
Math
(a) We have to solve the initial value problem given by: `x'=10y` `y'=10x` with initial conditions: `x(0)=3` `y(0)=4` Now we can write the above problem in matrix form as shown:...

Math
According to the variations of parameters method, the formula for the particular solution of the equation of the form `y''+q(x)y' + r(x) y = g(x)` is `Y_P = y_1 int y_2(x)g(x)/(W(y_1, y_2))...

Math
The divergence theorem states that `int_S (F*hatn) dS = int_V (grad*F) dV,` where `S` is a closed surface, `V` is the volume inside it and `F` is a good enough vector field defined inside `S` and...

Math
This function is odd, thus its Fourier expansions contains only `sin(nx)` terms, i.e. `a_n=0, ngt=0.` This expansion has the form `sum_(n=1)^(oo) b_n sin(nx)` where `b_n = 1/pi int_(pi)^(pi)...

Math
We are asked to graph the polar function `r=sqrt(theta), 0<= theta <= 2pi ` , along with its vertical and horizontal tangents. We can graph by plotting points: `theta: ` r:0...

Math
We are given the following coordinate pairs: (17,36), (27,25), (37,20), (47,12), (57,10), (67,7), and (77,5), where the xcoordinate represents the age in years and the ycoordinate the percent of...

Math
The method of disk or washers is based on breaking up the solid obtained by the revolution of the curve into either thin disks or thin washers (rings), and integrating their area to obtain the...

Math
The method of shells is based on breaking up the solid obtained by the revolution of the curve into thin cylindrical shells, and integrating the surface area of the shells to obtain the volume of...

Math
i, j, k are unit vectors. r(u,v)=(b+a*cos(u))*cos(v) i + (b+a*cos(u))*sin(v) j + a*sin(u) k 0<=u,v<=2pi and a=1/4 and b=1 The surface area is: A=double integral of magnitude of (r_u x r_v)...

Math
Please look at the pictures attached.

Math
Denote the given points: First, find the unit vector orthogonal to (for a plane it is a constant vector). An orthogonal vector is Its unit vector is a) Therefore the equation of the plane is or b)...

Math
For each `x,y` the point `(x, y)` is the point where a vector starts and F(x, y) is a pair of its components. For `x=y=0` we have `F(x,y)=vec0` and we cannot draw it with an arrow because it has no...

Math
First octant means `xgt=0, ygt=0, zgt=0.` The formula `y=27/8 x^3` may be written as `y=(3/2 x)^3.` We need the (simple) inverse formulas, too: `x=2/3 root(3)(y), z=1y/2.` Because `zgt=0` we see...

Math
Hello! As I understand, `r = sqrt(x^2 + y^2).` The volume is the integral by dx, dy and dz of the function that is constantly equal to `1.` To set up the integral we need to understand the limits...

Math
How can I find the probability (in fractions)? S={3,5,6,8,9,12,13,14,15,16} A={3,12,5,13}...
I am assuming that we are selecting one item from a sample space of s={3,5,6,8,9,12,13,14,15,16}. We have event spaces: A={3,5,12,13} B={3,6,14,15} C={5,8,9,12,16} We can compute the probabilities...

Math
In the attached image, the boxplot for Region C is both skewed and has an outlier. In statistics, a boxplot is a way to show numerical data in their quartiles. A boxplot consists of a rectangle...

Math
Hello! Although the range is from `1` to `6,` the mean and the distances from the mean are computed using the vertical values (those below `0.2`). They are relative frequencies (probabilities) of...

Math
Here is a view showing the local maximum:

Math
The negative trend in this scatterplot indicates that the two variables, weight and desired weight change, are negatively correlated. That means as a student's weight goes up, his or her desired...

Math
Well, you can use the information to describe the data. Or, use the data to draw a normal curve, since you know the mean and standard deviation. Now, actually, since these values are suppose to...

Math
We are asked to graph the function `f(x)=1/2(x^22x+15) ` : The graph will be a parabola (f(x) is a quadratic polynomial), opening up (the leading coefficient is positive.) To find the yintercept...

Math
If a computer loses 30% of its value each year, then every year the value of the computer is 70% of its original value. That is, if original value of the computer is $100, then in one year it will...

Math
I will use trigonometric substitution for this integral. First, substitute to eliminate the square root. Then evaluate the integral. Lastly, convert back to the original variable. Please see the...

Math
The volume can be found by using the shell method and multiplying2pi*x and f(x) then integrating over the bounds of x.

Math
This problem is solver by first converting your angle from degrees to radians. Then use the formula for the area of a sector of a circle.

Math
This is an application of the arc length formula. It needs the angle in units of radians. So first convert the angle from degrees to radians. Then apply the arc length formula.

Math
The average rate of change is the slope of the secant line between `x_2` and `x_1` . The slope of the secant line, `m_s`, is found in the attached image.

Math
The given point (1, 2) lies on the graph of the given function: f(1) = 3  5 = 2. In this situation, the equation of the tangent line at this point is y = f'(1)(x  (1)) + f(1). Because...

