
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
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)`...

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
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....

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
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`...

Math
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...

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
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`...

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
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)`...

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...

Math
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...

Math
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)` ,...

Math
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...

Math
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))` ....

Math
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...

Math
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...

Math
We often use degrees to measure angles, where a degree is 1/360th of the angular distance around a circle. (So 90 degrees is 1/4 of the angular measure around the circle.) Radians are another way...

Math
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...

Math
I assume that you are asking about results that either lie outside a confidence interval, or results during a hypothesis test that lie in the critical region (tail.) When creating a confidence...

Math
This is an excellent question to ask, and it happens to be one of those questions where the answer is actually very clear. Statistics wins. Whether you see yourself in a career based in the...

Math
`log_4(128)` To evaluate, factor 128. `= log_4 (2^7)` Then, apply the formula of change base `log_b (a) = (log_c (a))/(log_c (b))` . `= (log_2 (2^7))/(log_2 (4))` `= (log_2 (2^7))/(log_2 (2^2))` To...

Math
`log_125 (625)` To evaluate, factor 625. `=log_125 (5^4)` Then, apply the formula of change base `log_b(a) = (log_c(a))/(log_c(b))` . `= (log_5 (5^4))/(log_5 (125))` `= (log_5 (5^4))/(log_5...

Math
`log_8 (32)` To evaluate, factor 32. `=log_8 (2^5)` Then, apply the formula of change base `log_b(a) = (log_c(a))/(log_c(b))` . `= (log_2 (2^5))/(log_2 (8))` `=(log_2(2^5))/(log_2(2^3))` Also,...

Math
`log_27 (9)` To evaluate, factor the 9. `= log_27 (3^2)` Then, apply the formula of change base `log_b (a) =(log_c(a))/(log_c(b))` . `= (log_3(3^2))/(log_3 (27))` `=(log_3 (3^2))/(log_3 (3^3))` To...

Math
We are asked to graph the following function: `y=log_5(x+1)3` The base function is `y=log_5(x)` and the graph will be a translation of 1 unit left and 3 units down of the graph of the base...

Math
We are asked to graph the function `y=log_6(x4)+2 ` : The graph is a translation of the graph of `y=log_6x ` 4 units right and 2 units up. The domain is x>4 and the range is all real numbers....

Math
We are asked to graph the function `y=log_4(x+2)1 ` : This is a translation of the graph of ` y=log_4x ` 2units left and 1 unit down. The domain is x>2 and the range is all real numbers. There...

Math
We are asked to graph the function `y=log_3x+4 ` : The graph is a translation of the graph `y=log_3x ` up 4 units. Some points on the graph: (1/27,1),(1/9,2),(1/3,3),(1,4),(3,5),(9,6) The domain is...

Math
We are asked to graph the function `y=log_2(x3) ` : The graph is a translation of the graph of `y=log_2x ` 3 units right. Some points on the graph:...

Math
We are asked to graph the following function: `y=log_(1/5)(x)` The domain is x>0 and the range is all real numbers. The graph is decreasing and concave up on the domain. Some points on the...

Math
We are asked to graph the following function: `y=log_(1/3)(x)` The domain is x>0 and the range is all real numbers. The graph of the function is decreasing and concave up on its domain. Some...

Math
We are asked to graph the following function: `y=log_6(x)` The domain is x>0 and the range is all real numbers. The graph is increasing on the domain, and the graph is concave down on the...

Math
We are asked to graph the following function: `y=log_4(x)` The domain is x>0 and the range is all real numbers. The graph increases on its domain and is concave down on its domain. Some points...

Math
`2(log_3 (20)  log_3 (4)) + 0.5log_3(4)` First, apply the differencequotient rule of logarithm `log_b (m/n) = log_b(m)  log_b(n)` . `= 2 (log_3 (20/4))+0.5log_3(4)` `=2log_3(5) + 0.5log_3(4)`...

Math
`ln40+2ln(1/2) + lnx` First, apply the logarithm rule `log_b (a^m)=m*log_b(a)` . `= ln40 + ln(1/2)^2 + ln x` `=ln40+ ln(1/4) + lnx` And, apply the rule `log_b(m*n) = log_bm +log_b n` . `= ln (40 *...

Math
`5log_4(2) + 7log_4(x) + 4log_4(y)` To express this as one logarithm, first apply the rule `log_b a^m = m*log_b(a)` . `= log_4(2^5) + log_4(x^7) + log_4(y^4)` `= log_4(32) + log_4(x^7)+log_4(y^4)`...

Math
We are asked to solve `5^(2x)+20*5^x125=0 ` : Rewrite as ` (5^x)^2+20*5^x125=0 ` and let `y=5^x ` to get ` y^2+20y125=0` and (y+25)(y5)=0 so y=25 or y=5. y cannot be 25 as `5^x>0 ` for all...

Math
We are asked to solve `2^(2x)12*2^x+32=0 ` : Rewrite as `(2^x)^212*2^x+32=0 ` and let `y=2^x ` ; then `y^212y+32=0 ` and (y8)(y4)=0 so y=8 or y=4. y=8 ==> ` 2^x=8 ==> x=3 ` y=4 ==>...