
Math
Since the problem provides the coordinates of the vertices, hence, you should evaluate the area of triangle using the following formula: `A = (1/2)*[[x_A,y_A,1],[x_B,y_B,1],[x_C,y_C,1]]` `A =...

Math
You should find the equation of the line that is perpendicular to the segment that connects the point `(2,2)` to the point on the line `y=4` . You need to find first the equation of the segment...

Math
You need to determine the internal dimensions of the tank, converting to cm the given thickness of 0.02 m such that: height = `100  2 = 98 cm` wide = `75  2*2 = 71 cm` deep = `50  2*2 = 46 cm`...

Math
The average rate of change of a variable X over a given duration of time t is the difference between the initial value of the variable `X_0` and the final value of the variable `X_t` divided by t....

Math
Let the distance to Las Vegas is X km. In the return trip the jet took 12 hours at an average speed of 309km/h. Distance = speed * time X = 309km/h*12h = 3708km So for the going trip if the...

Math
f(x)= (tan^4(4x)+sin(7x^34x+1)) Here we need to know the following. `(d(tanf(x)))/dx = sec^2f(x)*f'(x)` `(d(sinf(x)))/dx = cosf(x)*f'(x)` f’(x) `=...

Math
`9cos^2(x)2cos(x)3=0` Let cosx = t Then; `9t^22t3 = 0` `t = [(2)+sqrt((2)^24*9*(3))]/(2*9)` So t = 0.699 OR t = 0.476 If t = 0.699; cosx = 0.699 `cosx = cos 0.253pi` General solution for...

Math
The function f(x) is such that f'(x) = x^3 and the line x + y = 0 is a tangent to the graph of the function. f(x) = `int f'(x) dx` => `int x^3 dx` => `x^4/4 + C` As the line x + y = 0,...

Math
For a function f(x) the tangent at any point is given by f’(x). F’’(x) = 6x F’(x) = `int f’’(x) dx` = `int 6x dx` = `3x^2+C` where C is a constant F(x) = `int f’(x)` dx = `int...

Math
The function f(x) has to be determined such that f''(x) = 5x and f(x) has a relative maximum at (2, 3) f''(x) = 5x f'(x) = `int f''(x) dx` => `int 5x dx` => `(5*x^2)/2 + C` As...

Math
`G(x)=2sin^(1) (sqrtx/2)` `F(x)=sin^(1)((x2)/2)` To determine the derivative of the two functions, use the formula `d/(du)sin u = 1/sqrt(1u^2)*u'` . So G'(x) is: `G'(x) =...

Math
`G(x)=2sin^(1)(sqrtx/2)` (1) To determine G'(x), use the formula `d/(du) sin^(1)u= 1/sqrt(1u^2)*u'` . `G'(x)= 2*1/sqrt(1(sqrtx/2)^2)*(sqrtx/2)'` `G'(x)=2/sqrt(1x/4)*1/2*1/(2sqrtx)`...

Math
1) You need to differentiate the given function with respect to x, using the chain rule, such that: `F'(x) = (1/(sqrt(1  (x2)^2/4)))*((x2)/2)'` `F'(x) = 2/(2sqrt(4  (x2)^2)) => F'(x) =...

Math
`y =x^2+4` The first derivative of a function will give the gradient of the tangent line to the function at the point considered. So let us say the tangent line is y = mx+c It is given that...

Math
The function f(x) = `x + sqrt x` . The slope of the line tangent to f(x) at the point where x = a is given by f'(a). f'(x) = `1 + 1/(2*sqrt x)` At x = 25, f'(x) = `1 + 1/10` The slope of the...

Math
(a) d/dx ((e^(x^3))+(log base 3 of pi)) Let y = ((e^(x^3))+(log base 3 of pi)) We know ; `(d(e^f(x)))/dx = e^f(x)*f'(x)` d(a)/dx = 0 where a is a constant. `dy/dx = e^(x^3)*3x^2+0 =...

