   1. Main fact: If f '(x) is positive (negative) for a < x < b, then f increases (decreases) on that interval
   2. finding extrema
   1. find places where f ' is equal to zero
   2. test to see that f ' changes sign as x values move by these numbers
   2. describing concavity, inflection points
   1. Main fact: If f ''(x) is positive (negative) for a < x < b, then f is concave up (down) on that interval
   2. finding inflection points
   1. find places where f '' is equal to zero
   2. test to see that f '' changes sign as x values move by these numbers
   3. Be able to take information like that you gather in determining extrema, inflection points and sketch a reasonable graph.
2. 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
   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
3. Optimization
   1. set things up
   1. define appropriate variables
   2. determine which variable is to be optimized (dependent variable)
   3. determine function for dependent variable
   1. 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))
   2. domain of the function you've written?
   3. some specific scenarios/quantities which we have encountered: economic order quantity, economic lot size, revenue, demand (often linear in our problems), carrying costs, profit
   2. differentiate your function
   3. find critical numbers (places where derivative is zero)
   4. 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
   5. give answer(s) with appropriate units

Exponentials and Logarithms

1. exponentials with base e
1. b^x may always be written as e^kx for an appropriate value of k
2. 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)
3. is invertible
2. the natural log function
1. is the inverse of e^x (All we know about this function follows from this)
1. ln e^x = 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 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