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MathLet `s` represent the side length of the cube. We want to estimate the relative error in measuring the volume if the error in measuring the sides is 2%. The propagated error can be estimated by...

MathLet `y=a_0+a_1x+a_2x^2+a_3x^3+a_4x^4+...` So: `y'=a_1+2a_2x+3a_3x^2+4a_4x^3+...` `y''=2a_2+6a_3x+12a_4x^2+...` `y(0)=1` tells us that `a_0=1` and `y'(0)=2` tells us that `a_1=2` Plugging...

MathThe man deposit $500. Interrest rate is 5%. At the end of 1st year he will have 500*(1+5/100) So at the start of second year he has 500*(1+5/100) At the end of second year he will have...

MathIf y=mx+c is straight line and y=m'x+c is another straight line they will be perpendicular when m*m' = 1 2x+5y = 12 5y = 122x y = (2/5)x+(12/5) Lets say the line perpendicular...

MathFind the intervals of concavity for `f(x)=2sinx` : `f'(x)=2cosx` `f''(x)=2sinx` `2sinx=0==>x=pi` on `(0,2pi)` On `(0,pi),2sinx<0` (Try a test value like `x=pi/2==>2sin(pi/2)=2`...

MathTo find the intervals of concavity, we need the second derivative. In this case, we can simplify the function before taking derivatives. `f(x)=(x2)(x+4)` ` ` `=x^2+2x8` Now differentiate...

MathYou need to determine the first derivative and then you can find the second derivative and to solve the equation f''(x) = 0. f'(x) = (x^3  x^2 + x  3)' => f'(x) = 3x^2  2x + 1 f''(x) = (...

MathYou need to solve the equation f''(x) = 0 to be able to tell what are the intervals of concavity of the function if x in [0,2], hence, you need to find the first derivative such that: f'(x) = (x^2...

MathIt is given that `y = (x  1)(x + 3)(y  1)` The derivative `dy/dx` is determined using implicit differentiation. `dy/dx = (x + 3)*(y  1) + (x  1)(y  1) + (x  1)(x + 3)(dy/dx)` =>...

MathDifferentiate `yx=sinxy` : Use implicit differentiation `d/(dx)[yx]=d/dx[sinxy]` `(dy)/(dx)1=cosxy(y+x(dy)/(dx))` `(dy)/(dx)1=ycos(xy)+x(dy)/(dx)cos(xy)` `(dy)/(dx)[1xcos(xy)]=ycos(xy)+1`...

MathTo find the derivative of `y=x^x` , you need to use logarithmic differentiation. Start with the function, then take logarithms of both sides then simplify. `y=x^x` take logs `ln y=ln x^x` use...

MathIt is given that `y + x = y^2` . The expression for `dy/dx` can be determined using implicit differentiation. `dy/dx + 1 = 2y*(dy/dx)` => `(dy/dx)*(2y  1) = 1` => `dy/dx = 1/(2y  1)` The...

MathThe derivative `dy/dx` of `y = x^2` has to be determined. For `y = x^n` , the derivative `dy/dx = n*x^(n  1)` For `y = x^2` , `dy/dx = 2x` The derivative of `y = x^2` is `dy/dx = 2x`

MathThe integral `int sin x + 3*cos x dx` has to be determined. `int sin x + 3*cos x dx` => `cos x + 3*sin x + C` The integral `int sin x + 3*cos x dx = cos x + 3*sin x + C`

MathThe integral `int e^x + sqrt 3*x dx` has to be determined `int e^x + sqrt 3*x dx` => `e^x + sqrt 3*(x^2/2) + C` The integral `int e^x + sqrt 3*x dx = e^x + (sqrt 3/2)*x^2 + C`

MathThe integral `int x^(1/3) + 1 dx` has to be determined. `int x^(1/3) + 1 dx` => `(x^(1/3 + 1))/(1/3 + 1) + x + C` => `(x^(2/3))/(2/3) + x + C` => `(3/2)*x^(2/3) + x + C` The...

MathThe anti derivative of f(x) = 9 has to be determined. `int f(x) dx` => `int 9 dx` => `9x + C` For `f(x) = 9` , `int f(x) dx = 9x + C`

MathCyclists A and B rode along a straight one way road starting from the same location and they both...To solve this problem, we need to use the speed formula `v=d/t` which can be rearranged as `d=vt`. Cyclist A: For the first 1/2 hour A has gone 5 miles. In the time up to the 1/2 hour, the...

MathYou may find x intercepts solving the equation `(1/2)(1x)(x^2x2) = 0` such that: `(1/2)(1x)(x^2x2) = 0 =gt 1x = 0 =gt ` `x = 1 `` `` x^2  x  2 = 0` You need to use quadratic formula such...

MathA plausible graph: Thus no beer from midnight to 8pm; 2 beers each from 89 and 910pm, 1 beer each from 1011 and 11pm12am. (I'm not much of a drinker) (a) `int_0^(23)f(t)dt` represents the...

MathThe graph y= f(x) passes through the points (1, 5) and (3, 7). The tangent line to y= f(x) at (3,...We are given that `f(1)=5,f(3)=7` and the tangent to the graph of `f(x)` at (3,7) is `2x+13` . One possibility for `f(x)` is `f(x)=3/2x^2+7x1/2` The graph of `f(x)` and the given tangent line:...

MathThe graph of `y = (x  1)/(3x + 1)` has to be plotted. This has a vertical asymptote at x = 1/3 and a horizontal asymptote at y = 1/3. The required graph is:

MathTo graph polynomials we know that there are no asymptotes (since polynomials are very wellbehaved) so we just need to use the remaining graphing techniques. (1) Intercepts. The function is in...

