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Calculus

MathThe integral `int 1/(1+sqrt x) dx` has to be derived. `int 1/(1+sqrt x) dx` Let `x = u^2` `(dx)/(du) = 2u` `dx = 2u*du ` `int 1/(1+sqrt x) dx` = `int 1/(1+u) * 2u du` Let `1 + u = y => dy = du`...

MathThe definite integral `int_0^8 (1+x^2)/(x^3 + 3x) dx` has to be determined. First determine the integral `int (1+x^2)/(x^3 + 3x) dx` and find the value of the same between x = 0 and x = 8 `int...

MathThe yintercept of the tangent to the curve y = e^x*sin x at the point where x = 0 has to be determined. For a curve y = f(x), the slope of the tangent to the curve at a point `(y_0, x_0)` is...

MathTo find the gradient or slope of the tangent line at a given point on the curve, we need to take the derivative of the equation of that curve, which gives the slope of the tangent line at any...

Math`f(x)=ax^3+6x^2+bx+4` Differentiating with respect to x, `f'(x)=3ax^2+12x+b` Differentiating again with respect to x, `f''(x)=6ax+12` f has a relative minimum at x=1 and a relative minimum at...

MathThe surface area of a closed cylinder is 100 cm^2. What is the height of the cylinder that can be...The surface area of a closed cylinder is 100 cm^2. The material used to build this cylinder is used to create another cylinder such that the volume is maximized. Let the radius of the ends of the...

CalculusGiven, velocity of water from the pipe, `v = 1064p^(1/2)` Differentiate both sides with respect to t, `(dv)/(dt)=1064*1/2*p^(1/2)*(dp)/(dt)` `=532* p^(1/2)*(dp)/(dt)` Put the given values:...

CalculusA rectangular sheet of perimeter 27 cm and dimensions x cm by y cm is to be rolled into a cylinder. What values of x and y give the largest volume? Perimeter of rectangularsheet= 27 cm But...

CalculusA man 6 ft tall walks at a rate of 2 ft/sec away from a lamppost that is 24 ft high. At what rate...Let x be the distance of the man from the lamp post and y be the distance of the tip of his shadow from the lamp post (please refer to the attached image). Triangles `Delta SBL` and `DeltaHFL` are...

CalculusFrom a thin piece of cardboard 30 in. by 30 in., square corners are cut out so that the sides can...Let h be the length (in inches) of the cutout portion of the square corners. Eventually, that would be the height of the cardboard box. Both sides of the cardboard would reduce by 2h, owing to the...

CalculusThe company wishes to manufacture a box with a volume of 44 cubic feet that is open on top and is twice as long as it is wide. Let the length of the box that uses minimum amount of material be L....

CalculusGiven equation of the curve is: `x^4y^4=16` Use implicit differentiation to solve for dy/dx: `d/(dx)(x^4y^4)=d/(dx)(16)` `rArr x^4*4y^3(dy)/(dx)+4x^3y^4=0` `rArr...

CalculusUsing the funcamental theorem of calculus to find the area enclosed between the xaxis and a. y =...A. Area=`int_(x=1)^(x=4)ydx` `=int_1^4 1/sqrt(x)dx` `={2sqrt(x)}_1^4` `=2(21)=2` `` Area is 2 square units. B. area=`int_1^3(x^23x+2)dx` `=(x^3/33x^2/2+2x)_1^3` `=(927/2+6)(1/33/2+2)`...

CalculusYou need to evaluate the partial derivative `f_x` differentiate the given function with respect to x, considering y as constant, such that: `f_x = (del f(x,y))/(del x)` `f_x = 1/(sqrt(1 ...

Matha) Using geometry, the object we wish to find the surface area of is a cone with no base and no 'nose' (a 'conical fustrum'). The formula for the surface area of this cone with no base and no...

MathWe are given the function `y = 3lnx  3x^2` and asked to fund the number of extreme points. An extreme point is an absolute maximum or absolute minimum, absolute meaning that it is finite in...

Calculus`f(x;y)=x/y` it's defined `D=<<AA x in RR >> xx<< AA y in RR  y!=0 >>` We se function is not limited for, x wit a finte value and y too much close to zero on the left...

