CHEM3440Lec2F06
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CHEM3440Lec2F06

Course Number: CHEM 3440, Fall 2006

College/University: University of Guelph

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Statistics and Analytical Chemistry Random noise is part of any analytical measurement. Precise and Accurate answers demand a statistical analysis of the data. 1. How much analyte is present? 2. Is one test protocol better than another? 3. Are the results from this lab the same as from another? 4. Are the current results consistent with previous results? CHEM*3440 Chemical Instrumentation Topic 2 Statistics for...

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and Statistics Analytical Chemistry Random noise is part of any analytical measurement. Precise and Accurate answers demand a statistical analysis of the data. 1. How much analyte is present? 2. Is one test protocol better than another? 3. Are the results from this lab the same as from another? 4. Are the current results consistent with previous results? CHEM*3440 Chemical Instrumentation Topic 2 Statistics for Analytical Methods What You Should Know Calculate Mean Standard Deviation Work well with a spreadsheet program Find the slope and intercept of a linear least squares t to a collection of data points with a spreadsheet program What We Want To Learn Answer How much analyte is present? Calibration Curve Standard Addition Internal Standard Calculate correct condence limits. Understand when to use each experiment. Recognize when experimental data is awed due to instrument failure. Linear Least Squares Fits What is meant by linear? What is meant by least squares? How do we determine the goodness-of-t? What is the variance-covariance matrix? How do we derive error limits or condence intervals for interpolated data? What are the dangers of extrapolation? Linear Several meanings 1. Polynomial where the highest power of the independent variable is 1. 2. In a multivariable function, each term depends on only one of the variables. 3. In a multiparameter function, each term depends on only one of the parameters In our case, it is the last description that is applicable. Linear cont 1 Typical equation is y = mx + b This is linear in polynomial power, variable, and parameters. Can also perform Linear Least Squares analysis on y = ax3 + bx2 + cx + d It is still linear in variables and parameters. Cannot perform Linear Least Squares on y = ae-bx since its is not linear in parameters. Least Squares Given a set of data points, nd equation for linear model (always linear in parameters, often linear in polynomial power) that minimizes the error between the points and the line. 20 15 10 5 0 0 3.75 7.50 11.25 15.00 Least Squares cont 1 Should minimize error in both x and y. Demands iterative solution. Instead, we assume error in x is negligible and ascribe the error to the dependent variable y. Matrix Formulation of Least Squares Measured data come in a set of ordered pairs (xi, yi). Fit the data to a selected model. Can t to any linear function, but most common to t to a straight line: y = mx + b. Assume error is all in y (the dependent variable). For each xi there is a measured value, yi, and a t value yi,t = mxi + b. The difference between the measured yi and the tted yi,t is the error for the point, ei. Strategy 1: Choose m and b so that the sum of the ei"s is 0. (Won"t work.) Strategy 2: Choose m and b so that the sum of squares of errors, ei2, is a minimum. (Works!) Write out the expression for the sum of the square of the errors, differentiate with respect to m and b, set derivatives to 0, leads to the following matrix equation: Error in y. Error in x and y. A f i i 2 i p= y i /y n /x b / x / x p dmn = e / x y o i i Matrix cont 1 Task is to nd the parameter vector p. Invert the matrix and multiply both sides. Variance of the Fit How well does the data follow the model equation (a straight line)? The sum of the squares of the errors divided by the degrees of freedom. b 1 / x 2 - / x i / yi i d n = p = A- 1 y = D e oe m / x i yi o - / xi n w h e r e D = d e t (A) = n / x 2 i - _ / xi i 2 s2fit = / ^y - y i o i , fit h2 = / ^ y - mx - b h i i 2 n-2 Form the sums, multiply out the two equations to obtain the two parameters for the straight line, b (the y-intercept) and