## Calculus Questions and Answers

Calculus

### Evaluate the limit of y=(2x^2-x-3)/(x+1), x-->-1 using L'Hopital theorem.

We have to find the value of lim x--> -1 [(2x^2-x-3)/(x+1)] If we substitute x = -1 in (2x^2-x-3)/(x+1) we get 0 / 0. So L'Hopital's theorem can be applied. We now have: lim x--> -1 [(4x...

Calculus

### What is the definite integral of y=3x^6+3x^-1-cosx if the limits of integration are 0 and 1.

The function given is f(x) = 3*x^6 + 3*x^(-1) - cosx. We need the definite integral of f(x) between the values of x = 0 and x = 1. F(x) = Int [ f(x)] => Int [ 3*x^6 + 3*x^(-1) - cos x dx] =>...

Calculus

### Find f(x) if f'(x)=11e^x/(11+e^x).

We have to find the integral of f'(x)=11e^x/(11+e^x) f'(x)=11e^x/(11+e^x) let 11 + e^x = y e^x dx = dy Int [ 11e^x/(11+e^x) dx] => Int [ 11dy/y] => 11*ln |y| + C substitute y = 11 + e^x =>...

Calculus

### Determine the gradient vector of the function f(x,y)=x^3*y^2-2x at the point (2,4)?

The gradient vector is a vector whose first component is the derivative of f(x, y) with respect to x and the second is the derivative of f(x, y) with respect to y. Here f(x, y) = x^3*y^2 - 2x...

Calculus

### What is the antiderivative of the function f(x) given by f(x)=x*e^8x ?

The antiderivative of f(x)=x*e^8x can be found using integration by parts. Int [u dv ]= u*v - Int[ v du] Let u = x => du = 1 dv = e^8x => v = e^8x/8 Int [ x*e^8x ] = (x*e^8x)/8 - Int...

Calculus

### What is the integral of y=(e^(13x)+sin x)/10

If you mean that you required the integral of y = (e^(13x)+sin x)/10, the result is. y = (e^(13x)+sin x)/10 Int [ y dx] = Int [ (e^(13x)+sin x)/10 dx] => (1/10)* Int [ (e^(13x)+sin x) dx] =>...

Calculus

### Evaluate the limit of the fraction (x^2-4)/(x^2+x-6) as x approaches to 2.

We have to find: lim x-->2 [(x^2-4)/(x^2+x-6)] substituting x= 2 gives the indeterminate form 0/0. So we can substitute the numerator and the denominator with their derivatives using l'Hopital's...

Calculus

### Calculate the second order linear derivatives fxy, fyx for the given function f(x,y)=x^3+8xy

I assume fyx means differentiation of f with respect to y followed by differentiation with respect to x, and vice versa for fyx. fxy = d[d(x^3+8xy)/dx]/dy => fxy = d(3x^2 + 8y)/dy => fxy = 0...

Calculus

### math Given I = Integral x^n/(x+1) demonstrate that I <= 1/(n+1). The limits of integration are x=0 to x=1 and n is...

`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...

Calculus

### Antiderivatives Determine the antiderivative of y=1/(x-1)(x+4).

We have to find the antiderivative of y=1/(x-1)(x+4) First let's express y as partial fractions. y = A / (x - 1) + B/(x + 4) => y=1/(x-1)(x+4) = A/(x - 1) + B/(x + 4) => Ax + 4A + Bx - B = 1...

Calculus

### Find derivative of the fraction (3x^3+3)/(3x^3-3) .

We have to find the derivative of y = (3x^3+3)/(3x^3-3) We use the quotient rule which state that the value of [f(x) / g(x)]' = [f'(x)*g(x) - f(x)*g'(x)]/ [g(x)]^2 Substituting the functions we...

Calculus

### What is the most considered method to find a function when knowing the integrand?

The only method to find the original function if you know the integrand is by the use of integration. This is because differentiation and integration are inverse functions. Though the process of...

Calculus

### What is the indefinite integral of the function 3sinx-4tan^2x ?

We have to find the integral of f(x) = 3*sin x - 4*(tan x)^2 f(x) = 3*sin x - 4*(tan x)^2 => 3*sin x - 4*[(sec x)^2 - 1] Int [ f(x) dx] = Int [3*sin x - 4*[(sec x)^2 - 1] => 3*Int [ sin x dx]...

