## Calculus Questions and Answers

Calculus

### Integral problem. Solve the integral f(x)=lnx.

The integral of ln x can be found using integration by parts. Int [ u dv ] = u*v - Int[v du] let u = ln x => du = 1/x let dv = 1 => v = x Int [ ln x * 1] = x*ln x - Int[x/x] => x*ln x - x...

Calculus

### What is the indefinite integral of 1/x(ln^2x + 9) x > 0 ?

We need to find the integral of f(x) = [1/(x(ln x)^2 + 9)] Let ln x = y => dy/dx = 1/x => dy = dx / x Int [f(x) dx] = Int [ 1/x*((ln x)^2 + 9) dx] => Int [ 1/ (y^2 + 9) dy] => [arc tan...

Calculus

### How can y=(x-1)^2+2(x-1)(x+2)+(x+2)^2 be differentiated in two different ways?

We have to find the derivative of y=(x-1)^2+2(x-1)(x+2)+(x+2)^2 Using the product rule and the chain rule: y' = 2(x - 1)*1 + 2[(x - 1)*1 + (x + 2)*1] + 2( x + 2)*1 => y' = 2x - 2 + 2x - 2 + 2x +...

Calculus

### Find the slant asymptote of f(x)=x^3/(x+2)^2

To find the slant asymptote of x^3 / (x + 2)^2 we have to divide x^3 by (x + 2)^2 (x^2 + 4x + 4) | x^3...........................................| x - 4 ...........................x^3 + 4x^2 + 4x...

Calculus

### What is the limit of the fraction bn/2^n if bn=1+2+2^n, n-->infinite?

We need to find the value of lim n--> inf [ bn/2^n] Also, we have bn = 1 + 2 + 2^n lim n--> inf [ bn/2^n] => lim n--> inf [(1 + 2 + 2^n)/2^n] => lim n--> inf [1/2^n + 2/2^n +...

Calculus

### functions Use the first principle to determine f'(3) whether f(x)=-5x^2.

First find the derivative of f(x)=-5x^2. From first principles this is equal to lim h-->0 [(f(x + h) - f(x))/h] f(x) = -5x^2 lim h-->0 [(-5(x + h)^2 + 5x^2)/h] => -5*lim h-->0 [((x +...

Calculus

### What is the limit of the fraction (x^3-2x^2)/(x-2) for x-->2 ?

We have to find the value of lim x-->2 [(x^3-2x^2)/(x-2)] IF we substitute x=2, we get the indeterminate form 0/0. We can use the l'Hopital's Rule and replace the numerator and denominator with...

Calculus

### What is the antiderivative (cos2x-cos^2x)^-1 ?

We have to find the anti-derivative of [ cos 2x - (cos x)^2]^-1 f(x) = [ cos 2x - (cos x)^2]^-1 => f(x) = [(cos x)^2 - (sin x)^2 - (cos x)^2]^-1 => f(x) = -(sin x)^-2 => f(x) = -1 / (cosec...

Calculus

### Find the slope of the tangent to the curve y=sin x at the point (pi/6, 1/2) on the curve.

To find the slope of the tangent to the curve at the point ( pi/6 , (1/2) we have to find the first derivative of the curve and substitute the value of x at that particular point. y = sin x =>...

Calculus

### Determine the indefinite integral of y=x/square root(x^2+9), using substitution.

We have to find the integral of y= x / sqrt(x^2+9) Int [ x/ sqrt (x^2 + 9) dx] let x^2 + 9 = u, du / 2 = x dx => (1/2)*Int [ 1/ sqrt u du] => (1/2)*Int [ u^(-1/2) du] => (1/2) u^(1/2) /...

Calculus

### What is the integral of cos 2x?

We have to find the integral of cos 2x. Int[ cos 2x dx] let 2x = u du = 2*dx dx = (1/2)*du => Int [ cos u * (1/2) du] => (1/2) sin u + C substitute u = 2x => (1/2) sin 2x + C The required...

