
MathMake a substitution `y = 2x,` then `dy = 2 dx` and the limits of integration for `y` are from `0` to `3.` The integral becomes the table one: `int_0^3 1/(1+y^2) (dy)/2 = 1/2 (tan)^(1)(y) _(y=0)^3...

MathMake a substitution `x = 3y,` then `dx = 3 dy` and the limits of integration for y are from 0 to 1/18. The integral becomes a table one: `int_0^(1/18) (9 dy)/(sqrt(99y^2)) = int_0^(1/18) (3...

MathHello! It is obvious that there is no linear formula exactly connecting these x's and y's, if we consider the slopes between neighbor points: `(155  134)/(31  30) = 21,` `(165  155)/(33  31) =...

MathHello! Suppose that the two numbers are both >=2 (if one of the numbers is 1, the problem is trivial). Then both have its unique decomposition into prime factors. If we find this decomposition,...

MathParent functions are the most basic, or "simplest" form of a given function with no transformations placed upon them. For example, `y=4x^2+2x+5` `is a quadratic function. The parent is: ` `y=x^2`...

MathAn operation follows the Commutative property if changing the order of the numbers in the operation does not change the outcome. Thus, addition (and multiplication) are commutative (i.e., a+b=b+a...

MathFor a stem and leaf plot, the number to the left of the bar is generally the stem and the numbers to the right of the bar are the leaves. The stem is the first digit(s) and the leaves are the next...

MathTo do this, the longest rod would be the diagonal of the box. Now, this is different than the diagonal of one side of the box. I am talking about the diagonal that would go through the middle of...

MathWell, at some point, all subjects can be related. For instance, you could use math quite easily in social sciences, studying peoples and populations, for instance. The "variable" would be how you...

MathThis is a typical problem involving percents. There are a few different way to approach such problems. Recall that percent is a numerator (top) of a fraction with the denominator (bottom) equal to...

MathThis all depends, on all kinds of factors. For instance, even if each plant is planted separately, away from each other, they may be so far away from each other than one may be more/less sun...

MathIn order to factor this expression, first open all parenthesis. The given expression will then become `a^2b + ab^2 + b^2c +bc^2 + ac^2 + a^2c + 3abc` . We need to group these terms in a way so that...

MathWhat does the variable n in mathematics usually mean? The variable can mean many things. 1) Maybe the variable is used in an equation only to represent an unknown, such as: `10 = 6n+4` `2) The...

Math`y=3ln(6t+1)` To take the derivative of this, refer to formula: `d/(dx) (ln u) = 1/u * (du)/dx ` Applying that, the derivative of the function will be: `d/(dt)(y) = d/(dt)[3ln(6t+1)]`...

MathThe general methods are the disk method and the shell method. The disk method adds disks with the radius the height of the representative rectangle from the axis to the curve and height either dx...

MathHello! We'll use an identity `a^3  b^3 = (a  b)(a^2 + ab + b^2)` in the form `a  b = (a^3  b^3) /(a^2 + ab + b^2)` for `a = root(3)((n+1)^2)` and `b = root(3)((n1)^2).` This way we...

MathThe given function is: `h(t) = 20t5t^2` where h(t) represents the height of the stone above the cliff. Since the cliff is 60m above the sea, when the stone hits the beach, the value of h(t) is...

MathIn solving trig, or other equations, finding the "roots" means to find the solutions. This also means find the values when "theta" or "x" is zero. Also known as the xintercepts. Trig equations...

MathGiven the vertices of a right triangle are A(1,0), B(7,0) and C(1,8), how would you find the area...The area of a right triangle can be found using the formula `A = 1/2 ab` , where a and b are the lengths of the sides that form the right angle. This is an application of a more general formula...

MathFor this one, I always go back to the lines of multiples. For instance, for 6, the line of multiples would be:6, 12, 18, 24, 30, . . .And, you are trying to think of another one where the LCM...

MathYou would add area of all the sides up. Now, that's literally the formula I like to consider. And, that will work for any noncircular prism, period. Now, special formulas do exist for specific...

MathHello! Actually, the probability density of the normally distributed random variable is known. The probability in question is the integral of this probability density over the given interval....

MathThe work done by the force field in moving the particle along a path is a circulation, or line integral, of this force field around the path. The circulation is defined as `W = int_C vecF* dvecs`...

MathHello! The domain of this function is all real numbers, because `tan^(1)(x),` and therefore `tan^(1)(x+1),` is defined for all `x in RR.` By the definition, `tan^(1)(y)` is the number `w in...

MathWe have to find the square root of `1+2i` i.e. `\sqrt{1+2i}` We will find the square roots of the complex number of the form x+yi , where x and y are real numbers, by the following method: Let...

MathDetermine the area "S" which the triangle "R" Project vertically upon the hemisphere. Check the... (1 image)Hello! If a surface is given as an image of a scalar function `y=f(x,z),` defined on some region `D` on `(x,z)` plane, then the corresponding surface area is `int int_D sqrt(1+((del f)/(del...

MathGreen's theorem (1 image)Hello! 5b. Green's theorem gives us a possibility to compute the area of a plane region integrating along its boundary. Actually, it can help for more complex tasks then computing area. There are...

Matha) A curl is the vector derivative of a vector field. It can be denoted as `vec grad xx vec F` , where `vec F` is the vector field. ` The curl is calculated as threedimensional determinant: i...