Calculus

### Definite integral Determine the definite integral for y=x+1/x. x=1 and x=3

We have to find the definite integral for y=x+1/x between x=1 and x=3. First let's find the indefinite integral of y Int [ y dx] = Int [ x + 1/x dx] => x^2 / 2 + ln |x| +C For x = 3 this is...

Calculus

### Use chain rule of differentiation and find derivative of f(x)=(x^3+4)^4?

The chain rule of differentiation gives the derivative of f(g(x)) as f'(g(x))*g'(x) For f(x) = (x^3 + 4)^4 we can take u(x) = x^3 + 4 f(x) = (u(x))^4 f'(x) = 4*(u(x))^3*u'(x) => f'(x) = 4*(x^3 +...

Calculus

### Solve the limit of fraction (tanx + sin2x)/ln(x+1)

The required limit that x tends to is not specified. I take it to be 0 as substituting x = 0, makes the expression indefinite with 0/0. We can use the l'Hopital's rule here and substitute the...

Calculus

### Evaluate the limit of expression 4cosx-6sin^2x+3tan^2x. x-->x0 , x0=60 degrees

We have to find the value of lim x--> 60 degrees[ 4cos x - 6(sin x)^2 + 3*tan x)^2] Here we see that cos 60 , (sin 60)^2 and (tan 60)^2 are defined, so we can just substitute x with 60....

Calculus

### What is the area between the curve y=cos3x*sin-x, x axis and lines x=o and x=pi?

We have to find the definite integral of y= - cos 3x * sin x, in the limit x = 0 to x = pi. Int [ - cos 3x * sin x] => Int [ cos 3x * sin dx] => Int [ (4(cos x)^3 - 3 cos x) sin x dx] let cos...

Calculus

### Let F(s) = s+2/s+10. Find the maximum of F(s) on the interval [1,4].

F(s) = s + 2/s + 10. To find the maximum value of F(s) in the interval [1,4], we differentiate F(s) and equate the derivative to 0. Also, at the point of maximum value F''(s) is negative. F(s) =...

Calculus

### What is the derivative of the expression (10+lgx^10+e^10x)^10?

We have to find the derivative of y = (10 + lg (x^10) + e^10x)^10. We use the chain rule to find the derivative of y. y' = 10 * (10 + lg (x^10) + e^10x)^9 * (10 / x + 10*e^10x) => 10 * (10 + lg...

Calculus

### Find the limit of function f(x) given by f(x)=ln(x-1)/(x-2), x->2?

We have to find the value of lim x-->2 [ ln (x - 1)/(x - 2)] substituting x = 2, we get the indeterminate form 0/0; that permits the use of l'Hopital's Theorem and we can substitute the...

Calculus

### Prove that the functions f(x)=x+2, if x=<1 and f(x)=x^2. if x>1 are discontinuous.

The function f(x) is defined such that f(x) = x + 2 , if x <= 1 and f(x) = x^2, if x > 1. At the point x = 1, if we approach from the left lim x--> 1- [ f(x)] = 1 + 2 = 3. If we approach...

Calculus

### Determine derivative of y=(x+3)^2+2(x+3)(x-4)+(x-4)^2

We have to find the derivative of y = (x+3)^2 + 2(x+3)(x-4) + (x-4)^2 y = (x+3)^2 + 2(x+3)(x-4) + (x-4)^2 Use the relation [x^n]' = n*x^(n - 1) and the chain rule y' = 2( x + 3) + 2*( x + 3) + 2* (...

Calculus

### What is the limit of the function sin x/squareroot(x^2+1), x->+infinite?

We have to find the value of lim x--> inf+ [sin x / sqrt (x^2 + 1)] sin x is a periodic function the value of which oscillates between -1 and 1 as x tends to infinity. It is not possible to...

Calculus

### Calculate area bounded by the curve y=cosx/(sin^2x-4), x axis and the lines x = 0 and x = pi/2 .

The area bound by the curve y = cosx/(sin^2x-4), the x axis and the lines x = 0 and x = pi/2 is the integral of y between the limits x = 0 and x = pi/2. y = cos x/ ((sin x)^2 - 4) let sin x = y...

Calculus

### If u=(fog)(x) and v=(gof)(x), verify if u'=v'? f(x)=3x+2,g(x)=x^2+1

We have the functions f(x) = 3x+ 2 and g(x) = x^2 + 1 u = fog ( x) = f(g(x)) => f(x^2 + 1) => 3(x^2 + 1) + 2 => 3x^2 + 3 + 2 => 3x^2 + 5 v = gof(x) = g(f(x)) => g( 3x + 2) => (3x...

