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Example 1 I will obtain a job as a high school math teacher within three months after graduating with my Bachelor of Science in Education Specific The goal of becoming a high school math teacher is welldefined;And = 2 2 2 2!Examples Let S = ℤ and define R = {(x,y) x and y have the same parity} ie, x and y are either both even or both odd The parity relation is an equivalence relation 1 For any x ∈ ℤ, x has the same parity as itself, so (x,x) ∈ R 2 If (x,y) ∈ R, x and y have the same parity, so (y,x) ∈ R 3

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Y=a(1+r)^t solve for r-Example 12 Find the length of ~r(t) =~it2~j t3~k for 0 6 t 6 1 This is straight forward calculations L = Z 1 0Example Example 321 Let R be the relation on the set R real numbers defined by xRy iff x−y is an integer Prove that R is an equivalence relation on R Proof I Reflexive Suppose x ∈ R Then x−x = 0, which is an integer Thus, xRx II Symmetric Suppose x,y ∈ R and xRy Then x − y is an integer

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For t and r must be the same Example of Exponential Growth The 00 census found that a US Population of about 281 million with an estimated growth rate of 07% Q = Q0 x (1 r)t 4/1/10 2 Forms of the ExponentialY = a(1 r)t Example 1 Exponential Growth 1a) The prize for a radio station contest begins with a $100 gift card Once a day, a name is announced The person has 15 minutes to call or the prize increases by 25% for the next day Write an equation for the amount of the gift card in dollars after t days with no winners How much will the giftAs examples, y = x 3 4x C is the general solution of example a above, and y 1 = ½ x 2 C is the general solution of example b above, shown as the collection of red graphs below The collection of graphs of all particular solutions of a DE completely fill up space, and no two graphs overlap, much as the layers of a dagwood sandwich

61 INTRO TO LINEAR TRANSFORMATION 191 1 Let V,W be two vector spaces Define T V → W as T(v) = 0 for all v ∈ V Then T is a linear transformation, to be called the zero transMeasurable Success can be measured by the number of applications, interviews and job offersY = a (1 r) t = 1800 (1 011) t = 1800 (111) t Find the value in 30 years y = 1800 (111) t = 1800 (111) 30 ≈ 41,613 After 30 years, the painting will be worth approximately $41,6 Create a table of values to graph the function t y (t, y) 0 1800 (0, 1800) 8 4148 (8, 4148) 16 9560 (16, 9560) 24 22,030 (24, 22,030) 32 50,770 (32, 50,770)

Theory used in this question1 Method of charging (induction)https//youtube/drfZbYMf50k=====🔴 ELECTRIC CHARGES ANDQuestion Use the compound interest formula A=P(1r/n)^nt to answer the following questions In the formula, A is the amount of money in the savings account, P is the principle, r is the interest rate, t is the time that the money is in the account, and n is the number of times the money is compounded per yearEXAMPLE 1 r(t)(4 t5)i tetj sin(4t)k (a) Find the derivative of (b) Find the unit tangent vector at the point t = 0 SOLUTION (a) According to this theorem, we differentiate each component of r r(t) = and r(o)j 4k, the unit tangent (b) Since r(0)= vector at the point (4, 0, 0) is j 4k

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Ttest(x, y = NULL, mu = 0, varequal = FALSE) arguments x A vector to compute the onesample ttest y A second vector to compute the two sample ttest mu Mean of the population varequal Specify if the variance of the two vectors are equal By default, set to `FALSE` Onesample ttestWe use the formula d = r * t, where d is the distance, r is the speed and t is time to solve the following problems Example 1 A car travels 1 miles at the speed of 48 miles per hour Find the time taken for the journey using the formula d = rt Solution Step 1 Distance, d = 1 miles;Y = a(1 r)t Example 1 Exponential Growth 1a) The prize for a radio station contest begins with a $100 gift card Once a day, a name is announced The person has 15 minutes to call or the prize increases by 25% for the next day Write an equation for the amount of the gift card in dollars after t days with no winners How much will the gift

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Independentsamples ttest using R, Excel and RStudio (page 2) On the previous page you learnt about the type of research where an independentsamples ttest can be used and the critical assumptions of the independentsamples ttest that your study design, variables and data must meet in order for the independentsamples ttest to be the correct statistical test for your analysis= 1 r Subtracting 1, you get that the rate is just over 003 or 3/10 of 1% The function that would model your growth is y = 484(1003)^t To do #2, find out how many years each year is after 1980, and plug those numbers in for "t" For example, to find 1985, you'd take 484(1003)^5 To do #3, "double" the original population would be 968 million peopleSo jjjj mav= 2 >1 = jjAjj2 mav Remark Not all submultiplicative norms are induced norms An example is the Frobenius

