
Math
You need to remember that the tangent line to the curve, at a point, is given by the derivative of the function at that point. You also need to remember that `f'(x_0) = m` , where m represents the...

Math
You need to know that the product cancels if one of its factors cancels. Hence, cancelling the first factor yields: `sin 2x =0 =gt 2x = sin^1(0) + n*pi =gt 2x = 0 + n*pi =gt x = npi/2` Cancelling...

Math
We have to find the value of tan (pi/2  arc tan(1/2)) use the relation tan (pi/2  x) = cot x tan (pi/2  arc tan(1/2)) => cot (arc tan (1/2)) => 1/tan ( arc tan (1/2)) => 1/(1/2) =>...

Math
We have to find x that lies in the range 0 < x < 360 such that sin 2x = (1/2)*sqrt 3 sin 2x = (1/2)*sqrt 3 => 2x = arc sin ( sqrt 3/2) => 2x = 60 => x = 60/2 => x = 30 In...

Math
Each of the figures of which the area has to be determined can be divided into rectangles and triangles. The area of a rectangle with sides of length a and b is A = a*b. And the area of a triangle...

Math
Let the point of intersection of x + 3y + 8 = 0 and x  7y + 9 = 0 be (X, Y). This gives us the following set of simultaneous equations to solve for X and Y: X + 3Y + 8 = 0 ...(1) X  7Y + 9 = 0...

Math
We have to find the points of intersection of x^2 + 2y^2 + y  1 = 0 with the line x + 2y = 0. x + 2y = 0 => x = 2y => x= 2y Substitute this in the other equation x^2 + 2y^2 + y  1 = 0...

Math
The point of intersection of the lines 3x + 2 = 18 and x  y = 7 has to be determined. 3x + 2 = 18 => 3x = 16 Substitute x = 16/3 in x  y = 7 => y = 16/3  7 => y = 5/3 The point of...

Math
Let the point be (x,y). Now the slope of the tangent to a point on any curve is the first derivative of the curve. Here y = 1 + 1/x => y' = 1/x^2 Now, the slope of a line between (x1, y1) and...

Math
For the curve y^2 + 4x^2 = 8, the slope of the tangent to the curve at x = c is given by `dy/dx` at x = c. Using explicit differentiation, `2y*(dy/dx) + 8x = 0` => `dy/dx = (8x)/(2y) =...

Math
The point on the curve 3x^2 + 2y = 8 is required where the tangent makes an angle of 45 degrees with the xaxis. The slope of the tangent is tan 45 = 1. The slope is also the value of `dy/dx` at...

Math
The matrix `A = [[2,1],[1,4]]` In the elementary matrix `E_1` the first row if A is multiplied by `1/6` . `E_1` = `[[1/3,1/6],[1,4]]` `E_1^1` = `1/(4/3 + 1/6)*[[4, 1/6],[1, 1/3]]` =...

Math
The center of mass of the rectangle (1,3) (4,3) (4,5) (1,5) has to be determined. The diagram of the rectangle is: The coordinates of the center of mass of the rectangle is given by: `M_x =...

Math
You need to notice that the function will have two vertical asymptotes at `x = +1` , since the function is not defined at `x = +1.` You need to check if the function has maximum and minimum...

Math
You need to evaluate the function `(fog)(x)` using the property of composition of two functions, such that: `(fog)(x) = f(g(x))` Replacing g(x) for x in equation of the function f(x) yields:...

Math
Let the time taken for B to pass A be t. Then when B passes A, has travelled 1000 m more than A. Therefore, Distance travelled by B = Distance travelled by A +1000 To find distance travelled by B,...

Math
`lim_(x>5) arctan((x^225)/(5x^225x))` We can simplify the rational expression by factoring `(x^225)/(5x^225x) = ((x5)(x+5))/(5x(x5))=(x+5)/(5x)` as long as `x!=5` So `lim_(x>5)...

Math
The plane through the points A(2, –4, 0), B(–4, 0, –4), C(–1, 4, 1) has to be determined. `vec (AB)` = [(–4) – 2]i + [0 – (–4)]j + [(–4) – 0]k = 6i + 4j  4k `vec(AC)` = [(–1) – 2]i + [4 – (–4)]j...

Math
Cedric has 4 pencils, a blue pen, and a black pen on his desk. He draws 2 of them from the desk without replacement. The total number of ways of picking 2 items is C(6,2) = 15. The number of ways...

Math
To get the phase shift and vertical shift, you must get the equation in the following form: `f(x) = acos(b(xphi))+ c` Where a denotes the amplitude, b denotes the factor by which we divide `2pi`...

Math
The question does not give all the information needed to answer it, but I will try and shed some light on this topic. A segment that has a slope of eight over zero is going to be vertical. The...

Math
The period of the function f(x) = sin x is `2*pi` . For the function `y = 2*sin(3*A)` the period is `2*pi/3` The period of the function `y = 2*sin (3*A)` is `(2*pi)/3`

Math
You should remember that the period of tangent function is `pi` . Your tangent function is `tan (x/2)` , therefore you should divide the period pi by `x/2` . Use the notation p for period. `p =...

Math
The period of a function f(x) = sin x is `2*pi` For sin(x/4) the period is 2*pi*4 = `8*pi` The period of the function `y = sin (x/4)` is `8*pi`

Math
The period of the function tan(2*x + 7) has to be determined. For f(x) = tan x, the period is `pi` . If the period of the function is A, `2*(x + A) + 7 = 2*x + 7 + pi` => `2*x + 2*A + 7 = 2*x +...

Math
We have the graph y = 2sinx(2pix) + 1 a) the period is 1  this is the length of one complete phase b) the amplitude is 2  this is the height of the peaks from the centre of the wave c)...