MathThe antiderivative of a function is equivalent of the integral of that function. Therefore the antiderivative of `(x^2+2x+2)^(1/2)` with respect to `x` is equivalent to `int (x^2+2x+2)^(1/2)dx`...

MathThe formula for the volume of a sphere is `V = 4/3pir^3` Think of this as adding two hemispheres together, where each of those hemispheres is the sum of many many circle slices/discs from `x=0` ro...

Calculus1) a) `y = 3/x` , `x>0` The area under the graph is given by `int_0^infty 3/x dx = 3log(x)_0^infty = 3[lim_(x> infty)log(x)  lim_(x>0)log(x)] ` `= lim_(x> infty) x` b) `y = 12x`,...

CalculusWe know by defnition of inverse function. `f^(1)(f(x))=x` Let `g=f^(1)` ,then `g(f(x))=x` (i) differentiating (i) with respect to x ,implicitly `g'(f(x))f'(x)=1` `g'(f(x))=1/(f'(x)) ``(ii)`...

Math`int_3^6 1/10x(6x^215)dx` First, factor out 1/10. `=1/10int_3^6x(6x^215)dx` Then, distribute x to (6x^215)dx. `=1/10int_3^6(6x^315x)dx` Then, integrate each term inside the parenthesis uisng...

CalculusHi, djshan, Sorry, but I'm not too sure what you want us to do here. Are we going to graph this? Find the intercepts? Find the zeros? Something else? I would assume we are graphing it. To...

MathThe solid takes the form of a scaled (by ` ``80pi`) bivariate bell (Normal/Gaussian) curve where the variance of the two variables `x` and `y` is `sigma^2 =2` and the correlation `rho=0` . Ignoring...

MathWe have the surface `f(x,y) = 64 + x^2  y^2` and the constraint `x^2+y^2 <=1` , `x` and `y` `in R` This constraint can be rewritten as `x^2 = 1  y^2` and `y^2 <=1` Substituting for...

MathI think you mean differential of y = x² sinx. First we find the derivative of sin2x. Using double angle formula sin2x= 2sinx.cosx, we get d(sin2x)/dx = d(2sinx cosx)/dx =...

Calculuslet x+1=t ,dx=dt , x^(n)=(t1)^(n)=t^(n)c(n,1)t^(n1)(1)+ c(n,2)t^(n1)(1)x^(2)+..............+c(n,n1)t (1)^(n1)+ (1)^(n) int(x^(n)/(x+1))dx=int (( t^(n)c(n,1)t^(n1)(1)+...

Calculus`int_0^1 x^n/(n+1)=x^(n+1)/(n+1)^2_0^1=1/(n+1)^2` So your integral is equal to `1/(n+1)^2` which is less than `1/(n+1)` because n is a natural number. I'm sorry because the above formula is hard...

CalculusThe integral `int (x*e^(2x))/(2x+1)^2 dx` has to be determined. Use integration by parts. Let `x*e^(2x) = u` `du = 2*x*e^(2x) + e^(2x) dx` =>` du = e^(2x)*(1 + 2x) dx` `dx/(2x+1)^2 = dv`...

CalculusI'm assuming that when you say you have a graph you mean you have graphical representation (picture) of a function which by itself doesn't mean much (you can't do anything better than guessing)....

CalculusWe have to find the value of lim x>0+ [(x^x)^2)] lim x>0+ [(x^x)^2)] => lim x>0+ [e^(ln ((x^x)^2))] => lim x>0+ [e^(ln ((x^2x))] => lim x>0+ [e^(2x*ln x)] As the...

CalculusWe'll have to differentiate with respect to x, using the product rule: (u*v)' = u'*v + u*v' Let u = cos x and v = ln x y' = (cos x)'*(ln x) + (cos x)*(ln x)' y' = sin x*ln x + (cos x)/x The first...

CalculusWe'll rewrite the given expression, isolating dy: dy = (3x^5 + 6e^2x)dx We'll integrate both sides: Int dy = Int (3x^5 + 6e^2x)dx We'll use the property of integral to be additive: Int dy = Int...

CalculusSince the function that has to be differentiated is the result of composition of 2 functions, logarithmic and linear functions, we'll apply the chain rule and we'll differentiate with respect to...

