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Section 142 Line Integrals Consider a smooth plane curve Cde ned by R~(t) = hx(t);y(t)i, where a t b A partition of the interval a;b a= t 01115 Substitute y = erx into the given differential equation to determine all values of the constant r for which y = erx is a solution of the equation y′′ y′ − 2y = 0 Solution The first and second derivatives of y = erx are y′ = rerx, y′′ = r2erx If we plug these into the differential equation we get r2e rxrerx − 2e = (r2 r −2)erx = 0 As erx 6= 0 for any r or x1 C is the initial amount t is the time period (1 r) is the growth factor, r is the growth rate The percent of increase is 100 r y = C (1 r) t E XPONENTIAL G ROWTH M ODEL W RITING E XPONENTIAL G ROWTH M ODELS A quantity is growing exponentially if it increases by the same percent in each time period 2

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What is y=a(1+r)^t-Growth/Decay Factor % of increase or decreaseIntegrate f(xyz)x y C1r(t) = t i tj, 0sts2 C2 r(t) = 21 4tk 0sts2 z over the path from (0,0,0) to (2,4,2) given by Type an exact answer) Get more help from Chegg Get 11 help now from expert Calculus tutors Solve it with our calculus problem solver and calculator

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123 Use the equation for arc length of a parametric curve;Y = 2,500(1 04) 2 Plug in valuesA is false You can test this by having C=100, r=05 and t=1 and then test it for t=2 y = 100 (105)^1 = 105 y = 100 (105)^2 = y is increasing from left to right

M y C h a r t P o r t a l A c t i v a t i o n Options for Login to MyChart 1 From the Baptist Health website https//wwwbaptisthealthcom/ select MyChart across theM y C h a r t C O V I D V a c c i n e S c h e d u l e y o u r 1 s t d o s e v ac c i n e 1 Once logged into MyChart, choose Visit 2 Select Schedule An Appointment 3 Choose the COVID Vaccine option 4 Select the location closest to you to see available times based on supply, then Continue 5This is general relativity As a consequence of introducing an arbitrary function, there may be freedom to explain the

Suppose y(t) = tr, we have y0(t) = rtr−1 and y00(t) = r(r − 1)tr−2 Thus t2y00(t) 7ty0(t) 8y(t) = (r(r − 1) 7r 8)tr = (r2 6r 8)t r Thus y = t is a solution of t2y00(t) 7ty0(t) 8y(t) = 0 if r 2 6r 8 = (r 2)(r 4) = 0 The roots of r 6r 8 = 0 are −2 and −4 Therefore the general solution is y(t) = c 1t −2122 Find the area under a parametric curve;124 Apply the formula for surface area to a volume generated by a parametric curve

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) = u;u2 cos( );u2 sin( ) for 0 6 u6 1, 0 6 6 2ˇ This is the surface you get by rotating y= x2, 0 6 x6 1, around the yaxisIf the density at a point on this surface is given by 24x5 4x3, nd the mass Remark The density has been chosen so that a simple substitution can be used to evaluate(b) The space curve C x=cost;y=sin2 t;z=sint;−1123 Use the equation for arc length of a parametric curve;

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We use the extended form of Green's theorem to show that ∮ C F · d r ∮ C F · d r is either 0 or −2 π −2 π —that is, no matter how crazy curve C is, the line integral of F along C can have only one of two possible values We consider two cases the case when C encompasses the origin and the case when C does not encompass the origin Case 1 C Does Not Encompass the OriginStokes' Theorem Let S be a piecewise smooth oriented surface with a boundary that is a simple closed curve C with positive orientation (Figure 679)If F is a vector field with component functions that have continuous partial derivatives on an open region containing S, then ∫ C F · d r = ∬ S curl F · d S ∫ C F · d r = ∬ S curl F · d SCompounded more frequently than once per year So, the formula y = a(1 r)t must be modifi ed for compound interest problems CCore ore CConceptoncept Compound Interest Compound interest is the interest earned on the principal and on previously earned interest The balance y of an account earning compound interest is P = principal (initial amount)

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Solving it we get r = 3=4 as a repeated root, so the general solution is given by y(t) = c122 Find the area under a parametric curve;L0 6= lim k!1jf(t) k xj C0, which means by the de nition that the two norms are not equivalent 2 Solve Pugh's Chapter 5, Problem #9;15;16;18 9) Give an example of two 2 2 matrices such that the norm of the products is less than the product of the norms Consider the following two matrices A= 1 0 0 0 B= 0 0