Math
`f(x)=2ln(secx+tanx)` To start, take the first derivative of f(x). Use the formula `d/(du) lnu =1/u*u'` . `f'(x) = 2*1/(secx+tanx)*(secx+tanx)'` `f'(x)=2/(secx+tanx)*(secx+tanx)'` Then, take the...

Math
`f(x) = (2sqrtx) lnx` To solve for f'(x), let's use the product formula of derivatives which is `(u*v)' = uv' + u'v ` . So let, `u=2sqrtx` and v`=lnx`...

Math
`f(x) = 3cos(x)*sin^(1)(x)` To determine f'(x), use the product formula of derivatives which is `(uv)' = uv'+u'v` . So let, `u=3cos(x) ` and `v=sin^(1) (x)` To...

Math
`f(x)= sec^(1) (7^x)` To take the derivative of f(x) use the formula `d/(du)sec^(1)u= 1/(usqrt(u^21))*u'` . `f'(x) = 1/(7^xsqrt((7^x)^21))*(7^x)'` `f'(x)= 1/[7^xsqrt(7^(2x)1)]*(7^x)'` To take...

Math
`y= tan^1sqrt((4x^2)1)` `tany = sqrt((4x^2)1)` Derivate both sides with respect to x; `Sec^2y*(dy)/dx = 1/{2*[ sqrt((4x^2)1)]}*8x` `dy/dx = 4x/[ sqrt((4x^2)1)*sec^2y]` `Sec^2y` `= 1+tan^2y`...

Math
The function `f(x)=e^(3x)+7cos x*sin x`. The derivative of f(x) is: f'(x) = `3*e^(3x) + 7*(cos^2x  sin^2x)` => 3*e^(3x) + 7*cos 2x f''(x) = `9*e^(3x)  14*sin 2x` The second derivative of...

Math
Here we need to know the following; If f(x) and g(x) are two function of x then; `(d((f(x)*g(x))))/dx = g(x)*(d f(x))/dx+ f(x)*(d g(x))/dx` `(d(f(x)/g(x)))/dx =...

Math
`f(x) = 8x^5sqrtx+ 4/(x^2sqrtx)` To simplify f(x), express the radicals as exponents. Note that `sqrtx=x^(1/2)`. `f(x) = 8x^5*x^(1/2) + 4/(x^2*x^(1/2))` Then, use the rule of exponents for...

Math
`tan^2 x=3tanx` To start, subtract both sides by 3tanx. `tan^2x3tanx=0` Factor left side. `tanx(tanx3) = 0` Set each factor to zero and solve for x. >> `tan x = 0` `x=...

Math
Solving an inequality is similar to solving an equation except when multiplying or dividing by a negative changes the inequalities. `6(2x5)<165x` distribute on the left side...

Math
A function `h(x)` is even if `h(x)=h(x)` , or odd if `h(x)=h(x)` . Substitute `x` into the function and we get: `h(x)` `={(x)^3}/{6(x)^25}` `={x^3}/{6x^25}` `=h(x)` The function is...

Math
Graph `g(x)=4^(x1)+5` The base function is `y=4^x` The exponent `x1` shifts the base function 1 unit to the right. (Horizontal translation) The term +5 shifts the graph up 5 units. (vertical...

Math
Given the function: `f(x)= 4x^2  x  3` We need to find the domain of the function f(x). The rule of the domain of polynomial is clear. If the function is a polynimial, then, the domain is all...

Math
The average rate of change is the difference in yvalues divided by the difference in xvalues. This means that `{Delta y}/{Delta x}={f(x)f(1)}/{x1}` `={x^3x(1^31)}/{x1}` `={x^3x}/{x1}`...

Math
When reflecting the function in the xaxis, we replace x by x, to shift down by 7 we subtract 7 from the function and to shift left by 8, we replace x by x+8. This means that the function `y=2^x`...

Math
For the function `f(x)=5(x+1)^2+1` , we can identify each transformation from the function `g(x)=x^2` . The 5 is a vertical stretch by a factor of 5, the 1 inside the brackets is a shift of 1 to...

Math
The transformations are applied to the base graph `f(x)=x` . The 4 is a vertical reflection and vertical stretch of a factor of 4. The +2 is a vertical shift up by 2. The transformations can...

