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# Review Sheet for MATH 250A at UA

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KARMA
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Date Created: 02/06/15
MATH 250a Fall Semester 2007 Section 2 J M Cushing Tuesday August 28 httpmathariz0naeducushing250ahtml Example Deposit PO in a bank account at r interest compounded annually How many years does it take for the deposit to double Solve for t 1r 2 This requires the inverse of the exponential function that is requires the logarithm function DEFINITIONS a A function is oneone invertible if each element in the range corresponds to only one element in the domain DEFINITIONS a A function is oneone invertible if each element in the range corresponds to only one element in the domain b For a oneone function the inverse function rnaps each range elernent back to its domain elernent DEFINITIONS a A function is oneone invertible if each element in the range corresponds to only one element in the domain b For a oneone function the inverse function rnaps each range elernent back to its domain elernent Graphically a oneone invertible function has a monotone graph either increasing or decreasing y yfx DEFINITIONS a A function is oneone invertible if each element in the range corresponds to only one element in the domain b For a oneone function the inverse function rnaps each range elernent back to its domain elernent Graphically a oneone invertible function has a monotone graph either increasing or decreasing y inverse function DEFINITIONS a A function is oneone invertible if each element in the range corresponds to only one element in the domain b For a oneone function the inverse function rnaps each range elernent back to its domain elernent We denote the inverse function of a oneone function y f x by y f 1x DEFINITIONS a A function is oneone invertible if each element in the range corresponds to only one element in the domain b For a oneone function the inverse function rnaps each range elernent back to its domain elernent We denote the inverse function of a oneone function y f x by y f 1x NOTE By definition ff 1xx DEFINITIONS a A function is oneone invertible if each element in the range corresponds to only one element in the domain b For a oneone function the inverse function rnaps each range elernent back to its domain elernent We denote the inverse function of a oneone function y f x by y f 1x NOTE By definition ff 1xx and f 1fxx DEFINITIONS a A function is oneone invertible if each element in the range corresponds to only one element in the domain b For a oneone function the inverse function rnaps each range elernent back to its domain elernent Graphically a oneone invertible function has a monotone graph either increasing or decreasing y yfx DEFINITIONS a A function is oneone invertible if each element in the range corresponds to only one element in the domain b For a oneone function the inverse function rnaps each range elernent back to its domain elernent Graphically a oneone invertible function has a monotone graph either increasing or decreasing y 3235 to get the graph of the inverse re ect through the line y x DEFINITIONS a A function is oneone invertible if each element in the range corresponds to only one element in the domain b For a oneone function the inverse function rnaps each range elernent back to its domain elernent Graphically a oneone invertible function has a monotone graph either increasing or decreasing y 3235 I DEFINITIONS a A function is oneone invertible if each element in the range corresponds to only one element in the domain b For a oneone function the inverse function rnaps each range elernent back to its domain elernent Graphically a oneone invertible function has a monotone graph either increasing or decreasing y yIf IOC Ptat agt0 is oneone invertible provided a i 1 Piat agt0 is oneone invertible provided a i 1 DEFINITION The inverse of Pl 2 at 0 lt a 1 is the logarithmic function with base a and is denoted by P1t10gat Piat agt0 is oneone invertible provided a i 1 DEFINITION The inverse of Pl 2 at 0 lt a 1 is the logarithmic function with base a and is denoted by P1t10gat NOTE PltP1xgt x means alogax x Piat agt0 is oneone invertible provided a i 1 DEFINITION The inverse of Pl 2 at 0 lt a 1 is the logarithmic function with base a and is denoted by P1l10gat NOTE P 13 1 x means alogax x P1ltPx x means 10galtaxgt x Review of Basic Algebraic Properties of Logarithmic Functions logts logt l log 5 Review of Basic Algebraic Properties of Logarithmic Functions logts log t log S log log t log S s Review of Basic Algebraic Properties of