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P = C (1 r) t Continuous Compound Interest When interest is compounded continually (ie n > ), the compound interest equation takes the form P = C e rt Demonstration of Various Compounding The following table shows the final principal (P), after t = 1 year, of an account initially with C = $, at 6% interest rate, with the given1980 ≈ t Round Exponential growth equation #1 – y = a(1 r) t ex Bob Industries bought a plasma for $2500 It is expected to appreciate at most 4% per year What will the plasma be worth in 2 years?1 = A Design 2 = B Design 3 = C Design 4 = D Design Nominal Capacity 007 = 7,000 BTU/h 009 = 9,000 BTU/h 012 = 12,000 BTU/h 019 = 19,000 BTU/h 024 = 24,000 BTU/h 030 = 30,000 BTU/h 036 = 36,000 BTU/h 042 = 42,000 BTU/h 048 = 48,000 BTU/h 060 = 60,000 BTU/h Bold = R22 Refrigerant Italic = R410A Refrigerant Discharge Air T = Top Return Air L

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HalfLife Cesium137 has a halflife of 30 years Find the amount of cesium137 left from a 100milligram sample after 180 years Equation Growth or Decay?Solving it we get r = 1=3 as a repeated root, so the general solution is given by y(t) = c 1e t=3 c 2te t=3 Q 8) Find the general solution to 16y00 24y0 9y = 0 Answer The characteristic equation is 16r2 24r 9 = 0;Learning Objectives 121 Determine derivatives and equations of tangents for parametric curves;

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Consider the curve C parametrized by x = t and y = (9 − t^2)1/2 for −3 ≤ t ≤ 3?1 t e(x t)2 Note that the denominator vanishes only if t = e(x t)2, and (x t)2 0, so the denominator can only vanish if t 1 In particular, u is a C1, indeed C1, function on R x 0;1) t On the other hand, for x = 1, as t !1 , u(x;t) = 1 t e(1 t)2!1, ie the solution blows up at T = 1 (at x 0 = 1) Problem 5 Solve u t uu x = 0;Learning Objectives 121 Determine derivatives and equations of tangents for parametric curves;

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T = A for all t , start from the perperiod budget constraint A t 1 = Y ' C (1 r ) A t and rewrite it as an expression of (constant) consumption C = Y ' (1 r ) A t A t 1 C = Y ' rA t ( A t 1 A t ) To hold consumption constant, there are two options 1Parametrize the curve x= t, y= t, with tfrom 0 to 1 Put everything in terms of t dx= dt dy= dt M= x2y= t3 N= x 2y= t Now we put this into the integral I= Z C Mdx Ndy= Z 1 0 t3 dt tdt= Z 1 0 t3 tdt= 1 4 This is a di erent value from Example GT3, which leads to the important principle Important principle for line integralsSolving it we get r = 1=3 as a repeated root, so the general solution is given by y(t) = c 1e t=3 c 2te t=3 Q 8) Find the general solution to 16y00 24y0 9y = 0 Answer The characteristic equation is 16r2 24r 9 = 0;

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T = Thymidylic acid U = Uridylic acid I = Inosylic acid should be obvious codes F = Phe = Phenylanine N = Asn = Asparagine R = Arg = Arginine Y = Tyr = Tyrosine are phonetic codes R = A or G = puRine Y = C or T = pYrimidine K = G or T = Keto M = A or C = aMino S = G or C = Strong base pair W = A or T = Weak base pair double base codes DCurves in R2 Three descriptions (1) Graph of a function f R !R (That is y= f(x)) Such curves must pass the vertical line test Example When we talk about the \curve" y= x2, we actually mean to say the graph of the function f(x) = x2That is, we mean the setWe don't see it, but there's a 1 there, times 100 In the second hour, 0965 to the second power, times 100 So in general, in the nth hour let me do this in a nice bold color in the nth hour, we're going to have 0965 to the nth power, times 100 left of our radioactive substance And oftentimes you'll see it written this way

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Further, since sin2 t has values in 0,1 only, the particle moves to and fro along the line segment joining (1,0,1) and (1,0,1)(−1,1,−2) = (0,1,−1) Example 43 Find the tangent vector and the equation of the tangent to the helixR(t) = < 0,1,2 > t < 3,−1,−2 > Parametric equations of L x = 3t, y = 1 − t, z = 2 − 2t Example 12 Find equations of the planes parallel to the plane x 2y − 2z = 1and two units away from it The distance D between parallel planes axbyczd1 = 0 and axbyczd2 = 0 is d2 − d1 q124 Apply the formula for surface area to a volume generated by a parametric curve

