
MathWe need to find the equation of the graph that passes through the point (0,2). The slope of the tangent line to the graph of y(x) is the derivative y' and it is given by the equation `y' = x/(4y)`...

MathThis differential equation can be solved by separating the variables. `(dr)/(ds) = e^(r  2s)` Dividing by e^r and multiplying by ds results in the variables r and s on the different sides of the...

MathFor the given problem:` yy'2e^x=0` , we can evaluate this by applying variable separable differential equation in which we express it in a form of `f(y) dy = f(x)dx` . Then, `yy'2e^x=0` can be...

MathFor the given problem: `12yy'7e^x=0` , we can evaluate this by applying variable separable differential equation in which we express it in a form of `f(y) dy = f(x)dx` . Then, `12yy'7e^x=0` can...

MathFor the given problem: `yln(x)xy'=0` , we can evaluate this by applying variable separable differential equation in which we express it in a form of `f(y) dy = f(x)dx` . to able to apply direct...

MathTo be able to evaluate the problem: `sqrt(14x^2)y'=x` , we express in a form of `y'=f(x)` . To do this, we divide both sides by `sqrt(14x^2)` ....

MathThe general solution of a differential equation in a form of` f(y) y'=f(x)` can be evaluated using direct integration. We can denote `y'` as `(dy)/(dx) ` then, `f(y) y'=f(x)` `f(y)...

MathThe general solution of a differential equation in a form of `y' = f(x) ` can be evaluated using direct integration. The derivative of y denoted as` y'` can be written as `(dy)/(dx)` then `y'=...

Math`(dr)/(ds)=0.75s` This differential equation is separable since it has a form `N(y) (dy)/dx=M(x)` And, it can be rewritten as `N(y) dy = M(x) dx` So separating the variables, the equation...

MathThe general solution of a differential equation in a form of can be evaluated using direct integration. The derivative of y denoted as `y'` can be written as `(dy)/(dx)` then `y'= f(x) ` can be...

Math`(dy)/dx = (6x^2)/(2y^3)` This differential equation is separable since it can be rewritten in the form `N(y) dy=M(x) dx` So separating the variables, the equation becomes `2y^3dy = (6x^2)dx`...

Math`(dy)/dx = 3x^2/y^2` This differential equation is separable since it can be rewritten in the form `N(y)dy = M(x) dx` So separating the variables, the equation becomes `y^2dy = 3x^2dx` Taking the...

Math`(dy)/dx = x/y` This differential equation is separable since it can be rewritten in the form `N(y)dy = M(x)dx` So separating the variables, the equation becomes `ydy = xdx` Integrating both...

MathFormula for compounding n times per year: `A=P(1+r/n)^(nt)` Formula for compounding continuously: `A=Pe^(rt)` A=Final Amount P=Initial Amount r=rate of investment expressed as a percent...

MathFormula for compounding n times per year `A=P(1+r/n)^(nt)` Formula for compounding continuously `A=Pe^(rt)` A=Final Amount P=Initial Amount r=rate of investment expressed as a decimal n=number of...

MathThe formula in compounding interest is `A = P(1 + r/n)^(n*t)` where A is the accumulated amount P is the principal r is the annual rate n is the number of compounding periods in a year, and t is...

MathThe formula in compounding interest is `A = P(1 + r/n)^(n*t)` where A is the accumulated amount P is the principal r is the annual rate n is the number of compounding periods in a year, and t is...

MathThe formula in compounding interest is `A = P(1 + r/n)^(n*t)` where A is the accumulated amount P is the principal r is the annual rate n is the number of compounding periods in a year, and t is...

MathThe formula in compounding interest is `A = P(1 + r/n)^(n*t)` where A is the accumulated amount P is the principal r is the annual rate n is the number of compounding periods in a year, and t is...

MathFormula: `y=Ce^(kt)` `1/2C=Ce^(k*1599)` `1/2=e^(k*1599)` `ln(1/2)=1599klne` `ln(1/2)=1599k` `ln(1/2)/1599=k` `k=4.3349x10^4` `y=Ce^(kt)` `y=Ce^[(4.3349x10^4)(100)]` `y=.9576C` Final...

Math`(dy)/dx=58x` This differential equation is separable since it has a form `N(y) (dy)/dx=M(x)` And, it can be rewritten as `N(y) dy = M(x) dx` So separating the variables, the equation...

Math`(dy)/dx = x + 3` This differential equation is separable since it can be rewritten in the form `N(y)dy = M(x) dx` So separating the variables, the equation becomes `dy = (x+3)dx` Integrating both...

MathIn order to use integration to solve this differential equation, multiply both sides of the equation by dx: `dy = (10x^4 2x^3)dx` . Now we can integrate both sides, using the formula for the...

MathHello! Probably we need to find all the terms of this progression. Recall that each next term of a geometric progression is obtained by multiplying the previous term by the quotient, denote it as...

MathThis question requires you to set up a system of equations. First, you must identify your variables. Since we want to know how many adult tickets and how many child tickets were bought, those...

MathHello! Actually, the function `x` (absolute value of `x` ) is defined as a piecewise function: `x = {(x if xgt=0),(x if xlt0):}` There are two "pieces" on which this function is defined using...

MathAn expression represents a rational number if the simplified result yields a number of the form`p/q` where p and q are integers and `q!=0` A square root is rational only if the expression in the...

