FormulaSheetExam - Formula Sheet for Final Exam Hi class...

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Formula Sheet for Final Exam Hi class, Here are some formulas that will be handy to know for the exam. As always, you’ve seen these formulas before, but here they all are in the same place. This sheet consists of the other three tossed together plus the later material. Also be sure to check out the resources on the main 1M03 page. J. Compound Interest: A = Pe rt Here P is the amount of money that you start with, r is the interest rate, t is time, and A is the amount of money that you have after time t. Notice that your book has a formula for “present value,” which is just P = Ae rt . This is the same formula as above; just divide both sides by e rt . Exponential Growth: The formula for exponential growth is Q ( t ) = Q 0 e kt where Q 0 is the initial amount that you start with. Exponential Decay: The formula for exponential decay is: Q ( t ) = Q 0 e kt So the formulas are the same, but there is just a minus sign in this one. Half-Life: Usually in the case of exponential decay, one refers to the half- life, which is the time it takes for half the material to decay. If the half life is h, then the k in the above formula is given by k = ln 2 h Notice that there is a similar formula for “doubling time,” in the case of expo- nential growth. Here, if the money doubles in time d, then k = ln 2 d . 1
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Derivatives: Here are the derivatives of the functions that you should know: Leibniz notation: d dx ( e x ) = e x d dx (ln x ) = 1 x d dx ( cx n ) = cnx n 1 d dx ( b x ) = b x ln b d dx (log b x ) = 1 x ln b d dx ( c ) = 0 Here c is a constant, and n is an integer. Notice that the formulas with b in them reduce to the formulas that you know when b is the natural base, i.e. when b = e. So, d dx ( e x ) = e x ln e = e x , since ln e = 1 . Here are the exact same formulas, except in the prime notation that many of you prefer: Prime notation: ( e x ) = e x (ln x ) = 1 x ( cx n ) = cnx n 1 ( b x ) = b x ln b (log b x ) = 1 x ln b ( c ) = 0 2
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Derivative Rules: Here are the rules for derivatives that you should know: Leibniz notation: Sum / DiFerence: d dx ( f ± g ) = df dx ± dg dx Pull out constants: d dx ( cf ) = c df dx Product rule: d dx ( fg ) = df dx g + f dg dx Quotient rule: d dx p f g P = df dx g f dg dx g 2 Chain Rule: df dx = df du du dx Here c is a constant, and f and g are functions. If you have trouble remembering the quotient rule, remember that you can just apply the product rule to fg 1 . (where the
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FormulaSheetExam - Formula Sheet for Final Exam Hi class...

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