MathThe differentiation of an integral is used with the FTC formula `d/{dx}int_a^x f(t)dt=f(x)`. However, with differentiating this integral, we need to switch the limits of the integral and then apply...

MathTo differentiate an integral use the FTC formula `d/{dx} int_a^x f(z)dz=f(x)` . However, in this case, we need to use the chain rule since the lower limit is not just x, and also split the...

MathTo differentiate an integral, we use the FTC formula `d/{dx} int_a^x f(t)dt=f(x)`. In this case, we also need to use the chain rule, since the upper limit of integration is not just x. `d/{dx}...

MathWhen differentiating an integral we have the FTC formula `d/{dx} int_a^x f(s)ds=f(x)`. In this case, we need to manipulate the integral before applying the FTC. `d/{dx} int_x^6 cos(sqrt{s^4+1})ds`...

MathWhen differentiating an integral, you have the FTC formula `d/{dx}int_a^x f(t)dt=f(x)`. In this case, we can apply the FTC directly to get `d/{dx} int_2^x cos(t^4) dt` `=cos(x^4)` The derivative...

MathYou should remember that `x^2 + y^2 = (x + y)^2  2xy` , hence, you need to substitute `(x + y)^2  2xy` for `x^2 + y^2` in the given condition such that: `(x + y)^2  2xy = 34xy =gt (x + y)^2 =...

MathYou should notice the discontinuity of the function at x=0 and you need to prove that the antiderivative of the function exists. Differentiating the product `x^2*cos(1/x)` yields: `(x^2*cos(1/x))'...

Mathcos3x+cos7x=cos2x cos3x+cos7xcos2x = 0 We know that `cos A +cos B = 2cos((A+B)/2)*cos ((AB)/2)` cos7x+cos3xcos2x = 0 2cos((7x+3x)/2)*cos((7x3x)/2)  cos2x = 0 2cos5x * cos2xcos2x = 0...

MathIf `6i` is a zero of the function, then `(x6i)` will divide `f(x)`. Even better, `f(x)` has only real coefficients. (No complex ones, such as `2+3i` ). This means that all the complex zeros come...

Math`(3x5)/(x+3)lt=2` To start, change the inequality sign to "`=`" .Then, make the right side zero by subtracting both sides by 2. `(3x5)/(x+3)2=0` Simplify left side. `(3x5)/(x+3)...

MathLet us say box contain x number of apple, bag contain y number of apple and string bag contain z number of apple. In first day shop keeper receives (9x+3y) apples. When he repack he use 11 string...

MathAmong the properties of triangles are: (I) The sum of three interior angles of the triangle is `180^o` . (II) The measure of an exterior angle is equal to the sum of two remote interior...

MathSuppose you had the system of equations: `a+b=3` `2a+2b=6` These two equations actually are "the same" (if one is true the other must be true... once you know one of these, the second doesn't...

MathFor a population proportion, the standard error is given by : `E=z_(alpha/2)sqrt((hat(p)hat(q))/n)` where: `E` is the standard error `z_(alpha/2)` gives the area (probability) that the error...

MathWe are given a table of values: (1,16.5),(2,18.3),(3,21.8),(4,25.6),(5,29.2),(6,32.6),(7,32.6),(8,32.4),(9,30.3),(10,26.4),(11,21.3),(12,18.0) where the independent variable is the month and the...

MathTo do this question we need to know the following combination. (a^2)^m = a^(2m) `(4^(1/2*x)2^x)` `= (2^2)^(1/2*x)2^x` `= 2^(2*1/2*x)2^x` `= 2^x2^x` ` = 0` `int (4^(1/2*x)2^x) dx` = `int 0...

MathWhen an investment compounds semiannually, then investment grows according to the formula `A=P(1+r/2)^{2t}` where P is the principal, r is the interest rate pa and t is the number of years of...

MathEvaluate `1.0698 : 0.001` (1) Using long division  the divisor is 0.001, and the dividend is 1.0698 To use long division, it is usually simplest to make the divisor a whole number. In order to...

MathThe water company charges an initial fee of $40, which covers the first 4000 gallons, and $.005 per gallon beyond that. The monthly charge would be C=.005(x4000)+40 or C=.005x+20 where x is the...

MathSketch the graph of `y=2cos(xpi/4)` on the interval `[0,2pi]` : The base graph is `y=cosx` . There is a vertical stretch of factor 2 (The amplitude is doubled) and a horizontal shift of `pi/4`...

MathThe three types of row operations are: 1) switching two rows2) multiplying a row by a constant3) adding a multiple of one row to another row To switch two rows, say row 1 and row 2, the matrix...

Mathis its equation

Matha(x) = 5 sin x b(x) = 4 cos(2x+pi/3) Combine wave is a(x)+b(x) a(x)+b(x) `= 5 sin x+ 4 cos(2x+pi/3)` `= sqrt(5^2+4^2)[(5/sqrt(5^2+4^2) )sin x+ 4/(sqrt(5^2+4^2))cos(2x+pi/3)]` `=...

MathThe graph of the given equation will intercept x when y = 0 Simply speaking if `ax^2  8x +4 = 0` has no solutions then the xaxis would not be intercept. For `ax^2  8x +4 = 0` not to have...

MathThe intersection points of any two functions `f(theta)` and `g(theta)` are given by solving the equation formed by making the two functions equal to each other. That is `f(theta)=g(theta)` . Since...

MathThe yintercept is found by setting x=0 and solving for y. For the line `x4y=16` , we find the yintercept by solving the equation (after setting x=0) `4y=16` divide both sides by 4...

MathYou need to post separate posts if you have more than one question. #1: Solve `T(n)=3T(n1)+1` as a first order linear recurrence. The linear recurrence `T(n)=aT(n1)+b` has the solution given by...