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1 is often assessed by comparing it with thenull model, that re ects no dependence of y on the explanatory variables The model formula for the null model is > y ~ 1 signifying that we use a constant to approximate y The natural constant is the mean of y Functions in R that generate models report the statistics thatY = 2,500(1 04) 2 Plug in values y = 2,500(104) 2 Add the numbers in the parenthesisUsing the growth formula we have y = a(1 r) x where a = 1 (we start with 1 bacteria), and r = 100%, since the amount doubles y = 1(1 100) x = 2 x (same result) Notice that the graph is a scatter plot You cannot have a fractional part of a bacteria The dotted line is the exponential function which contains the scatter plots (the model)

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• A Solve V0 (1 r)T = 2V 0 • => (1 r)T = 2 • => Tln(1 r) = ln(2) • => T = ln(2)/ln(1r) • = 069/ln(1 r) ≈ 070/r for r not too big Present Value of Future Cash Flows • A cash flow is a sequence of dated cash amounts received () or paid () C0, C1, , CT C h t i d iti h h E Zivot 06 RW Parks/LF Davis 04Exponential 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?Y Write the exponential growth function= a(1 r)t = 150,000(1 008)t Substitute 150,000 for a and 008 for r Add= 150,000(108)t The festival attendance can be represented by y = 150,000(108)t b The value t = 4 represents the fi fth year because t = 0 represents the fi rst year y = 150,000(108)t Write the exponential growth function

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Curves 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 setAt any point along the path, the (small) tangent vector ${\bf r}'\,\Delta t$ gives an approximation to its motion over a short time $\Delta t$, so the work done during that time is approximately ${\bf F}\cdot{\bf r}'\,\Delta t$;Parametrize 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 integrals

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1 Loop Examples 11 Example Sum Primes Let's say we wanted to sum all 1, 2, and 3 digit prime numbers To accomplish this, we could loop through all 1, 2, and 3 digit integers, testing if each is a prime number (using the isprime function) If and only if a particular value is prime, then we'll add it to our running totalIn this section we solve linear first order differential equations, ie differential equations in the form y' p(t) y = g(t) We give an in depth overview of the process used to solve this type of differential equation as well as a derivation of the formula needed for the integrating factor used in the solution processAn equation for the depreciation of a car is given by y = A(1−r)t y = A (1 − r) t, where y = y = current value of the car, A = A = original cost, r = r = rate of depreciation, and t= t = time, in

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Precalculus Examples Popular Problems Precalculus Solve for t A=P(1r/n)^(nt) Rewrite the equation as Divide each term by and simplify Tap for more steps Divide each term in by Cancel the common factor of Tap for more steps Cancel the common factor Divide byDefinition In the case of a space curve, the radius of curvature is the length of the curvature vector In the case of a plane curve, then R is the absolute value of ≡ =, where s is the arc length from a fixed point on the curve, φ is the tangential angle and κ is the curvature Formula In 2D If the curve is given in Cartesian coordinates as y(x), then the radius of curvature isFV = C 0 * (1 r)^5

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# A formula y ~ x # A converted formula y = a_1 a_2 * x This is an example of a simple conversion y ~ x gets translated into y = a_1 a_2 * x To see and understand what R actually happens, you can use the model_matrix() function This function creates a design or model matrix by, for example, expanding factors to a set of dummy variables, depending on the contrasts, and expanding interactions similarlyOneSample ttest The onesample ttest, also known as the singleparameter t test or singlesample ttest, is used to compare the mean of one sample to a known standard (or theoretical / hypothetical) mean Generally, the theoretical mean comes from a previous experiment For example, comparing whether the mean weight of mice differs from 0 mg, a value determined in a previous studySuppose that the path of an object is given by a vector function ${\bf r}(t)$;

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Section 31 Parametric Equations and Curves To this point (in both Calculus I and Calculus II) we've looked almost exclusively at functions in the form \(y = f\left( x \right)\) or \(x = h\left( y \right)\) and almost all of the formulas that we've developed require that functions be in one of these two formsThis equation gives the amount B that the person still needs to repay after t years B = A (1 r/n) NT P (1 r/n) NT 1 (1 r/n) 1 where B = balance after t years A = amount borrowed n = number of payments per year P = amount paid per paymentTo find the interest rate (r) in the formula a=p(1r)^t, you need to know the values of a (amount), p (principal) and t (time) You would take a and divide it by p You will then take that result and take the t root of it You then subtract that answer by 1 to get your interest rate in decimal form Here is an example You need to find the interest rate of an account that grew from $4000 to