Calculusf(x)=ax^4+bx^2+c What are a,b,c if f(0)=2 , f'(1)=26 and the definite integral of f(x) for x=0 to...We'll determine f(0). To get f(0), we'll replace x by 0 in the expression of f(x): f(0) = a*0^4 + b*0^2 +c f(0) = c But f(0) = 2 (from enunciation) => c = 2 Now, we'll calculate f'(x): f'(x) =...

CalculusareaCalculate the area of the surface between the graph of f and the lines x = 1, x = 2. f(x) = 1...Int f(x)dx = Int dx/(x+2)(x+3) We'll apply LeibnizNewton rule to calculate the definite integral. We'll decompose the fraction into partial fractions. 1/(x+2)(x+3) = A/(x+2) + B/(x+3) 1 = A(x+3) +...

CalculusWe'll differentiate the given function with respect to x: df/dx = d/dx {[3(1/x)] / (x1)} df/dx = d/dx [3/(x1)]  d/dx [1/x(x1)] d/dx [3/(x1)] = [(x1)*d/dx(3)  3*d/dx(x1)]/(x1)^2 d/dx...

CalculusWe'll have to differentiate the given function y with respect to x. y = sin (cos x) We'll differentiate both sides: dy = [sin (cos x)]'dx We'll differentiate using chain rule, since the given...

CalculusIf we'll replace x by 0, we'll get the indetermination "0/0" type. Therefore, we can use L'Hospital's rule, to determine the limit of the quotient of derivatives. lim f(x)/g(x) = lim f'(x)/g'(x)...

CalculusWe'll have to apply product rule and chain rule to determine the 1st derivative of the given function: y' = (2x)'*ln(2x)*sin(2x) + (2x)*[ln(2x)]'*sin(2x) + (2x)*ln(2x)*[sin(2x)]' y' =...

CalculusThe dimensions of the rectangle, at the time t, are x and y cm. We'll compute the area of the rectangle, at the time t: A = x*y cm^2 (1) We'll have to determine dA/dt if we want to know how fast...

CalculusLet f(x) = x^x If we'll put x = 1 => f(1) = 1^1 =1 We'll rewrite the function whose limit has to be found out: lim (f(x)  1)/(x  1) By definition, the derivative of a function f(x), at the...

CalculusThe function f(x) = x^(sin x) Let y = f(x) = x^(sin x) Take the natural log of both the sides ln y = ln [ x^(sin x)] => ln y = sin x * ln x Differentiate both the sides with respect to x =>...

CalculusTo determine the primitive function y, we'll have to compute the indefinite integral of the function. If dy/dx = 1/sqrt(36x^2) => dy = dx/sqrt[(6)^2  x^2] We'll integrate both sides: Int dy =...

CalculusTo determine the primitive function f(x), we must calculate the indefinite integral of f'(x). We'll apply substitution technique, replacing ln(2x+3) by t. ln(2x+3) = t We'll differentiate both...

CalculusWe'll write the denominator as the result of expanding the square: x^2 + 4x + 4 = (x+2)^2 We'll rewrite the integral: Int f(x)dx = Int dx/(x+2)^2 We'll use the techinque of changing the variable....

CalculusUse StolzCesaro theorem for proving the convergence of the sequence if an=(182+2*3+..+n*(n+1))/n^3.I'll solve this problem, considering that the term "182"i s 1*2 (since the "*" symbol and the digit 8 are on the same key and the rest of the terms are written accordingly). The Stolz Cesaro...

CalculusTo determine the antiderivative of a given function, we'll have to calculate the indefinite integral of that function. We'll apply integration by parts. First, we'll recall the formula: Int udv =...

CalculusWe'll manage the right side of the expression and we'll differentiate tan(x/2). [tan(x/2)]' = 1/2*[cos(x/2)]^2 We'll rewrite the expression: (1+sinx)/(1+cosx)=1/2*[cos(x/2)]^2 + tan(x/2) We'll...

CalculusWe have to find lim x> inf. [(3x^24x+1)/(8x^2+5)] substituting x = inf., gives the indeterminate form inf./inf., we can use l'Hopital's rule and substitute the numerator and denominator with...