m (the slope). Degrees of freedom are the number of data points minus the number of parameters determined (in this case, 2). ! = n - p R2, the correlation coefcient is NOT a meaningful goodness-of-t measure. Use variance of t as evaluation. An excellent article presents this material: C. Salter, J. Chem. Ed. 77, 1239-43 (2000) Variance-Covariance Matrix Answers what is the variance in the t parameters (m and b). Quanties the correlation between the parameters. Functions of the Fit Parameters Usually, determining m and b is just the start of an analysis, not the end. Different experiments will use m and b differently to derive a property of interest. Some function F(b,m) will provide the desired result (often the concentration of an unknown). Complete analysis demands that one not only provide the unknown concentration, but one must also assign condence limits to the result. Common mistake: We have the errors in b and m from the diagonal elements of the variance-covariance matrix. The function F uses b and m. Well-known error propagation rules can give a variance to the result. But this ignores covariance, which can either over- or underestimate the true error. s2fit / x2 - / xi i V = s2fit A- 1 = D e o - / xi n Diagonal elements are the variances of the parameters. Off-diagonal elements are the covariance between the parameters. s2 = V11 = b s2fit / x2 i D s2fit n s2 = V22 = D m Errors in Functions of the Fit Parameters Correct error calculation. The variance in the function F(b,m) is given by Example: Interpolation x(y) Calibration curve y(x). x could be analyte concentration, y detector response. Consider case of the curve being a rst-order polynomial. For a selected ysel value, the corresponding interpolated x value, xint, is given by inverting the LSQ-t equation of the calibration curve. s / x - / xi o d F s2 = d T Vd F = D d T e F F F - / xi n 2 fit 2 i The elements of the vector dF are the partial differentials of F with respect to the t parameters. The vector dFT is just the transpose of dF. y = mx + b ( x int = y sel - b m 2b p dF = f 2F 2m This is the key equation in error analysis. 2F Find partial derivative vector for this function F(b,m) = (ysel - b)/m. y -b y se b 2 a selm k 2 a ml - m k 2F 1 = = =- m 2b 2b 2b y -b 2 a selm k ^ y sel - b h 2F = =m2 2m 2m J N 1 2F O 2b p = K - m ` dF = f K ^ y sel - b h O 2F KO m2 2m L P Example: Interpolation x(y) cont1 Find the variance in the interpolated result s2xint. Solve the matrix equation. Three Analytical Experiments The three most important analytical experiments are: 1. Calibration Curve Used when experimental conditions are well controlled, in a laboratory or factory environment where the solution containing the analyte is known and controlled. 2. Internal Standard Used for experiments where the detection volume and efciency is difcult to control and reproduce. Chromatography and mass spectrometry are common examples. 3. Standard Addition Used when sample matrix is complex with unknown interferences. Environmental samples are of this nature. Find the unknown concentration and determine the variance in the result so that condence limits can be assigned to the answer. s2x int J 1N s2fit y sel - b m / x2 - / xi K - m O 1 i T = d F Vd F = D c- m - m2 e o K ysel - b O - / xi n KO m2 P L Remember the rules for matrix multiplication. Solve to obtain s2fit n ^ y sel - b h2 y -b s2x = D m2 c - 2 selm / xi + / x2 m i m2 int Recall that the interpolated x-value is xint = (ysel -b)/m so that s2fit s2x = D m2 _ nx2nt - 2x int / xi + / x2 i i i int This expression includes the effect of correlation between the t parameters. Calibration Curve - Experiment Form a series of solutions of known concentration of the analyte. Choose a concentration range to include that expected for any unknown. Measure their response with a particular instrument. Each solution measured once or several times; overall there are n measurements to which a straight line will be t. Use LS algorithm to nd the parameters b and m of the line. Compare variance of t with that expected for this instrument. Graph the data to make sure it is linear throughout the range studied. Measure unknown several times (p times). Calibration Curve - Variance Slight variation to variance equation; measurement of y introduces