Logarithmic Functions logts logt i logs log logt logs s 10 ts 2510 t g g loglzo Review of Basic Algebraic Properties of Logarithmic Functions logts logt i logs log logt logs s 10 ts 2510 t g g loglzo common mistakes logt s i log tlog s logt s i log t log 5 Example Deposit PO in a bank account at r interest compounded annually How many years does it take for the deposit to double Solve for t lrgtt2 Example Deposit PO in a bank account at r interest compounded annually How many years does it take for the deposit to double Leta l r and solve for t at2 Example Deposit PO in a bank account at r interest compounded annually How many years does it take for the deposit to double Letazl l r and solve for t at2 Take loga of both sides Example Deposit PO in a bank account at r interest compounded annually How many years does it take for the deposit to double Leta l r and solve for t at 2 Take loga of both sides logaat loga 2 Example Deposit PO in a bank account at r interest compounded annually How many years does it take for the deposit to double Leta l r and solve for t at 2 Take loga of both sides logaltatgt loga 2 tloga a loga 2 Example Deposit PO in a bank account at r interest compounded annually How many years does it take for the deposit to double Leta l r and solve for t at 2 Take loga of both sides logaat loga 2 tloga a loga 2 t loga 2 Example Deposit PO in a bank account at r interest compounded annually How many years does it take for the deposit to double Answer t loga 2 a l r Example Deposit PO in a bank account at r interest compounded annually How many years does it take for the deposit to double Answer t loga 2 a l r For example if r 005 then r log105 2 years Example Deposit PO in a bank account at r interest compounded annually How many years does it take for the deposit to double Answer t loga 2 a l r For example if r 005 then t log105 2 s P Example Deposit PO in a bank account at r interest compounded annually How many years does it take for the deposit to double Answer t loga 2 a l r For example if r 005 then t log105 2 s For example if r 007 then tlog107 2 22 Example Deposit PO in a bank account at r interest compounded annually How many years does it take for the deposit to double Answer t loga 2 a l r For example if r 005 then t log105 2 s For example if r 007 then tlog107 2 22 Not convenient to use a different base for each problem Example Deposit PO in a bank account at r interest compounded annually How many years does it take for the deposit to double Answer t loga 2 a l r For example if r 005 then t log105 2 s For example if r 007 then tlog107 2 22 Not convenient to use a different base for each problem Most commonly used bases are a 10 and e Example Deposit P0 in a bank account at r interest compounded annually How many years does it take for the deposit to double 7 Answer I loga 2 a lr For example if r 005 then I logL05 2 m 7 For example if r 007 then I log1072 m 7 Not convenient to use a different base for each problem I Most commonly used bases are a 10 and e 9 1 Solve for t but using logs of base 10 at2 Where a1r Solve for t but using logs of base 10 2 where a1r If we want to work in base 10 we rewrite this exponential function to base 10 Solve for t but using logs of base 10 at 2 Where a 1 l r a 10quot Solve for t but using logs of base 10 at 2 Where a 1 l r a 10quot Recall by definition of logs a 101 gwa Solve for t but using logs of base 10 at 2 Where a 1 l r a 10quot Recall b definition of logs a 101 gwa Solve for t but using logs of base 10 at 2 Where a 1 l r a 10quot Recall by definition of logs a lologloa at lologloa Solve for t but using logs of base 10 at 2 Where a 1 l r a 10quot Recall by definition of logs a lologloa at 101 gw t at 1010g10 a Solve for t but using logs of base 10 2 Where a1r 6110 Reca by definition of logs lologlo a I Ologlo a Solve for t 1010g10at 2 Where a H r Solve for t 1010g10at 2 Where a H r Take 10g10 of both sides 10g1010lt1 gwquotgtt 10g10 2 Solve for t 1010g10at 2 Where a H r Take 10g10 of both sides log10 100 g10 0t 10g10 2 10g10 at 10g1010 10g10 2 Solve for t 1010g10at 2 Where a H r Take 10g10 of both sides 10g1010lt1 gwquotgtt 10g10 2 10g10 at10g10 2 Solve for t 1010g10at 2 Where a H r Take 10g10 of both sides 10g1010lt1 gwquotgtt 10g10 2 10g10 at10g10 2 t 10g10 2 loglo a Example Deposit PO in a bank account at r interest compounded annually How many years does it take for the deposit to double tloga2 azl l r or in terms of base 10 t log10 2 10g10 a Example Deposit PO in a bank account at r interest compounded annually How many years does it take for the deposit to double tloga2 azl l r or in terms of base 10 t log10 