Calculus

### Simplify the fraction (4x^2-1)/(4x^2-4x+1) and find the derivative of the fraction.

We have the fraction (4x^2-1)/(4x^2-4x+1) and we have to find its derivative. (4x^2-1)/(4x^2-4x+1) Use the relation x^2 - y^2 = (x - y)(x + y) and (x -y)^2 = x^2 + y^2 - 2xy => (2x - 1)(2x + 1)...

Calculus

### What is the minimum value of 4x^2-8x-18 ?

To find the minimum value of y = 4x^2 - 8x -18, we need to find the first derivative and equate it to 0. We solve for x and determine the value of the function for the value of x that we find. y =...

Calculus

### What is f(x) if derivative is f'(x)=sin2x/(sin^2 x-4)?

We need to find f(x) given that f'(x) = sin 2x /((sin x)^2 - 4) let ((sin x)^2 - 4) = y dy = 2*sin x * cos x dx =>dy = sin 2x dx Int [ sin 2x /((sin x)^2 - 4) dx] => Int [ 1/y dy] => ln...

Calculus

### How to find derivative of f(x)=(2x^2+1)/(2x^2-1)

We have to find the derivative of f(x)=(2x^2+1)/(2x^2-1) We use the quotient rule here, which gives the derivative of f(x)/g(x) as (f'(x)*g(x)* - f(x)*g'(x))/(g(x))^2 f(x)=(2x^2+1)/(2x^2-1) f'(x) =...

Calculus

### what is dy/dx if y=(3x^4+4x^3+1)(3x^2+2x^3-1)?

We have y = (3x^4+4x^3+1)(3x^2+2x^3-1) To find the derivative dy/dx we use the product rule. y' = (3x^4+4x^3+1)'(3x^2+2x^3-1) + (3x^4+4x^3+1)(3x^2+2x^3-1)' y' = (12x^3+12x^2)'(3x^2+2x^3-1) +...

Calculus

### Evaluate the integral using the fundamental theorem of calculus.y=4x^t + 5x^-1 + 6sinx.

We have to find the integral of y=4x^t + 5x^-1 + 6sin x. I assume t to be a constant. Int[ y dx] => Int[ 4x^t + 5x^-1 + 6sin x dx] => Int[4x^t dx] + Int[5x^-1 dx]+ Int [6sin x dx] =>...

Calculus

### Find the antiderivative of the function given by f(x)=(ln(2x-5))/(2x-5)?

We have to find the anti derivative of f(x) = (ln(2x-5)) / (2x-5). This can be done using substitution Int [ (ln(2x-5)) / (2x-5) dx] let ln(2x - 5) = t => dt/dx = 2/(2x - 5) => dt / 2 =...

Calculus

### Verify if the value of the limit of the function f(x)=(x^2-6x+5)/(x-1) is 4 x-->1

We have to verify the value of lim x--> 1[ (x^2 - 6x + 5)/(x - 1)] Substituting x = 1 we get the indeterminate form 0/0. So, we can use l'Hopital's rule and substitute the numerator and...

Calculus

### Evaluate the limit of I(a), if a goes to infinite. I(a) is definite integral of (2x-3)*e^x, ifx=-a to x=0

First I(a) needs to be determined. That is given as the definite integral of (2x - 3)*e^x for x = -a to x= 0 Int [ (2x - 3)*e^x dx] use integration by parts. Int[ u dv ] = u*v - Int [ v du] 2x - 3...

Calculus

### Find the derivative of the function y=(3+e^x+8x^3)^2*(2x+e^2x)

We have to find the derivative of y=(3+e^x+8x^3)^2*(2x+e^2x) We can use the product rule here y=(3+e^x+8x^3)^2*(2x+e^2x) y' = [(3+e^x+8x^3)^2*(2x+e^2x)]' =>y' = [(3+e^x+8x^3)^2]*(2x+e^2x)]' +...

Calculus

### Verify the relation f'(x)/f(x)=1+lnx, if f(x)=e^(ln((x)^x))?

The function f(x) = e^(ln((x)^x)) taking the log to the base e of both the sides we get: ln f(x) = ln x^x differentiate both the sides f'(x)/f(x) = [ln x^x]' => f'(x)/f(x) = [x*ln x]' =>...

Calculus

### Evaluate the limit of the function (x^2-3x+2)/(x^2-4) using L'Hospital rule.

As you have not specified what x is tending to, I take it as 2. We have to find the value of lim x--> 2 [(x^2-3x+2)/(x^2-4)] If we substitute x = 2, we get the indeterminate form 0/0. So we use...