additional source of error. Variance is square of error. Variance in xcc arising from the measurement of y is 2x 2 c CC m s2y, meas 2y The variance of the measured average is s2y, meas ^y = / - y h i 2 p-1 The derivative has the value 1/m. The expression for the overall variance in the result xcc and the solution is xCC ^ y - bh = m 2xC 2 T s2x = c 2yC m s2y, meas + dCC VdCC s2fit 1 1 2 = m2 ' p + D _ nxCC - 2xCC / xi + / x2 i 1 i CC Internal Standard - Experiment Internal standard needs to be chemically similar to analyte but distinguishable by analytical technique. Best if an atomic isomer. Make a series of solutions with a known concentration of both analyte and internal standard. Keep standard concentration relatively constant while varying analyte concentration throughout the expected unknown concentration range. Measure the response of both analyte and internal standard in each solution. Form a response curve of the ratio of analyte to internal standard response vs. analyte concentration. This least squares response curve is then treated identically to the calibration curve. Standard Addition - Experiment Standard is same molecule as analyte. Two common approaches: 1. Form a series of solutions with same amount of unknown but increasing amounts of standard added. Solutions then are all diluted to the same volume. (Constant volume experiment.) 2. A single unknown solution is measured and then increasing amounts of standard are added, the signal being measured for each addition. Each measurement has a different solution volume, but only one container needs to be used. (Variable volume experiment.) A good reference for this is M. Bader, J. Chem. Ed. 57, 703-706 (1980). Constant Volume n separate containers. Put volume Vx of unknown (whose as yet unknown concentration is Cx) in each. Add volume of standard Vs(2) whose concentration is Cs to second container (leave the rst container with only unknown in it so that Vs(1) = 0). Add a larger volume Vs(3) to third, etc. Dilute all solutions with solvent to an identical nal volume, Vf. Measure response Rn of each solution. Determine linear response, least squares t parameters b and m. 1. nx = VxCx 2. nx = VxCx 3. nx = VxCx 4. nx = VxCx etc. ns(1) = 0 ns(2) = Vs(2)Cs ns(3) = Vs(3)Cs ns(4) = Vs(4)Cs ntotal = nx ntotal = nx + ns(2) ntotal = nx + ns(3) ntotal = nx + ns(4) Canalyte(1) = nx/Vf Canalyte(2) = (nx + ns(2))/Vf Canalyte(3) = (nx + ns(3))/Vf Canalyte(4) = (nx + ns(4))/Vf Constant Volume cont1 Instrument responds linearly to analyte concentration. k represents the responsivity of instrument. nt n n Vx C Vs C R = kC analyte = k ; Votal E = k ;V x + V s E = k ; V x + V s E f f f f f kV C kV C kC kV C R = Vx x + Vs s = c V s m Vs + c Vx x m = mx + b = y f f f f kV C kC ` b = Vx x m= V s f f Solve two expressions for k, equate, and solve for unknown concentration Cx. bV f k =V C and x x bV f mV f `V C = C x x s bC s ` C x = mV x mV f k= C s Constant Volume cont2 What is the variance in this result? Find the partial differentials with respect to b and m, solve the matrix equation. Recall that x is the volume of added standard and y is the instrument response for that particular solution. Constant Volume cont 3 Expand this matrix equation to nd the variance in the extrapolated result. 2C x 2 c bC s m Cs = = mV 2b 2b mV x x 2C x 2 c bC s m bC s = = - m2 V 2m 2 m mV x x We know the variance of the t from the least squares routine that determines b and m. Recall the matrix equation for the variation of the result. s 2 x, ext J Cs N s2fit Cs / x2i - / xi oK mV x O bC s = D c mV - m 2 V m e K O x x - / xi n K- b2C s O L m Vx P J Cs N s2fit Cs / x2i - / xi oK mV x O bC s s = D c mV - m 2 V m e K O x x - / xi n K- b2C s O m Vx P L J Cs N bC s 2 K mV / x 2 + m 2 V / x i O i s fit C s bC s x x = D c mV - m 2 V m K O x x K - C s / x i + nbC s O 2 mV x m Vx P L s2fit C2 bC2 bC2 nb 2 C 2 s s s 2 = D ' m2 V2 / xi + m3 V2 / xi + m3 V2 / xi - m 4 V2s 1 2 x, ext x x x x s2fit C2 2bC2 nb 2 C 2 = m2 D 'V2s / x2 + mV2s / xi - m2 V2s 1 i x x x We now know the unknown concentration and the variance in that value, so an answer with condence limits can be given. Variable Volume This experiment only requires 1 container. Introduce volume Vx of unknown (with, as yet, unknown concentration Cx). Measure instrument response. Add a volume Vs(2) of standard of concentration Cs. Remeasure the response. Add more standard and remeasure. Variable Volume cont1 The response of the ith addition of standard is Vs (i) Cs Vx C x Ri = kC analyte (i) = k <V + V + V + V F x s (i) x s (i) Ri is dependent variable; Vs(i) is independent variable. Cannot rewrite this equation to linearize the variables. Non-linear problem. Can sort of linearize the problem. 