Math
Hello! This indefinite integral is simple if we note that `sin(6x) dx = 1/6 d(cos(6x)).` Formally, perform the variable substitution `u = cos(6x),` then `du = 6sin(6x)` and the integral...

Math
This is an integral of the form `oint_C f(z)/(za) dz` , where `f(z)=ze^z` and `a=i` in this case. If this meets the conditions of the Cauchy Integral Theorem we can use the Cauchy Integral...

Math
First, use the identity `sin(a + b) = sin(a) cos(b) + cos(a) sin(b),` which is true for any complex `a, b.` We obtain `sin(pi/2 + i ln(2)) = sin(pi/2) cos(i ln(2)) + cos(pi/2) sin(i ln(2)).` It is...

Math
Let the complex number `z=8` . Rewrite this in polar form as `z=8e^(i(0+2pi*n))` . `z^(1/3)=(8e^(i(0+2pi*n)))^(1/3)` `z^(1/3)=2e^(i(0+(2pi*n)/3))` Here, `theta` becomes 3 distinct angles for all...

Math
The volume between two surfaces can be calculated by `V=int int_D (z_(t o p)z_(b o t)) dA` Where `D` is the area contained by the boundary of the volume projected onto the xyplane. To find `D`...

Math
We are asked to write the complex number z=1i in polar form. The polar form of a complex number is `z=r(cos theta + i sin theta)` where r is the distance from the origin (the modulus or absolute...

Math
We are asked to determine the value for n such that the indicated sum is correct to 1 decimal place. `S_n=sum_(n=1)^(oo) n/(n^2+n^3)` The series converges (using the comparison test) and the sum is...

Math
a. For the Ratio Test, we need to examine the ratio of (k+1)th coefficient to kth coefficient, here it is `(1/((k+1)!)) /(1/(k!)) = (k!)/((k+1)!) = 1/(k+1).` The limit of this ratio is 0,...

Math
Your description of the median is correct except for one minor, but very important, point. The list must be ordered in order to compute the median. You were given 15,11,12,3,14,17 as your data set....

Math
Hello! I agree with your answer to the part a), the only possible series is `U_n = 4*3^(n1).` The question b) becomes simple if we recall the formula of the sum of `N` terms of a geometric...

Math
The function is obviously defined only for `v gt 0` and is continuously differentiable on this interval. When `v` approaches zero the function tends to `+oo,` when v tends to `+oo,` the function...

Math
Express `f(x) = x^2  4x + 9` as `x^2  2*x*2 + 2^2  2^2 + 9 = (x  2)^2  4 + 9 = (x  2)^2 + 5.` We used the formula `(a  b)^2 = a^2  2ab + b^2` in the reverse direction. We see that this...

Math
First, know that the vector that is in the direction of the line of intersection `(r)` is the cross product between the normal vectors of the planes since it is perpendicular to each of them. A...

Math
The distance d between two points between `P_1 = (x_1,y_1)` and` P_2 = (x_2,y_2)` is computed using the Pythagorean Theorem. The hypotenuse of a right triangle which has the sides `a = x_2...

Math
To solve this inequality, find where the expression under absolute value sign is nonnegative and where it is negative. `1/2 x  3 gt= 0` for `x gt= 6` and `1/2 x  3 lt 0` for `x lt 6.`...

Math
The closed interval `[a,b]` is the set of all real numbers `x` such that `alt= xlt= b` . This is written as, `[a,b]={x in R alt= xlt= b}` This is usually written this simply as `{x  alt= xlt=...

Math
The volume between two surfaces is `V=int int_D (f_(t o p)(x,y)f_(b o t t o m)(x,y))dA` Where `D` is the region between the two surfaces projected on the xyplane. `f(x,y)` is a parabolic surface...

Math
`V=int int_D z dA` `V=int_0^1 int_x^x (3+x^22y)dydx` `V=int_0^1(3y+x^2yy^2)x^x dx` `V=int_0^1(6x+2x^3)dx` `V=(3x^2+1/2x^4)_0^1` `V=3/2` Below is a plot from `0lt= x lt= 1` , where the red...

Math
We are asked to confirm that the volume of the figure known as Gabriel's Horn is finite. We will use the fact that ` int_1^( oo) (dx)/x^p={[[1/(p1),"if" p>1],["diverges", p<=1]] ` The solid...

Math
The volume underneath a surface `S(x,y)` can be calculated by `V=int int S(x,y) dA` Where the bounds of integration are the area under the curve. Therefore, `V= int int_A z(x,y) dA` `V=int_1^2...

Math
For 95% confidence interval we can calculate the true proportion as; `P = barp+1.96sqrt((barp(1barp))/n)` Where; P = true proportion of perry como fans `barp` = Observed proportion of perry como...

Math
Part (a) Yes, the covariance and correlation coefficient between two variables will always have the same sign. This is easily seen from the definition of correlation coefficient. Let `X` and `Y` be...