
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)...

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Please look at the pictures attached.

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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)...

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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...

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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...

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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...

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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...

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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...

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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...

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We are asked to graph `f(x)=(x1)^3(x+4)^2(x2) ` : First we will use only techniques from Algebra. There are a few key attributes of graphs of polynomials we are interested in: end behavior (how...

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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...

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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...

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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...

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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...

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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.

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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.

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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.

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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.

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We are going to make a tangent line of the form eq. (1): y(x)=m*x +b Such that y(x) goes through point (1,2). Therefore y(1)=2 must be satisfied. Here m is the slope that is tangent at point...

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...

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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...

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We must use the following relationship for a complex number `z` . `sin(z)=(e^(iz)e^(iz))/(2i)` `sin(pi/2+i ln(2))=sin(z)=(e^(i(pi/2+i ln(2)))e^(i(pi/2+i ln(2))))/(2i)`...

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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...

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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`...

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The complex number `z` has the form `z=x+iy` . In the complex plane where `x` is the real axis and `y` is the imaginary axis we have `x=1` and `y=1` . Find the hypotenuse....

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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...

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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,...

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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....

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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...

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Take the derivative of `P(v)` . `P'(v)=17v^2+3*10^3v^2` Set `P'(v)` equal to zero and solve for the critical values. `17v^2+3*10^3v^2=0` `17+3*10^3v^4=0` `3*10^3v^4=17` `v^4=17/3*10^3`...

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First, when we complete the square for a quadratic function of the form `f(x)=ax^2+bx+c` we get it into the form `(x+b/2)^2(b/2)^2+c` and then simplify. To do this we must add and subtract (to...

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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...

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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...

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Solve `1/2 x 3 lt=4` First, the absolute value means `4lt= 1/2 x 3 lt=4` Now continue to solve the inequality by adding `3` to all sides. `1lt= 1/2 x lt=7` Multiply by `2` . `2lt= x lt=14`...

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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=...

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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...

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`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...

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Gabriel's Horn is made by revolving the function `f(x)=1/x` about the xaxis, with the domain `1lt= x` . The volume can be found by slicing the horn up into infinitesimal circles of radius `f(x)`...

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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...

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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...

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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...

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A company manufactures headphones for $35. They want to make a profit of 24%, so they want to know how much they should charge for the headphones. One way to approach this problem is to reason...

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A fourth order taylor polynomial `p_4(x)` about `x_0=0` has the form `p_4(x)=y(x_0)+y'(x_0)x+(y''(x_0))/(2!)x^2+(y'''(x_0)) /(3!)x^3+(y^((4))(x_0))/(4!) x^4` We need the values `y(0)` , `y'(0)` ,...

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The formula for the Taylor polynomial of degree `n` centered at `x_0` , approximating a function `f(x)` possessing `n` derivatives at `x_0` , is given by...

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Solve `int 8/(16x^2)dx` Factor the denominator and pull the `8` outside the integral. `=8int 1/((4x)(x+4))dx` Preform partial fraction decomposition on `1/((4x)(x+4))` ....

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Your data set is [(0,0),(1,2),(2,16),(3,52),(4,118),(5,223)]. Determine the values of a and b...
We are given the data set (0,0),(1,2),(2,16),(3,52),(4,118),(5,223) and we are asked to fit these in a power model `y=ax^b ` : (1) The easiest way is to input the data in Excel or a graphing...

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There are probably very few people, other than math teachers, who routinely solve calculus problems on a daily basis. (Most of the applied fields would use computers, especially since most models...

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There are two questions here. I am answering the first of the questions (number 3 on the worksheet) as the policy is one question per post. The length AB is equal to the radius of the circles...

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Large numbers can be expressed in standard form, also known as scientific form, by using positive integral exponents.A number is in standard form when it is written as `a*10^n` , where a is a...