1. nx = VxCx 2. nx = VxCx 3. nx = VxCx 4. nx = VxCx etc. ns(1) = 0 ns(2) = Vs(2)Cs ns(3) = Vs(3)Cs ns(4) = Vs(4)Cs ntotal = nx ntotal = nx + ns(2) ntotal = nx + ns(3) ntotal = nx + ns(4) Canalyte(1) = nx/Vx Canalyte(2) = (nx + ns(2))/(Vx + Vs(2)) Canalyte(3) = (nx + ns(3))/(Vx + Vs(3)) Canalyte(4) = (nx + ns(4))/(Vx + Vs(2)) Ri ^V x + Vs (i) h = k 6V x C x + Vs (i) Cs @ = kV x C x + kC s Vs (i) Plot Ri(Vx + Vs(i)) against Vs(i). Such a plot would give a straight line with intercept b = kVxCx and slope m = kCs. With y(i) = Ri(Vx + Vs(i)) and x(i) = Vs(i) a LSQ analysis gives a value for m and b. Solve for unknown concentration Cx. b = kV x C x b k =V C x x b m `V C = C x x s and and so that m = kC s m k=C s bC C x = mVs x Variable Volume cont2 Variance-Covariance matrix arises from the linear problem. Can use it as if it were linear. Variance for b and m are found from the diagonal elements. Can nd an approximate variance for the result; neglect covariance. 2 sC = s2 + s2 b m x Variable Volume cont 3 Vs (i) Cs Vs (i) Cs Vs (i) Cs Vx C x Vx C R i = k < V + V + V + V F , k ; V x + V E = k ;C x + V E x s (i) x s (i) x x x under the assumption that Vx >>Vs(i) Can provide an answer with condence limits. BUT, don"t forget the many approximations (errors?) made in reaching the result. or Vx C n n R i = k ; V x + V i E = k ;C x + V i E x x x Another approximate approach: perform experiment where the added standard volumes are very small compared to the unknown volume. A concentrated standard would allow small added volumes or, if possible, standard could be added as a solid. Problem becomes linear in these approximations. where in this case we are adding moles of material directly (presumably by weighing methods). You should be able to easily see how these two equations are linear and can be solved to nd the unknown concentration and its variance, accurate, again within our approximations used. You should be able to prove that the solutions for Cx are, respectively bC C x = mVs x or b C x = mV x Standard Deviation Variance is the squared error parameter which sums between separate sources of error which are normally distributed. The standard deviation is the square root of the variance and has the same units as the property being measured. 0.45 0.4 Confidence Interval Must decide upon a condence level (e.g. 95%) Know the standard deviation s. Number of measurements is important; more measurements makes us more condent of the result. Find the Student"s t-factor (degrees of freedom and condence level). Gaussian Curve Normal Error Curve 1 S.D. 68.3% 0.35 0.3 2 S.D. 95.5% 0.25 0.2 0.15 C.I. =! ts n 3 S.D. 99.7% 0.1 0.05 0 -4 -3 -2 -1 0 1 2 3 4 Students t-Table Degrees of Freedom 1 2 3 4 5 6 7 10 60 50% 1.000 0.816 0.765 0.741 0.727 0.718 0.711 0.700 0.679 0.674 90% 6.314 2.920 2.353 2.132 2.015 1.943 1.895 1.812 1.671 1.645 95% 12.706 4.303 3.182 2.776 2.571 2.447 2.365 2.228 2.000 1.960 99% 63.656 9.925 5.841 4.604 4.032 3.707 3.500 3.169 2.660 2.576 99.9% 636.578 31.598 12.924 8.610 6.869 5.959 Reporting a Result From a series of measurements, nd the average value, xaverage. Use the matrix formulation for errors to nd the variance in the result, while accounting for covariance between the t parameters. The square root of the variance (s2) is the standard deviation (s). Select a condence level (C.L.). Find the appropriate Student"s t-value, given the C.L. and the degrees of freedom (usually n-1). Calculate the condence interval. 5.408 4.587 3.460 3.291 Answer = xaverage ! ts n #

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University of Guelph - CHEM - 3440
Current ConventionCONVENTION: Electrical current ows from a region of positive potential energy to a region of more negative (or less positive) potential energy.CHEM*3440Chemical InstrumentationTopic 3Truth is that electron ow is electrical current.