Math
We have to prove [(tan x)^2 + 1]/[1  (sin x)^2] = (sec x)^4 Start from the left hand side [(tan x)^2 + 1]/[1  (sin x)^2] tan x = sin x / cos x => [(sin x/cos x)^2 + 1]/[1  (sin x)^2] =>...

Math
Given the right angle triangle with two sides 3 and 4 To find the perimeter first we will need to determine the length of the third side which is the hypotenuse. We will use the formula to find the...

Math
Let the vertices of the triangle be : A(1,1) B( 2,0) and C ( 4,2) First we will calculate the length of the sides: AB = sqrt( 2+1)^2+ ( 01)^2 = sqrt(3^2 + 1) = sqrt( 10) ==> AB...

Math
`sin^2 (pi/6)  2sin (pi/6) cos (pi/6) + cos^2 (pi/6) ` First, we know that: `cos(x) = cos(x) ` `==gt cos^2 (pi/6) = cos^2 (pi/6)` Also, we know that: `2sinxcosx = sin2x` `==> 2sin (pi/6) cos...

Math
It is possible to write `cos(pi/2  x) = sin (x)` as the two trigonometric functions are shifted along the xaxis by `pi/2` `cos (pi/2) = 0` `cos(pi/2)  cos x = cos x = cos x` The right hand side...

Math
When simple interest is being calculated, the interest I is given by the formula I = P*r*t where P is the principle amount, r is the rate of interest per period and t is the number of periods....

Math
You should use the following logarithmic identity such that: `log_a b = (log_c b)/(log_c a)` `log_4 x = (log_2 x)/(log_2 4) => log_4 x = (log_2 x)/2` Substituting `(log_2 x)/2` for `log_4 x...

Math
Here we have log(x) 2 = 0.5, where the base of the log is x. log(x) 2 = 0.5 use the relation that log(a) b = c => b = a^c => 2 = x^(1/2) => 2^2 = x => x = 4 The solution is x = 4.

Math
490 g as a percentage of 7 kg is equal to `((490/1000)/7)*100` = `(0.49/7)*100` = `49/7` = 7 490 g is 7% of 7 kg.

Math
The predicted yield of the reaction is 21 g but only 3.8 g is recovered. This gives the percentage yield as `(3.8/21)*100 % ~~ 18.095 %` The percentage yield is approximately 18.095%

Math
The problem requires finding the integral of 16x * sin^3(2x^2 + 1)*cos (2x^2 + 1) with substituting 2x^2 + 1 with u Int [16x * (sin (2x^2 + 1))^3*cos (2x^2 + 1) dx] let u = 2x^2 + 1 du = 4x * dx...

Math
The percentage of 760 that 950 is can be determined by evaluating `(950/760)*100 = = 1.25 = 125 %` 950 is 1.25 or 125% of 760

Math
We have the present value and the future value in 3 years time so we can use the compound interest formula: F= P `(1+i)^n` where F is the future value (Rs 7290) P is the present value (Rs 5120) i...

Math
It seems that math is a difficult class in which to address learning styles. My memory of math classes is that they build on each other. Most of these math classes seem very linear in structure...

Math
We have to prove that (x+y)^5  (5yx^2+5xy^2)(x^2+xy+y^2) = x^5 + y^5. (x + y)^5 = x^5 + 5*x^4*y + 10*x^3*y^2 + 10x^2*y^3 + 5*x*y^4 + y^5 (5yx^2+5xy^2)(x^2+xy+y^2) = 5yx^4 + 5y^2x^3 + 5y^3x^2 +...

Math
`sqrt(18)/ sqrt(2)` rationalize the denominator (which will eliminate square root denominator) = `sqrt(18)/ sqrt(2) times sqrt(2)/sqrt(2)` = `sqrt(36)/ 2` A square root can be positive or negative...

Math
(a) 6x=27+3x Subtract 3x from both sides of the equation 9x=27 Divide both sides of the equation by 9 x=3 The solution is x=3: Check 6(3)=18=27+9=27+3(3)...

Math
One could get `4/3` from 4 by multiplying 4 by `1/3` : `4*1/3 = 4/3` Likewise, one could get `4/9` from `4/3` by multiplying by `1/3` : `4/3 * 1/3 = 4/9` If we multiply this by `1/3` ,...

Math
The partial fraction decomposition of ` 3/(x  3)^2` has to be determined. Let `3/(x  3)^2 = A/(x  3) + B/(x  3)^2` => `3/(x  3)^2 = (A(x  3))/(x  3)^2 + B/(x  3)^2` => 3 = Ax  3A +...

Math
You need to remember the form of the equation of the quadratic: `f(x) = ax^2 + bx + c` Hence, you need to calculate the parameters a,b,c. You should consider the given facts: `f(1)=4; f'(1)=4 ;...

Math
The difference of two cubes is `x^3a^3=(xa)(x^2+xa+a^2)` So `p^38=(p2)(p^2+2p+4)` Factoring `p^2+2p+4` using the quadradic formula is...

Math
We can solve for x and y using trigonometry. Here, tan 60 = y/x = √3 or, y = √3 x Similarly, tan 30 = y/(1000+x) = 1/√3 . or, √3 y = 1000 + x Substituting the value of y from previous equation, we...

Math
Since we have 2^x, we will be using ln which stands for logarithm base e. If we apply ln to both sides of the given equation we get: `ln(e^x)=ln(20)` logarithm and exponential with the same base...

Math
You need to remember the formula of `sin 2a = 2sin a*cos a` Hence, you need to prove that: `2 sin a* cos a = (2 tan a)/(1 + tan^2 a)` I suggest you to form the tangent function to the left side...