
Math
The function we have is f(x) = 2/e^(2x)  3/e^x f(x) = 2/e^(2x)  3/e^x => f(x) = 2*e^2x  3*e^(x) f'(x) = 2*2*e^2x  3*(1)*e^(x) => 4*e^2x + 3*e^(x) f''(x) = 8*e^2x  3*e^(x) 2*f(x) +...

Math
We have to solve 3^(3x3)*5^(x4)=15^2x/5^7 3^(3x3)*5^(x4)=15^2x/5^7 => (3^3x / 3^3)*(5^x / 5^4) = (5*3)^2x / 5^7 => (3^3x / 27)*(5^x) = (5*3)^2x / 5^3 => (3^3x / 27)*(5^x) = (5*3)^2x /...

Math
We have to simplify 1/(1+ sqrt 2) + 1/(sqrt 2 + sqrt 3) +...........+ 1/(sqrt 2000 + sqrt 2001) multiply the numrator and denominator of each term of the form 1/(sqrt a + sqrt b) by (sqrt a  sqrt...

Math
We have to factor x^5  4x^3  8x^2 + 32 x^5  4x^3  8x^2 + 32 => x^3( x^2  4)  8(x^2  4) factor out x^2  4 => (x^2  4)(x^3  8) use the relation a^2  b^2 = (a  b)(a + b) => (x ...

Math
We have to expand and simplify 3/4(3x+2)2 I hope I have the part in the numerator and the denominator right and the expression is (3/4)(3x + 2)  2 => (3/4)(3x)  (3/4)(2)  2 => 9x/4 ...

Math
We have to simplify 2a/5b times 3a/5b divided 5a/4b. 2a/5b times 3a/5b divided 5a/4b can be written as (2a/5b)*(3a/5b)/(5a/4b) (2a/5b)*(3a/5b)/(5a/4b) writing the terms that form the numerator and...

Math
4*cos(2x)*sin(2x) = 1 First we will divide by 2 . ==> 2cos(2x)*sin(2x) = 1/2................(1) Now we will use trigonometric identities to solve. We know that: sin2x = 2sinx*cosx ==> sin4x =...

Math
We have to find the distance between the points A and B which are the points of intersection of the line of the line r= (8,6,1) + s(2,2,1), and the xz and yzcoordinate planes respectively. The...

Math
A point on the on the y axis has an xcoefficient equal to 0. Let the point on the yaxis which is at a distance of 12/sqrt 13 from 3x  2y + 6 = 0 be (0, y). The distance between a line ax + by +...

Math
2tan^2 x 1 = 0 ==> 2tan^2 x = 1 ==> tan^2 x = 1/2 Now we know that tanx = sinx/cosx ==> tan^2 x = sin^2 x/ cos^2 x ==> (sin^2 x) / (cos^2 x) = 1/2 Now we know that cos^2 x = 1 sin^2...

Math
Let the points we have to determine be (x, y). The distance of a point (x1, y1) from a line ax + by + c = 0 is given by D = a*x1 + b*y1 + c/ sqrt (a^2 + b^2) Here we are given the line 5x  3y =...

Math
We have to determine whether log(4)60 is greater or log(3) 40. Let's consider log(4) 60 first. 4^2 = 16 , 4^3 = 64 as 60 lies between 16 and 64 => 2 < log(4) 60 < 3 Now for log(3) 40 3^2 =...

Math
We have to solve e^(0.5*ln(.25(x^2 + 4))) = 1 e^(0.5*ln(0.25(x^2 + 4))) = 1 use the property that a*log b = log b^a e^(ln(0.25(x^2 + 4))^(0.5)) = 1 e^ln a = a as ln represents logarithm to the base...

Math
We are given that for n terms of a geometric progression, s is their sum, p is their product and r is the sum of their reciprocals. Let the first term of the GP be a and the common ratio be r. The...

Math
If we'll put y = 0, we'll get x^2 >=0, for any real value of x. We'll divide the entire inequality by y^2: (x/y)^2 + 3(x/y) + 4 >=0 We'll put x/y = t t^2 + 3t + 4 >=0 We'll calculate the...