University of Guelph - CHEM - 3440
SignalIn All experiments there is: Sample Response: the instrument's response when the analyte is present.CHEM*3440Output ResponseBlank Response: the instrument!s response when the analyte is absent. The Signal is the difference between the sample and
University of Guelph - CHEM - 3440
Circuit Diagram of 741 Op AmpCHEM*3440Chemical InstrumentationTopic 5Operational AmplifiersThanks to Tony van Roon in the Electronics Shop for the use of this picture.Schematic of Op AmpWhile op amp is complex in detail, there are really only three
University of Guelph - CHEM - 3440
Digital Data DomainAn analog signal is continuously varying in time. A digital signal consists of discrete samples of an analog signal. It is discretized in both coordinates (time and signal amplitude).CHEM*3440Signal LevelSample Spectrum 10 9 8 7 6 5
University of Guelph - CHEM - 3440
Spectrum of Electromagnetic RadiationElectromagnetic radiation is light. Different energy light interacts with different motions in molecules.CHEM*3440Chemical InstrumentationTopic 7Radiofrequency Microwave Infrare d UV-Visible Far UV - X-ray Gamma R
University of Guelph - CHEM - 3440
UV-Visible Electronic TransitionsThis technique is mainly a study of molecules and their electronic transitions. Molar absorptivity (&quot;) ranges from 0 to 105.CHEM*3440Chemical InstrumentationTopic 8Transitions with &quot; &lt; 103 are considered to be of low
University of Guelph - CHEM - 3440
Types of LuminescenceFluorescence Happens quickly after the absorption of the initial photon (s to ps lifetime).CHEM*3440Chemical InstrumentationTopic 9Phosphorescence Happens slowly after the absorption of the initial photon (min to ms lifetime).Ch
University of Guelph - CHEM - 3440
Infrared is Rovibrational SpectroscopyWavelengths between 0.8 m to 1 mm. Associated with changes in nuclear motion (vibrations and rotations).CHEM*3440Chemical InstrumentationTopic 10In gas phase, rotational transitions are resolved; in liquid phase,
University of Guelph - CHEM - 3440
Raman SpectroscopyAnother spectroscopic technique which probes the rovibrational structure of molecules.CHEM*3440Chemical InstrumentationTopic 11C.V. Raman discovered in 1928; received Nobel Prize in 1931. Can probe gases, liquids, and solids. Must u
University of Guelph - CHEM - 3440
Fundamental SchemeIn general there are four steps associated with a mass spectroscopic experiment: ! generate gas-phase molecules from analyte (solid, liquid, solution, etc.)CHEM*3440Chemical InstrumentationTopic 12&quot; ionize those molecules # separate
University of Guelph - CHEM - 3440
X-Ray Energiesvery short wavelength radiation 0.1 to 10 nm (100 ) Visible - Ultraviolet (UV) - Vacuum UV (VUV) - Extreme UV (XUV) - Soft X-ray - Hard X-ray - Gamma Ray often report photon energies, rather than wavelength. (above range is from 125 keV dow
University of Guelph - CHEM - 3440
Thermal MethodsWe will examine three thermal analytical techniques: Thermogravimetric Analysis (TGA)CHEM*3440Chemical InstrumentationTopic 14Differential Thermal Analysis (DTA) Differential Scanning Calorimetry (DSC)TGA and DTA have found important
University of Guelph - CHEM - 3440
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University of Guelph - CHEM - 3440
1 CHEM*3440 FINAL EXAMINATION Fall 2004 Duration: 2 hours 1. (8 points) A certain molecule was analyzed by electrospray ionization mass spectrometry. There are a lot of overlapping bands, but a tetramer of the molecule appears with a residual charge of +7
University of Guelph - CHEM - 3440
CHEM 3440 Final Examination Fall 2005 Duration: 2 hours 1. (10 points) Chondrotinase is a protein which was analyzed by ESI FT ICR MS by Kelleher et al. in 1997. Some spectra are shown below. Mass here is reported in daltons (Da) = atomic mass unit (amu).