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1A curve parametrized by r(t), t2a;b is called closed if r(a) = r(b) (intersects at endpoints) A curve is called simple if it intersects itself only at the endpoints The following gure illustrates a simple closed curve,Z 1 0 r2 cos2 θr2 sinθdr = 2π Z Z π 0 cos2 θsinθdθ Z 1 0 r4dr = 2π ×2× 1· 1 3· 1 ×1 = 4π 15 Remarks 1 As V is a sphere it is natural to use spherical polar coordinates to solve the integral Thus, x = rcosφsinθ, y = rsinφsinθ, and z = rcosθ and dxdydz = r2 sinθ 2 R R R V 3x 2ydxdydz = 0 and R R R1 = A Design 2 = B Design 3 = C Design 4 = D Design Nominal Capacity 007 = 7,000 BTU/h 009 = 9,000 BTU/h 012 = 12,000 BTU/h 019 = 19,000 BTU/h 024 = 24,000 BTU/h 030 = 30,000 BTU/h 036 = 36,000 BTU/h 042 = 42,000 BTU/h 048 = 48,000 BTU/h 060 = 60,000 BTU/h Bold = R22 Refrigerant Italic = R410A Refrigerant Discharge Air T = Top Return Air L

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Write an equation relating x and y but without t and without square roots Write it so variable terms are on one side and any constant on the other sideQuestion Depreciation on a car can be determined by the formula V=C(1r)^t , where V is the value of the car after t years, C is the original cost, and r the rate of depreciation If a car's cost, when new, is $15,000, the rate of depreciation is 30%, and the value of the car now is $3,000, how old is the car to the nearest tenth of a year?Search the world's information, including webpages, images, videos and more Google has many special features to help you find exactly what you're looking for

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Baran Lab T h e C h e m i s t r y o f 1A c y l p y r i d i n i u m s D W Lin 1 N O R acyl transfer 1,4addition 1,2ad dition 1,2a it on N Nu O R N O R Nu NR O u * * * * * * * 1,4addition 1,2addition acyl transfer Be sure to review the previous Baran Lab group seminar on a related topic, "Iminium and Pyridinium Photochemistry" (J Richter(a) Use the chain rule to compute h′(1) (b) Find h(t) and compute h′(1) directly Solution • (a) Note that f is differentiable on R, since each of its component functions is differentiable, and gis differentiable on R3 since its partial derivatives exist and are continuous • The Jacobian matrix of f at t= 1 is df(1) = 1 2t 3t2 tRoots r1, r2 They give two distinct ‡ solutions y er1t 1 = and y er2t 2 = Therefore, a general solution of (*) is y Cy C y Cer1t Cer2t = 1 1 2 2 = 1 2 It is that easy Example y″ 5 y′ 4 y = 0 The characteristic equation is r2 5 r 4 = (r 1)(r 4) = 0, the roots of the polynomial are r = −1 and −4 The general solution is then y = C1 e −t C 2 e

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Exercise 362 Let R be the equivalence relation defined on the set of real numbers R in Example 321 (Section 32) That is, xRy iff x − y is an integer Prove that every equivalence class x has a unique canonical representative r such that 0 ≤ r < 1 That is, for every x there is a unique r such that x = r and 0 ≤ r < 1V < c(3,1,TRUE,2+3i) t < c(4,1,FALSE,2+3i) print(v&t) it produces the following result − 1 TRUE TRUE FALSE TRUE It is called Elementwise Logical OR operator It combines each element of the first vector with the corresponding element of the second vector and gives a output TRUE if one the elements is TRUEWhich of the following is false for an exponential growth function y= C (1r)t?

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R(t) = < 0,1,2 > t < 3,−1,−2 > Parametric equations of L x = 3t, y = 1 − t, z = 2 − 2t Example 12 Find equations of the planes parallel to the plane x 2y − 2z = 1and two units away from it The distance D between parallel planes axbyczd1 = 0 and axbyczd2 = 0 is d2 − d1 qSolving it we get r = 3=4 as a repeated root, so the general solution is given by y(t) = cY = C(1 r)t A population of 950 deer is decreasing by 32% every year How many deer will be left after 4 years?

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