Math
The graph of the function `y = tan(x/2) + 2` for `2*pi<=x<=3*pi` is required. The graph of y = tan x is: y = tan x has a graph: y = tan(x/2) has the following graph: And...

Math
The two lines given are 2x  3y = 18 and 6x  9y = 18 2x  3y = 18 => y = (2/3)x  6 6x  9y = 18 => y = (6/9)x  2 = (2/3)x  2 The slope of the lines is the same, therefore they are...

Math
The equation of the line passing through the points (2, 3) and (3, 4) has to be determined. The equation of a line passing through (a1, b1) and (a2, b2) is given by `(y  b2)/(x  a2) = (b1 ...

Math
The graph of the relation x^3 + y^2 + 6 = 0 is required. x^3 + y^2 + 6 = 0 => y^2 = x^3  6 => `y = sqrt(x^3  6)` and `y = sqrt(x^3  6)` The graph of these functions is:

Math
The roots of the cubic equation x^3+3x^246x+72 = 0 have to be determined. x^3+3x^246x+72 = 0 => x^3 + 9x^2  6x^2  54x + 8x + 72 = 0 => x^2(x + 9)  6x(x + 9) + 8(x + 9) = 0 => (x^2 ...

Math
The equation x^4  13x^2 + 36 = 0 has to be solved. This is a biquadratic equation with 4 roots. x^4  13x^2 + 36 = 0 => x^4  9x^2  4x^2 + 36 = 0 => x^2(x^2  9)  4(x^2  9) = 0 =>...

Math
Let the trapezoid be named ABCD with `bar(AB)bar(CD)` . (1) Since the trapezoid is convex, in order for a circle to be inscribable in the trapezoid we must have `AB+CD=AD+BC` (2) The midline...

Math
Notice that the given angle `hatA` is an included angle, hence, you may use the law of cosines to find the missing length such that: `(BC)^2 = (AB)^2 + (AC)^2  2AB*AC*cos 60^o` You need to...

Math
You should regroup the factors such that: `(((sin(pi/16)*cos(pi/16))*cos(pi/8))*cos(pi/4))` Notice that if you multiply `sin(pi/16)*cos(pi/16)` by `2` yields: `2sin(pi/16)*cos(pi/16) =...

Math
You should notice that `x o y = xy  3x  3y + 12` , hence, you need to substitute `x*x  3x  3x + 12` for `x o x` in equation `x o x = x` such that: `x*x  3x  3x + 12 = x => x^2  6x + 12...

Math
You need to evaluate the limit of definite integral such that: `lim_(x>oo) (1/x^2)* int_0^x f(t) dt` Since the problem provides the equation of the function, then you need to substitute `(cos t...

Math
You need to prove the given reccurence relation `I_n +I_(n1) = (a^n)/n` . Supposing that `I_n = int_0^a (t^n)/(t+1) dt` , hence, `I_(n1) = int_0^a (t^(n1))/(t+1) dt` You need to add the...

Math
You should remember how you may find the coordinates of the centroid of a triangle ABC such that: `x_G = (x_A + x_B + x_C)/3` `y_G = (y_A + y_B + y_C)/3` You may find the coordinates of vertices A...

Math
You need to express the general terms of this summation as `n(n+2)` , hence, opening the brackets yields: `n(n+2) = n^2 + 2n` `sum_(n=1)^10000 (n^2 + 2n) = sum_(n=1)^10000 n^2 + sum_(n=1)^10000...

Math
You should change the bases of logarithms using the logarithmic identity `log_a b = 1/(log_b a)` such that: `log_x 2 = 1/(log_2 x)` `log_(sqrt x) 2 = 1/(log_2 sqrt x) => log_(sqrt x) 2 =...

Math
Let, A  # of people in Room A B  # of people in Room B The total number of people in A and B is 1200. So we have, `A+B=1200` (EQ.1) > Then, let's consider the number of...

Math
This is a minimization problem hence, you need to find a function that models the distance between a point `A(x,y)` that lies on the given curve and the fixed point `(3,0).` You need to use the...