Lecture Notes, Final Review Sheet - Formulas PDF

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M119 Final review sheet This is NOT a complete list of formulas and concepts, but knowing these formulas and concepts is a great start to your studying! I would suggest trying to make your own formula sheet first and then using this to make sure you didn’t miss anything. Change is just subtraction while average rate of change is the slope formula: f (b ) ! f (a ) =m b!a This slope is the slope of the secant line. If we want the slope of the tangent line, we need the instantaneous rate of change or derivative. Words that will hint that you want the derivative are: slope (of the tangent line), rate (without percent or interest which is r), velocity, or changing. The unit for the derivative will always have the word per as m you need to divide. (e.g. miles per hour ) h For the second derivative, words like acceleration, concavity, or the use of two words from above (thus implying the derivative is taken twice) are a good thing to look for. The unit will also have the word per, but it will usually be per something squared since there m are to derivatives (e.g. miles per hour squared 2 ) h The chart below is critical to know: f(x)

f'(x)

f"(x)

means

positive

??

??

above the x-axis

negative

??

??

below the x-axis

zero

??

??

x-intercept

increasing

positive

??

increasing

decreasing

negative

??

decreasing

horizontal

zero

??

critical point

concave up

increasing

positive

concave up

concave down neither concavity

decreasing not changing

negative

concave down

zero

possible inflection point

If you see a chart remember that if you are asked to model it, it is probably either linear or exponential. Linear functions have a constant difference (subtraction) and can be represented by: y=mx+b Exponential functions have a constant ratio a and can either be represented as: y = Pa t = P(1 + r ) t or by y = Pe rt .

Power functions must pass through the origin and will fit the form y = kx p Since the r’s are slightly different (effective and nominal respectively), I would suggest always using the first unless you see the word continuous. r Variants of this include the formula for n statements a year: y = P(1 + ) nt (monthly n means n=12).

The log rules are also very important when working with exponential functions: 1) ln (AB) = ln A + ln B 2) ln (A/B) = ln A – ln B p 3) ln A = p ln A

4) log b A =

ln A (change of base) ln b

If you see the word proportional, you need to have a k. Remember that proportion or directly proportional means to multiply after the k, while inversely proportional tells you to divide after the k. For example: Y is inversely proportional to the square root of x 1 k = means Y = k * x x This relates to power functions which are just individual terms of a polynomial. They can always be written in the form Y = k * x p . Note that power functions are different than exponentials because the x is in the base for power and in the exponent (and hence the name) for exponentials. Local linear approximation is using the tangent line to guess what will happen near a point. Hence the formulas:

f ( x + ! x) " f ( x) + f ' ( x) * ! x is really the same as y = y1 + m * ( x ! x1 ) which is just the form of the tangent line: y ! y1 = m * ( x ! x1 ) in disguise. Notice that the m and the f’(x) are really the same once you plug in a value for x. From economics, we have:

" ( q ) = R ( q ) ! C (q ) and R( q) = pq Marginal reminds us that we need m or slope so:

MC = C ' (q ) MC = MR . MR = R ' (q ) which gives us that profit is maximized when either M! = 0 M! = ! ' (q ) If MR >MC, then you still profit from making more, and hence you should increase production. Likewise, if MR...