MathObtain the line integral (1 image)Given `f=x^2z ds` `x=cost, y=2t, z=sint ` for `0<=t<=\pi` We have to find the line integral i.e. `\int_{c} f(x,y,z)ds=\int_{c} x^2z ds` = `\int_{c} f(x(t),y(t),z(t))....

Matha) Divergence of a vector field is a scalar quantity that represents how the field spreads out, or "diverges", in different directions. It is usually denoted as `vecgrad*vecF` and is calculated as...

MathMatch equations "a" though "d" with vector fields "A" though "D", as shown on the attachment. (1 image)In order to match these twodimensional vector field with their equations, try to predict how the x and y components of each field will vary with the values of x and y, and then check which of...

MathHello! Actually, such a situation is typical. If `A` is a subset of `B` and `B` is a subset of `C,` then `A` is a subset of `C` (any element of `A` is an element of `B` and thus is an element of...

MathHello! The original number may be written as `bar(ab),` where `a` and `b` are onedigit natural numbers, `0lt=blt=9,` `1lt=alt=9.` The value of the number `bar(ab)` is obviously `b + 10a.` When we...

MathHello! When we define the square root function, the function which we start with is a square function `(y=x^2).` We want a function which, given an `y,` would return an `x` such that `x^2=y.` In...

MathHello! The resultant ground velocity of a jet is a vector sum of the jet's air velocity and wind ground velocity. To find its magnitude and direction, we consider the projections on the NS and WE...

Math`F(x) =int_0^(e^(2x)) ln(t+1)dt` `F'(x)=?` Take note that if the function has a form `F(x) = int_a^(u(x)) f(t)dt` its derivative is `F'(x)=f(u(x))*u'(x)` Applying this formula, the derivative of...

Math`F(x)=int_pi^(lnx) cos(e^t)dt` `F'(x)=?` Take note that if the function has a form `F(x)=int_a^(u(x)) f(t)dt` its derivative is `F'(x)=f(u(x))*u'(x)` Applying this formula, the derivative of the...

MathFind the derivative if `y=e^(2x)tan(2x) ` : Let u=2x and rewrite as: `y=e^utanu ` Use the product rule noting that `d/(du)e^u=e^udu,d/(du)tanu=sec^2udu ` : `(dy)/(du)=e^udutanu+e^usec^2udu ` Since...

MathFind the derivative if `y=e^x(sinx+cosx) ` : Use the product rule: `(dy)/(dx)=e^(x)(cosxsinx)+e^(x)(sinx+cosx) ` `(dy)/(dx)=2e^xcosx `

MathFind the derivative if `y=(e^(2x))/(e^(2x)+1) ` : Use the quotient rule to get: `(dy)/(dx)=((e^(2x)+1)(2e^(2x))(e^(2x)*2e^(2x)))/(e^(2x)+1)^2 ` `(dy)/(dx)=(2e^(4x)+2e^(2x)2e^(4x))/(e^(2x)+1)^2 `...

MathFind the derivative if ` y=(e^x+1)/(e^x1) ` : Use the quotient rule to get: `(dy)/(dx)=((e^x1)(e^x)(e^x+1)(e^x))/(e^x1)^2 ` `(dy)/(dx)=(e^(2x)e^xe^(2x)e^x)/(e^x1)^2 ` `...

MathWe are asked to differentiate `y=(e^xe^(x))/2 ` : We use the fact that if f(x), g(x) are differentiable functions of x then `d/(dx)(f(x)+ g(x))=d/(dx)f(x)+d/(dx)g(x) ` and ` d/(dx)e^u=e^u...

MathFind the derivative for `y=2/(e^x+e^(x)) ` : Using the quotient rule we get: `(dy)/(dx)=(2(e^xe^(x)))/(e^x+e^(x))^2 `

MathFind the derivative of `y=ln((1+e^x)/(1e^x)) ` : Use a property of the natural logarithm to rewrite as: `y=ln(1+e^x)ln(1e^x) ` If u is a differentiable function of x, then ` d/(dx)ln(u)=(du)/u `...

MathFind the derivative if ` y=ln(1+e^(2x)) ` : If u is a differentiable function of x then ` d/(dx)lnu=(u')/u, d/(dx)e^(u)=e^u *u' ` so we get: `(dy)/(dx)=(2e^(2x))/(1+e^(2x)) `

MathFind the derivative if `y=e^(3/t^2) ` : If u is a differentiable function of x then `d/(dx)e^u=e^u (du)/(dx) ` . Note that `d/(dt) 3/t^2=6/t^3 ` so we get: `(dy)/(dx)=6/t^3e^(3/t^2) `

MathFind the derivative if `y=(e^(t)+e^t))^3 ` : Use the power rule ( `d/(dx)u^n=n*u^(n1)(du)/(dx) ` where u is a differentiable function of x) to get: `(dy)/(dt)=3(e^(t)+e^t)^2(e^(t)+e^t) `

MathFind the derivative if ` y=x^2e^(x)` : Use the product rule to get: `(dy)/(dx)=x^2(e^(x))+2xe^(x) ` `(dy)/(dx)=e^(x)(2xx^2) `

MathFind the derivative if `y=x^3e^x ` : Use the product rule to get: `(dy)/(dx)=x^3e^x+3x^2e^x ` `(dy)/(dx)=e^x(x^3+3x^2) `

MathFind the derivative if `y=xe^(4x) ` : If u is a differentiable function of x then ` d/(dx)e^u=e^u (du)/(dx) ` , so using the product rule we get: `(dy)/(dx)=x(4)e^(4x)+e^(4x) `...
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