MathWe can compare/contrast areas of a semicircular region (involving the arcsin function) to areas under a hyperbola. The hyperbolic functions arise from this second application. Other applications...

MathHello! I suppose that the age of each child is a natural (whole) number. Then from (b) we get that Alejandra's age is a natural number greater than 8 but less than 10. There is the only such...

MathGiven `dy/dx = 1/sqrt(4x^2) ` and `y(0) = pi` We have to find y. So we can write, `dy=dx/sqrt(4x^2)` Integrating both sides we have, `y=int dx/sqrt{4x^2)+C` Now let `x=2sint ` So, `dx=2cost...

MathRecall that` int f(x) dx = F(x) +C` where: f(x) as the integrand function F(x) as the antiderivative of f(x) C as the constant of integration.. For the given problem, the integral: `int...

MathComplete the square at the denominator: `x^4 + 2x^2 + 2 = (x^2)^2 + 2x^2 + 1 + 1 = (x^2 + 1)^2 + 1.` Now we see the substitution `y = x^2 + 1,` then `dy = 2x dx,` and the integral becomes `int...

MathWe have to evaluate the integral: `\int \frac{1}{(x1)\sqrt{x^22x}}dx` We can write the integral as: `\int \frac{1}{(x1)\sqrt{x^22x}}dx=\int \frac{1}{(x1)\sqrt{(x1)^21}}dx` Let `x1=t` So...

MathRecall that `int_a^b f(x) dx = F(x)_a^b` : `f(x)` as the integrand function `F(x) ` as the antiderivative of `f(x)` "a" as the lower boundary value of x "b" as the upper boundary value of x To...

MathTo evaluate the given integral:` int 2/sqrt(x^2+4x)dx` , we may apply the basic integration property: `int c*f(x)dx= c int f(x)dx` . The integral becomes: `2 int dx/sqrt(x^2+4x)` We complete the...

MathBy completing the square and making simple substitution, we will reduce this integral to a table one. `x^24x = (x^2 + 4x + 4) + 4 = (x+2)^2 + 4 = 4  (x+2)^2.` Now make a substitution `y =...

MathWe have to evaluate the integral:`` `\int \frac{2x5}{x^2+2x+2}dx` We can write the integral as: `\int \frac{2x5}{x^2+2x+2}dx=\int\frac{2x5}{(x+1)^2+1}dx` Let `x+1=t` So, `dx=dt` Now we can write...

MathFor the given integral: `int 2x/(x^2+6x+13) dx` , we may apply the basic integration property: `int c*f(x) dx = c int f(x) dx` . `int 2x/(x^2+6x+13) dx =2 int x/(x^2+6x+13) dx` To be able to...

MathTo evaluate the given integral: `int_(2)^(2)(dx)/(x^2+4x+13)` , we follow the first fundamental theorem of calculus: If f is continuous on closed interval [a,b], we follow: ` int_a^bf(x)dx =...

MathTo be able to evaluate the given integral:` int_0^2 (dx)/(x^22x+2)` , we complete the square of the expression:`x^22x+2` . To complete the square, we add and subtract `(b/(2a))^2` . The...

MathRecall that `(arccos(x))' = 1/sqrt(1x^2)` and make the substitution `y = arccos(x),` then `dy = 1/sqrt(1x^2).` The limits of integration are from `arccos(0) = pi/2` to `arccos(1/sqrt(2)) =...

MathWe have to evaluate the definite integral: `\int_{0}^{1/\sqrt{2}}\frac{arc sinx}{\sqrt{1x^2}}dx` Let `t= arcsinx` Differentiating both sides we get, `\frac{1}{\sqrt{1x^2}}dx=dt`...

MathMake the substitution `u = sin(x),` then `du = cos(x) dx.` The integration limits for `u` are from `sin(0) = 0` to `sin(pi/2) = 1,` and the integral becomes `int_0^1 (du)/(1 + u^2) = arctan(1) ...

MathWe have to evaluate the integral: `\int_{\pi/2}^{\pi}\frac{sinx}{1+cos^2x}dx` Let `cosx=u` So, `sinx dx=du` When `x=\pi/2, u=0` `x=\pi, u=1` So we have,...

MathMake the substitution `y = e^(x),` then `dy = e^(x) dx` and `e^(2x) = y^2.` The limits of integrations for `y` become from `e^(ln2) = 1/e^(ln2) = 1/2` to `e^(ln4) = 1/e^(ln4) = 1/4.` The...

MathFor the given integral problem:` int_0^(ln(5))e^x/(1+e^(2x))dx` , it resembles the basic integration formula for inverse tangent: `int_a^b (du)/(u^2+c^2) = (1/c)arctan(u/c) _a^b` where we let:...

MathMake the substitution `u = sqrt(16x^2  5),` then `16x^2 = u^2 + 5,` `du = (32x)/(2sqrt(16x^2  5)) dx = (16 x dx)/sqrt(16x^2  5),` or `dx/sqrt(16x^2  5) = (du)/(16 x).` The limits of...

MathTo be able to solve for definite integral, we follow the first fundamental theorem of calculus: `int_a^b f(x) dx = F(x) +C` such that f is continuous and F is the antiderivative of f in a closed...

MathMake the substitution `u = sqrt(4x^2  9),` then `du = (4x)/sqrt(4x^2  9) dx.` Inversely, `dx =sqrt(4x^2  9)/(4x) du = u/(4x) du` and `4x^2 = u^2 + 9.` The limits of integration become from...

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...
rows
eNotes
search