University of Guelph - CHEM - 3440
CHEM*3440 Instrumental Methods Mid-Term Examination October 28, 2003 Duration: Two Hours 1. (5 points) A real battery actually behaves more like a voltage source in series with a resistance, called its internal resistance. (This arises from the fact that
University of Guelph - CHEM - 3440
CHEM*3440 Instrumental Analysis Mid-Term Examination Fall 2004 Duration: 2 hours 1. (10 points) An atomic absorption experiment found the following results for a [Pd] ppm series of standard solutions for dissolved palladium metal ion. The data were plotte
University of Guelph - CHEM - 3440
Chemical Instrumentation CHEM*3440Mid-Term Examination Fall 2005 TUESDAY, OCTOBER 25, 2005 Duration: 2 hours. You may use a calculator. No additional aids will be necessary as a series of data and equation sheets are included as part of the examination p
University of Guelph - CHEM - 3440
Chemical Instrumentation CHEM*3440 Mid-Term Examination Fall 2006 Tuesday, October 24Duration: 2 hours. You may use a calculator. No additional aids will be necessary as a series of data and equation sheets are included as part of the examination.1. (10
University of Michigan-Dearborn - CHEM - 434
Instrumental Analytical ChemistryToolsvarious optical and electronic instrumentations.Reportdata analysis and statistics (lecture 1 &amp; 2).Problemsanalyte identification, mixture separation.Basic Components of an InstrumentStimulus (energy source) S
University of Michigan-Dearborn - CHEM - 434
Lecture #13 Raman Spectroscopy and ApplicationReading: Chapter 18, page 481 497; Problems: 18-1, 3, 5. Basics of Raman spectroscopy; Instrumentation of Raman spectroscopy; Qualitative and quantitative applications; Resonance and Surface Enhanced Raman Sp
University of Michigan-Dearborn - CHEM - 434
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University of Michigan-Dearborn - CHEM - 434
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University of Michigan-Dearborn - CHEM - 434
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University of Michigan-Dearborn - CHEM - 434
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Kent State - MATH - 3450:149
From Section 3.7 Exercises; Precalculus by Coburn, 2nd Edition
Kent State - MATH - 3450:149
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Kent State - MATH - 3450:149
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Kent State - MATH - 3450:149
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Kent State - MATH - 3450:149
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Kent State - MATH - 3450:149
From Section 5.2 Exercises; Precalculus by Coburn, 2nd Edition
Kent State - MATH - 3450:149
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Kent State - MATH - 3450:149
From Section 5.4 Exercises; Precalculus by Coburn, 2nd Edition
Kent State - MATH - 3450:149
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Kent State - MATH - 3450:149
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Kent State - MATH - 3450:149