2 log10 a Similarly one calculates answer using base 6 1 2 l oge loge a Example Deposit PO in a bank account at r interest compounded annually How many years does it take for the deposit to double tloga2 azl l r or in terms of base 10 t log10 2 log10 a Similarly one calculates answer using base 6 1 2 l oge loge a logt log10 t common log Notation convention lnt loge t natural log Example Deposit PO in a bank account at r interest compounded annually How many years does it take for the deposit to double For example for r 005 ln2 t lnl05 m 1421 years Example Deposit PO in a bank account at r interest compounded annually How many years does it take for the deposit to double For example for r 005 ln2 t lnl05 m 1421 years Next goal The geometry graphs of exponential and log functions A Quick Review of Derivatives Derivative of f x at x a 2 11mm dx IHO h A Quick Review of Derivatives Derivative of f x at x a g hm fa hgt fa dx xza h gt0 h 1 df instantaneous rate of change of f x dx xza with respectx to atx a A Quick Review of Derivatives Derivative of f x at x a g hm fa hgt fa dx xza h gt0 h 1 df instantaneous rate of change of f x dx xza with respectx to atx a a f 2 Geometrlcall y dx slope of tangent toy f x at point x y a fa A Quick Review of Derivatives Derivative of f x at x a g hm fa h fa dx xza h gt0 h 3 gtOzgtgraphofyfxatxa dx xza ls increasing at x a A Quick Review of Derivatives Derivative of f x at x a g hm fa h fa dx xza h gt0 h 3 gtOzgtgraphofyfxatxa dx xza ls increasing at x a f graphofyfx atxa lt gt xza ls decreasing at x a A Quick Review of Derivatives Derivative of f x at x a g hm fa h fa dx xza h gt0 h 3 gtOzgtgraphofyfxatxa dx xza ls increasing at x a f graphofyfx atxa lt gt xza ls decreasing at x a Second derivative of f x 1 dzf dx dx cix2 A Quick Review of Derivatives Derivative of f x at x a g hm fa h fa dx xza h gt0 h 3 gtOzgtgraphofyfxatxa dx xza ls increasing at x a f graphofyfx atxa lt gt xza ls decreasing at x a A Quick Review of Derivatives Derivative of f x at x a g hmfah fa dxxza h gt0 h ra hof x atxa 3 gtOgtep y f dx xza ls increasing at x a h f txa f lt grap 0 y fxa xza ls decreasing at x a 2 ra hof x atxa moi gtOgtep y f dx xza ls concave up atxa A Quick Review of Derivatives Derivative of f x at x a g hmfah fa dxxza h gt0 h d ra hof x atxa 31 gtOgte p y f dx xza ls increasing at x a ra hof x atxa f lt g p y f xza ls decreasing at x a d2 rahof x atxa 4 gt0gte p y f dx xza ls concave up atxa d2 rahof x atxa lt gt g p y f dx xza ls concave down at x a A Quick Review of Derivatives Some Fundamental Derivatives dc dx O for any constant c A Quick Review of Derivatives Some Fundamental Derivatives dc dx dxm mxm 1 power rule dx O for any constant c A Quick Review of Derivatives Some Fundamental Derivatives d C O for any constant 0 dx 0 m x mxm 1 power rule dx dex A Quick Review of Derivatives Some Fundamental Derivatives d C O for any constant 0 dx 0 m x mxm 1 power rule dx dex x e dx A Quick Review of Derivatives Some Fundamental Derivatives dc dx O for any constant c mxm 1 power rule A Quick Review of Derivatives Some Fundamental Derivatives d C O for any constant 0 dx 0 m x mxm 1 power rule dx dex x 6 dx A Quick Review of Derivatives Some Fundamental Rules ltPQgt A Quick Review of Derivatives Some Fundamental Rules 1 dP dQ dxltPQgt dx dx A Quick Review of Derivatives Some Fundamental Rules 1 dPdQ ww wm 012 A Quick Review of Derivatives Some Fundamental Rules P iltpQgtddQ dx dx dx 0 dQ dP Product Rule P P dxlt Q dx Qa x A Quick Review of Derivatives Some Fundamental Rules P iltpQgtddQ dx dx dx 0 dQ dP Product Rule P F dxlt Q dx Qa x dP d dP Q 2 05 05 Quotlent Rule Q2 A Quick Review of Derivatives Some Fundamental Rules d P ltpQgtddQ dx dx dx 0 dQ dP Product Rule P P dxlt Q a x Qa x dP dQ Q P d P 05 05 Quotient Rule 61x Q Q2 d EHQ A Quick Review of Derivatives Some Fundamental Rules 1 dP dQ ww wm i dQ dP Product Rule dx Pg P dx Q dx dP dQ p d P Q 05 05 Quotient Rule 61x Q Q2 iPQ dPQ d Q Chain Rule dx dQ dx EXAMPLE f x e r constant A composite function f x P Q00 PQ 6Q Q06 m EXAMPLE f x e r constant A composite function f x P Q00 PQ 69 Q06 3906 Use Chain Rule dPQx dP dQ dx dQ dx EXAMPLE f x e r constant A composite function f x P Q00 PQ 6Q Q06 m Use Chain Rule dPQx dP dQ dx dQ dx EXAMPLE f x e r constant A composite function f x P Q00 PQ 6Q Q06 m Use Chain Rule dPQx dP dQ dx dQ dx 6 EXAMPLE f x e r constant de x 2 re dx EXAMPLE f x ax EXAMPLE f x ax fx elnax EXAMPLE f x ax fx elnax Use previous example with r Ina dax elnax Ina EXAMPLE f x ax fx elnax Use previous example with r Ina dax elnax Ina zaxlna

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