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Y = a(1 r)t Write exponential growth model = 609(1 )t Substitute 609 for a and for r = 609()t Simplify Using this model, you can estimate the world population in 05 (t = 5) to be y = 609()5 ≈ 646 billion b Use the table feature of a graphing calculator to determine that y ≈ 7 when t = 12 So, the world population was about 7 billion in 12Y = a(1 r)t EXPONETIAL DECAY FUNCTION y = a(1 – r )t EXPONENTIAL DECAY APPLICATIONS 5 The population of a town is 2500 and is decreasing at a rate of 35% per year Write an exponential decay function to find the population of the town after 5 years EXPONENTIAL GROWTH APPLICATIONS 1A better formula to use would be y = a ( 1 r k) k t, where k is the number of times the interest is compounded per year So, plugging in your information gives y = 3750 ( 1 06 12) 132 = $ Share

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For t and r must be the same Example of Exponential Growth The 00 census found that a US Population of about 281 million with an estimated growth rate of 07% Q = Q0 x (1 r)t 4/1/10 2 Forms of the ExponentialSpeed, r = 48 miles/hour Step 250 (298) Bilingual math tutor and much more See tutors like this See tutors like this I am assuming that you mean y= (14) t8 Using the laws of exponents, this equals 14 t * 14 8 This equals y = * (14) t By the way the number 4 is 40% Hope this helped

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Stokes' theorem relates a surface integral of a the curl of the vector field to a line integral of the vector field around the boundary of the surface After reviewing the basic idea of Stokes' theorem and how to make sure you have the orientations of the surface and its boundary matched, try your hand at these examples to see Stokes' theorem in actionFV = C 0 * (1 r) 1 * (1 r) 2 * * (1 r) T or FV = C 0 * (1 r)^T Where r is the interest rate, C 0 is the cash flow at time 0 (now), FV is the future value, and T is the number of periods Example Problem You are given $10,000 and you want to invest it in a 5 year CD that yields 4% interest per year How much money will you have in 5 years?The total work over some time period

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# A formula y ~ x # A converted formula y = a_1 a_2 * x This is an example of a simple conversion y ~ x gets translated into y = a_1 a_2 * x To see and understand what R actually happens, you can use the model_matrix() function This function creates a design or model matrix by, for example, expanding factors to a set of dummy variables, depending on the contrasts, and expanding interactions similarlyRewrite y=5(1/4)^{t/6} in the form y=a(1r)^t or y=a(1r)^t Round the value of r to the nearest tenthousandth Then tell whether the model represents exponential growth or exponential decayThe 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 −4t Suppose there are initial conditions y(0) = 1, y′(0) = −7 A unique particular solution can be found by solving for C1 and C2 using the initial conditions First we

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Example x 0 = 50 r = 4% = 004 t = 90 hours x(t) = x 0 × (1 r) t = 50×(1004) 90 = 1706 Exponent calculator1= p 2 for x= (1;0)T Example of a norm that is not submultiplicative jjAjj mav= max i;j jA i;jj This can be seen as any submultiplicative norm satis es jjjj jjAjj2 In this case, A= 1 1 1 1!Definition In the case of a space curve, the radius of curvature is the length of the curvature vector In the case of a plane curve, then R is the absolute value of ≡ =, where s is the arc length from a fixed point on the curve, φ is the tangential angle and κ is the curvature Formula In 2D If the curve is given in Cartesian coordinates as y(x), then the radius of curvature is

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Example 1 STOKES' THEOREM The projection D of S on the xyplane is the disk x2 y2 ≤ 1 So, using Equation 10 in Section 167 with z = g(x, y) = 2 – y, we have the following result Example 1 STOKES' THEOREM ( ) ( ) ( ) ( ) 21 00 23 1 2 0 0 2 12 0 23 1 2 curl 1 2 1 2 sinMeasurable Success can be measured by the number of applications, interviews and job offersExample 1 I will obtain a job as a high school math teacher within three months after graduating with my Bachelor of Science in Education Specific The goal of becoming a high school math teacher is welldefined;

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Rewrite y=5(1/4)^{t/6} in the form y=a(1r)^t or y=a(1r)^t Round the value of r to the nearest tenthousandth Then tell whether the model represents exponential growth or exponential decayUsing the growth formula we have y = a(1 r) x where a = 1 (we start with 1 bacteria), and r = 100%, since the amount doubles y = 1(1 100) x = 2 x (same result) Notice that the graph is a scatter plot You cannot have a fractional part of a bacteria The dotted line is the exponential function which contains the scatter plots (the model)

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