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Kent State - MATH - 3450:149
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Kent State - MATH - 3450:149
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Kent State - MATH - 3450:149
Bl,.,7.nc&quot;,JJaa,y^ AN^,^,.u,Loa' 6 +tt&quot;?3cfw_From Section 6.6 Exercises; Precalculus by Coburn, 2nd EditionY= -v43)n*:-7In\ ?rtqc-\tu,! &quot;oo4 )-, &amp;tV b) 2 &quot;-oolst3X&quot; t\^-'(-,h\-'r&quot; Zt;s i&lt;-.Osx-/- &quot;r'7 &quot;cfw_,(-' ).26.7l| T ar,&quot; r'ta-
Kent State - MATH - 3450:149
/z to6tf1,);nx 1C.&gt;tf )tLq -q -=(-,-JASaJ At4r,o?3tFrom Section 6.7 Exercises; Precalculus by Coburn, 2nd Edition5'l+ 7-g&quot;.-tyc-.&gt;f x : Z 2 |',nXcoS/ : + Lv-zL+ztl\&lt;.? t/, + flLv+O xrt', a./_*+T,Lz-l&quot;T'rfu- +LTILi7-cfw_2,L11,5.( a-7
Kent State - MATH - 3450:149
jtc-JAtu-7.S:,^ t?'9-,lo^ 3oo laALAr,tOFrom Section 7.1 Exercises; Precalculus by Coburn, 2nd Editiona)bl1- t;.92DZi3= Cs.-(=- ( ta,4 | b2,.4);46.7 : L*_.-_r'bx&quot;\'76.Lt8,1 Srnll&quot; L.tt7-7,&quot;,3;^C4 s.64z.uLl &amp;.&lt; si,^ tq &quot;4e,z
Kent State - MATH - 3450:149
qJAS'phtran,olo(atcL-UtFrom Section 7.2 Exercises; Precalculus by Coburn, 2nd EditionLcfw_t c*,,+Lu-.LU)&amp;)r-tn-( s) ( 6)- s' ) (&quot;' -4Lv-'7&quot;q- + q&quot;5- z(3 7 1.5) c.-9 7e&quot;/2c-.-t&quot;-zQ,&quot;71&quot;- u(gt)r3P&quot;(' ,&lt; S-9in 5\5?.&gt;: \r-o IO,1'1
Kent State - MATH - 3450:149
cfw_o-/&quot;1 .&gt;&quot;/'t1rt'16.)?.g.6-! c u!.*rs ?. t l+k) #'5 vcfw_/\q, 'z.f , 1,br ?/, ,t/ , urcfw_.\ 3X 1y =!'l /27/t'From Section 8.1 Exercises; ri,fS &quot;/ ,&quot;/, G/, ez,-/, ,/Edition Precalculus by Coburn, 2nd(z z)0&lt;_ ,G ltt -rt i,t. -t?i-/)
Kent State - MATH - 3450:149
/46lp-2,&quot;,1)J&quot;-r&quot;t*, p.cfw_&quot;^-o?l LFrom Section 8.2 Exercises; Precalculus by Coburn, 2nd Editionz-u-Z:*!I -:31cfw_ -z!t): -[d+y-'-.?.2,':-tr=y-)*3-e*-..cfw_.(,:-)= *r7)23--3(t-) rz(0-l:-?*,t.?,-v- -z,-1Y,t). -- _1-*:-t:.:.t- -.-,
Kent State - MATH - 3450:149
From Section 8.3 Exercises; Precalculus by Coburn, 2nd Editionfoq,;- !n.o?/8VZ?4 /z/tcU+c)Q-'z) cfw_+3/--zIiAi:?-fui1(*:ul -*lstuL'* j-7*lcfw_- L1J)lt' rL l+z d-ApVz!il]--/-g'-'(r.-71(r't&quot;s*+7/ -b(t-&quot;)st , t?/t - jx '7tx'- ?x
Kent State - MATH - 3450:149
.A-e,-'From Section 10.1 Exercises; Precalculus by Coburn, 2nd Editioni-l'i,*','f,.Q.'^teL;4,z-:.JLi-f-i-J'/.,.l.n.J:-g:o,)t5 (esLr-II Ii iIl itII !ajlA^ - (a^-t41, - (z - t?_cfw_za^-(a+7T+\&gt; llL\q6n-'cfw_rtuIa56l1't,
Kent State - MATH - 3450:149
l()/'l-.r' l/Jlo!yer-C*Jcr&quot;p.uiI cfw_ion*, 4*oA.)From Section 10.2 Exercises; Precalculus by Coburn, 2nd Editionf-l1ta;/_cfw_.'t-3 l, il-b,!:8-.r?-2,,L ')L7 1 -, bO UD 4I c q-'.&amp;.&quot;g.$.t'c117mn- Jl'an',.kL-t6- ,cfw_ l0*\ + l] , tL+ (
Kent State - MATH - 3450:149
l9 lc)10. aJa,wnFrom Section 10.3 Exercises; Precalculus by Coburn, 2nd EditiontGt (- a(4eCLt 4.-b- --g-:-:'b\ ;z t &lt;.+ -zt( 49Aj _- 3 K-33. -Gcfw_:a-^tFl-20, _ 4&quot;t_-= V ?_ ,.-C, \J.54?&quot;9:-6, J1.i, S- ,.5-:-v-1',a,16t LS -Ngit-Gnp