Math
We need to find the real values of a for which f(x)=18x^2  lnx >= a a is the minimum value of the function f(x) = 18x^2  ln x. At the minimum value of f(x) = 18x^2  ln x, f'(x) = 0 and f''(x)...

Math
If (2*sqrt x)(ln x  2) is the antiderivative of f(x) = (ln x) / (sqrt x), we have the derivative of (2*sqrt x)(ln x  2) given by (ln x) / (sqrt x) f(x) = (2*sqrt x)(ln x  2) f'(x) = [(2*sqrt...

Math
We have to solve cos 4x = 1 for values of x that satisfy 0<=x<=2*pi cos 4x = cos 2*2x => 1  2(sin 2x)^2 => 1  2*(2*cos x * sin x)^2 => 1  8*(cos x)^2*(sin x)^2 => 1  8*(1 ...

Math
The complex number we have is z = (8 + i)/(7  4i). z = (8 + i)/(7  4i) => z = (8 + i)(7 + 4i) / (7  4i)(7 + 4i) => z = (56 + 7i + 32i + 4i^2)/(49  16i^2) => z = (52 + 39i)/(49 + 16)...

Math
We have to solve sin 3x = sin x for 0 < x < pi. Use sin(A + B) = sin A cos B + cos A sin B sin 3x = sin (2x + x) => sin 2x * cos x + cos 2x * sin x => (sin x * cos x + sin x* cos x) cos...

Math
Looking the terms of the series given to us we can see that there is a common ratio between consecutive terms equal to (2x/3)/x = (4x/9)/(2x/3) = 2/3. The series is a geometric progression with the...

Math
We have uv=3i+2j and u+v=2i+3j, and we need to determine u^2  v^2. u^2  v^2 = (u  v)(u + v) => (3i + 2j)(2i + 3j) => 6i^2 + 9i*j + 4j*i + 6j^2 i^2 = 1 and j^2 = 1 , i*j = j*i = 0 =>...

Math
We have to find the value of sin(arc sin(1/4)) + cos(2*arc cos(1/4)) sin(arc sin(1/4)) + cos(2*arc cos(1/4)) => sin ( arc sin (1/4)) + 2*(cos (arc cos(1/4))^2  1 => 1/4 + 2*(1/4)^2  1 =>...

Math
First, we'll note this expression by E. E = arctan (1/3)+arctan(1/5)+arctan(1/7)+arctan(1/8) We'll put arctan (1/3) = a and arctan (1/5) = b. We'll subtract arctan(1/7)+arctan(1/8) both sides: E ...

Math
The range of a function f(x) is all the real values that f(x) can take for real values of x. Here we have the function y = cos 2x  4*sin x. To find the range we have to remember that x is the same...

Math
We have to find the value of E = cos x + 2*sin (x/2 + pi/12)*sin(x/2 pi/12) We use the formula 2 sin A sin B =  cos (A + B) + cos (A  B) E = cos x + 2*sin (x/2 + pi/12)*sin(x/2 pi/12)=> cos...

Math
We have to find the definite integral of y=x^3/square root(x^4+1) between the limits of integration: x=0 and x=1 let t = x^4 + 1 dt / 4 = x^3 dx Int [x^3/sqrt(x^4+1) dx] => Int [ (1/4)/ sqrt t...

Math
We'll build another string (bn), having as base of construction the original string (an). The string (bn) is higher than (an). If the limit of the string (bn) exists and it is finite, then this...

Math
We'll rearrange the denominator of the fraction, factorizing by cos x: y = tan x/[(cos x)^2*(sin x/cos x + 1)] But sin x/cos x = tan x y = tan x/[(cos x)^2*(tan x + 1)] We also know that 1/(cos...

Math
9cos(6x) = 11sin(6x) We will square both sides. ==> 81cos^2 (6x) = 121*sin^2(6x) Now we know that sin^2 + cos^2 = 1==> cos^2 = 1sin^ ==> 81(1 sin^2 (6x) = 121sin^2(6x) Open brackets:...

Math
Consecutive terms of a GP have a common ratio. Let the terms we have to find be a, ar and ar^2. The third term is 3.2 times the sum of the first two => ar^2 = 3.2(a + ar) => r^2  3.2r  3.2...