Calculus

### How do you find the extreme points of the function f(x) = 0.25x4 + 3x3 – 18x2 + 10 and classify them?

The function given is f(x) = 0.25x^4 + 3x^3 – 18x^2+ 10 At the extreme points f'(x) = 0 f'(x) = 4*0.25x^3 + 9x^2 - 36x x^3 + 9x^x - 36 = 0 => x(x^2 + 9x - 36) = 0 => x(x^2 + 12x - 3x - 36) =...

Calculus

### What is the second derivative of the function f(x)=2x^3+e^2x+sin 2x-lnx?

We have the function f(x) =2x^3 + e^2x + sin 2x - ln x. We have to find f''(x). f'(x) = 6x^2 + 2e^2x + 2 cos 2x - 1/x f''(x) = 12x + 4e^2x - 4 sin 2x + 1/x^2 The required second derivative f(x) =...

Calculus

### What are the critical points of h(x)= 5x+(4/x) and are they minimum or maxmum values.

We have the function h(x) = 5x + (4/x). The critical points of this function are at points where h'(x) = 0 h'(x) = 5 - 4/x^2 5 - 4/x^2 = 0 => x^2 = 4/5 => x= 2/sqrt 5 and -2/sqrt 5 h''(x) =...

Calculus

### Determine the antiderivative of y=sin x*cos^n x?

We have to find the anti derivative of y = sin x*(cos x)^n let cos x = y => dy/dx = -sin x => (-1)*dy = sin x dx Int [ sin x*(cos x)^n dx] => Int [ (-1)* y^n] => (-1)* y^(n+1) / (n +...

Calculus

### Solve the derivative of expression: tan^2(ln(square root(4e^x+2x^2))) + cot^2(ln(square root(4e^x+2x^2)))

We have the expression: tan( ln (sqrt (4e^x+2x^2)))^2 + cot (ln (sqrt (4e^x + 2x^2)))^2 We have to find the derivative of the expression. We use the chain rule and start from the innermost...

Calculus

### What is the derivative of y = ln[(x+1)^4 (x+7)^2 (x+2)^3]

We have to find the derivative of y = ln[(x+1)^4*(x+7)^2*(x+2)^3] First simplify the expression using the properties of logarithms: log(a*b) = log a + log b and log(a^b) = b*log a y =...

Calculus

### Find the limit of the function [(3x+2)/(2x+1)]^[(2x+1)/(x+4)]; x approaches to infinite.

As you have not specified what x tends to, I am taking it to be 0. The value of lim x-->0 [(3x+2)/(2x+1)]^[(2x+1)/(x+4)] is required. lim x-->0 [(3x+2)/(2x+1)]^[(2x+1)/(x+4)] substitute x =...

Calculus

### How to verify the inequality: {(x+1)*lnx}/2>x-1

We have to prove that [(x+1)*lnx]/2 > x-1 [(x+1)*lnx]/2 > x-1 => [(x+1)*lnx] > 2(x-1) => ln x > 2(x-1)/(x+1) => ln x - 2(x-1)/(x+1) > 0 If we find the derivative of ln x -...

Calculus

### Prove that the result of differentiating arc tan (1-x^2) + arc cot (1-x^2) is 0.

We have to prove that the derivative of arc tan (1-x^2) + arc cot (1-x^2) is 0. We know that the derivative of arc tan x = 1/[1 + (1-x^2)^2] and the derivative of arc cot x = -1 / [1 +...

Calculus

### What is the partial derivative fx of the function f(x,y)=x^2*2^(x*y)?

The function given has two variables x and y. We have to find the partial derivative f'x, which implies that y can be considered as a constant. f(x,y) = x^2*2^(x*y) f'x = [x^2]'*2^(x*y) +...

Calculus

### If the tangent to the curve f(x) = x^2 - 2x + 2 is perpendicular to the straight line 4y - x - 5 = 0, find the...

The first derivative of a curve at any point gives the slope of the tangent at that point. Here the curve is defined by f(x) = x^2 - 2x + 2 f'(x) = 2x - 2 As the tangent is perpendicular to the the...

Calculus

### What is dy/dx if square root x = 1+x^2*y?

What is dy/dx if square root x = 1+x^2*y ? The method required to answer this question is called implicit differentiation: `d/dx (x)^0.5 =d/dx (1+x^2*y)` (notice I converted the square root of x...

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