Kent State - MATH - 3450:149
Quiz1l ^'\O :)NuQuiz 1 All work must be shown to receive credit.&quot;,JAS,-ALA*oPrecalculus January 15,2010 L. Brubaker2. (2 points) Solve the inequality indicated using a number line. Write yow answer in interval notation.g(x) = -x' +10x -/- -5=*
Kent State - MATH - 3450:149
J boLtcfw_ a z:(s,oa'l&lt;-r r-\ '&quot;eJ&quot;Quiz 2Name:Quiz 2)fi9ouIPrc.&quot;1.&quot;t*AtauoJanuary 22,2010 L. BrubakerAll work must be shown to receive credit.l. (3 points) This function is not onerto-one. (a) Determine a domain restriction that preseryes all
Kent State - MATH - 3450:149
Quiz3Name:Quiz 3)Auul-l'lr-uP*.&quot;1.&quot;1.&quot;January 29,2010 L. BrubakerAll work must be shown to receive credit.2. (2 points) Find the domainof /(x) = t&quot;r,[=)/*\2O x&gt;-44- &lt; 76-I:_ij t( &gt;v/z\q-xr' zrl/, &gt; -ul&gt;p l*4D,. (- +,\)(l,rcfw_+ -,fr-c
Kent State - MATH - 3450:149
ID':(Quiz4Name:Quiz 4PrecalculusFebruary 5,2010 L. Brubakersteps. Do not use a calculator to do conversion.All work must be shown to receive credit. 2. (2 points) Convert the following. Show ALL a) 45J 45' 45&quot; to decimal degrees.b) 142.207 5Ll t
Kent State - MATH - 3450:149
)ur(z-''t&quot;I/'ii-fQuiz5 I /i-' l!,I'\h, l&quot;.aaIName: J A9o ^r ,/t_y'r&quot;.,0IQuiz 5PrecalculusFebruary 12,2010 L. Brubaker All work must be shown to receive credit. Do not use a calculator to do the graphing. lra) (2 points) Use a reference
Kent State - MATH - 3450:149
5r^ Oc-otocftita* A; irrauz( Qt-ol oc&quot;et dtilb1in L.o t. (osz6 &quot;. I(^&quot; 09Ac o.,(D(d,tcfw_^e .:SPd.*c&quot; OLa-ro+ t; s&lt;czJ Quiz c c-o( ta )- 6 (9a'PQuiz6,^(*),-t'&quot;dcotcfw_8l,t(-dt - -'ftu*&quot;, (-o) - _a_ &quot; EJ45u&quot;t I (A-r,\D Precalcu
Kent State - MATH - 3450:149
QutzTtra 'all; Quiz 7Nu*J /c*t*; &quot;, JA-S aPrecalculusMarch 5,2010 L. BrubakerAll work must be shown to receive credit.' 9',&lt; ( ! B) 7 ,:rT'I'ff:-;&quot;' &quot;[#)'&quot;'(#).'&quot;'(#)&quot;'(#) ; t',':?)lT;T i .,a ! btd 4:rp,A_;l2. (4 points) Givend2g^'&quot;'(.'+ ^ -+
Kent State - MATH - 3450:149
&quot;a ,'r'?,,a2cfw_.t-| ,o.cfw_)GO&lt;-oJ;/ v^1&quot;Quiz8 LT-Name:r' Laq*t-.J Arrr 4 r,t nroPrecalculusQuiz 8%wvrlwvo .ot[.or-March 12,2010 L. BrubakerAll work must be shown to receive credit.n urqrvr' 1. (1 point) Find the exact value of the
Kent State - MATH - 3450:149
Quiz 9q)a.Y 6j,All work must be shown to receive credit.Quiz 9loName:JA fr'-'r Ac4, ls.Precalculus March 26,2010L. Brubakerl;,t,il.'J']ilI&quot;#-t'n&quot;'T,&quot;l&quot;T'*;:)Tl vangleB:45&quot; sidea:32.8km t;* cfw_ (&quot; 2,1 .q' 1i^ 45&quot;Xtriangresexi'l:&quot;il'iithcompret
Kent State - MATH - 3450:149
Quiz 10glo*&quot;*&quot;,JAfu^' 4&amp;'44-PPrecalculus April9 ,2010Quiz 10L. BrubakerAll work mustbe shown to receive credit. l. (5 points) Solve using elimination. State if a system is inconsistent or dependent. For systems with linear dependence, write the a