Math
To obtain a random sample you only need to relate each of the samples that you have to choose from with an event that is random in nature. There are innumerable such events, some examples that you...

Math
We have to find the solution for the equation 3.5^(3x + 1) = 65.4 An easy way to determine the value for x is to use logarithms. Take the log of both the sides log 3.5^(3x + 1) = log 65.4 use the...

Math
We need to find the interval of the values of x where the function f(x) = x^3 + (3/2)x^2 + 6x + 27 is increasing. The function is increasing if the value of the first derivative is positive. Here:...

Math
You would measure distance from the front of the vehicle because if you had different cars, measuring from the back would result in a different lengths travelled, even if they all travelled the...

Math
We know, from Pythagorean identity, that: 1 + (tan x)^2 = 1/(cos x)^2 1 + (cot x)^2 = 1/(sin x)^2 We'll rewrite the denominators: 1 + 1 + (cot x)^2 = 1 + 1/(sin x)^2 2 + (tan x)^2 = 1 + 1 + (tan...

Math
We'll have to determine the gradient vector at the point (1 , 1). Since the function is of 2 variables, the gradient of the function is the vector function that is calculated using the formula...

Math
The range of a function is the value that the independent variables can take to give a real value for f(x). Here we have f(x, y) = sqrt [ 100  (x^2 + y^2)]. For f(x, y) to give real values 100 ...

Math
Given the curve y= x^2 + 3x +18 We need to find the x intercepts for y. The xintercepts are the values where the function meets the xaxis. Then the values of y= 0 Let us substitute with y= 0...

Math
Given the equation. x^2 + kx 6 = (x2)(x+3) We need to find the values of k. First we will simplify the right side by opening the brackets. ==> x^2 + kx 6 = x^2 +3x 2x  6 ==> x^2 + kx 6...

Math
Given the exponent equation 2^(3x1) = 16. We need to find the values of x that satisfies the equation. First we will simplify the right side. We know that 16 = 4*4 = 2*2*2*2 = 2^4 Then we will...

Math
Given that 0, 2, and 3 are the zeros of the function f(x). Then x, (x+2), and (x3) are the factors of f(x). Then f(x) = x(x3)(x+2). Now we need to find the solutions for f(x2) Then, we will...

Math
Given that log(x) (1/8) = 3/2 We need to find the values of x. ==> Let us simplify. First we will rewrite (1/8) = (1/2)^3 ==> (1/2)^3 = 2^3 ==> log(x) 2^3 = 3/2 Now we know that log...

Math
Let the first even integer be n. Then the second even integer be n+2 The third even integer is x+ 4 Given that the sum of the first and the third is 131 less than 3 times the second. ==> n +...

Math
Let us assume that the odd integer is n. Then the consecutive odd integer is n+2 Twice the consecutive is 2*(n+2) = 2n+4 Given that the sum of the integer and twice the consecutive is 3757 ==>...

Math
Let the first even integer be x, Then the next even integer is (x+2) and the third is (n+4). Given that the sum of the three even integers is 84. Then we will rewrite into an equation as follow: n...

Math
Let us assume that the first integer is n. Then the next integer is (n+1) and the third integer is (n+2) Given that the sum of the three integers is 366 Then we will write as algabraic expression....

Math
We need to find the sum S = 1 / (cos 15)^2 + 1 / ( sin 15)^2. I assume 15 is in degrees here. 1 / (cos 15)^2 + 1 / ( sin 15)^2 => [(sin 15)^2 + (cos 15)^2]/(cos 15)*(sin 15) => 1/ (cos...

Math
You are only allowed to ask one question at a time. I am responding to your first question. f(x) = tan(x^2 + 1) We need to determine f'(x) f'(x) = [sec (x^2 + 1)]^2 * (2x) The required derivative...

Math
To find the area between the curves, we first need to determine their points of intersection. y = x^2 + 6 and y = (x3)^2 + 4x  7 Equate the two x^2 + 6 = (x  3)^2 + 4x  7 => x^2 + 6 = x^2 +...