Calculus

### Given f(x)=12x^4+24x^2+56, what is x if f'(x)=0?

We have the function: f(x)=12x^4+24x^2+56 f(x) = 12x^4 + 24x^2 + 56 f'(x) = 48x^3 + 48x If f'(x) = 0 => 48x^3 + 48x = 0 => x^3 + 3 = 0 => x( x^2 + 1) = 0 x1 = 0 x2 = -sqrt (-1) => x2 =...

Calculus

### What is f(x) if f'(x)=25x^4+2e^2x ?

We have f'(x) = 25x^4 + 2e^2x f(x) = Int [ f'(x) dx] => f(x) = Int[ 25x^4 + 2e^2x dx] => f(x) = Int [ 25 x^4 dx] + Int [ 2e^2x dx] => f(x) = 25* x^5 / 5 + 2*e^2x/2 + C => f(x) = 5x^5 +...

Calculus

### How to determine the function of y and x if the relationship is non linear Hi, I have a graph of y and x and the...

You are not going to be able to write a linear equation non-linearly, because the linear part means that you are using a line. Even if you have a graph that is not a line, you can create an...

Calculus

### Determine the extreme value of the function x^2-2x+2.

The extreme value of a function is its value at the point where the value of the first derivative is equal to 0. f(x) = x^2 - 2x + 2 => f'(x) = 2x - 2 2x- 2 = 0 => x = 1 At x = 1, f(x) = 1^2...

Calculus

### Discover the minimum value of the function x^2+x-2.

To find the minimum value of x^2+x-2 we find the derivative and equate it to 0 to solve for x. Using the value of x in the function we can find the lowest value. f(x) = x^2 + x - 2 f'(x) = 2x + 1...

Calculus

### the region bounded by the curve y=(x^2) +1, the x-axis, y-axis and the line x=2 rotated completely about x-axis.in...

The solid that would be constructed by the curve y=(x^2) +1, the x-axis, y-axis and the line x=2 when rotated completely about x-axis, can be considered to be a series of cylinders of a height dx...

Calculus

### Evaluate the indefinite integral of the function 1-sin^2x .

We have to find the integral of 1 - (sin x)^2.1- (sin x)^2 = (cos x)^2(cos x)^2 = (1/2)(1 + cos 2x)So, Int[(cos x)^2 dx]=> Int[(1/2)(1 + cos 2x) dx]=> (1/2)Int[1 dx] + (1/2)[Int(cos 2x...

Calculus

### What is the minimum value of the function y=e^ln(x^x) ?

We have the function y = e^ln(x^x) take the log of both the sides ln y = ln (e^ln(x^x)) => ln y = ln(x^x) => ln y = x*ln x differentiate both the sides (1/y)dy/dx = ln x + x/x dy/dx = y*(ln x...

Calculus

### Integral Evaluate if the integral of y=1/(x^1/3) is convergent or divergent when x = 1 to x = infinite.

We have the function y = 1/(x^1/3) = x^(-1/3) The integral of the function for x = 1 to x = inf. is convergent if the value is finite. Int [ x^(-1/3) dx] => x^(2/3) / (2/3) For the limits x = 1...

Calculus

### 1) What are the areas bounded by the lines and the x axis? In each case x>0 a) y = 3/xb) y=12xc) y=(1/12)x2)Find...

1) 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`,...

Calculus

### Calculate the limit of the fraction (f(x)-f(1))/(x-1) if f(x)=x^300+x+1, x-->1

We have f(x) = x^300 + x + 1 and we have to find lim x-->1 [ (f(x)-f(1))/(x-1)] => lim x-->1 [ x^300 + x + 1 - 3)/ (x-1)] for x = 1 we have the form 0/0, so we can use l'hopital's rule...

Calculus

### First derivatives. Find the first derivative of v(t) = ( 1 + 3^t) / 3^t.

We need the first derivative of v(t) = ( 1 + 3^t) / 3^t v(t) = ( 1 + 3^t) / 3^t => 1/3^t + 1 => (1/3)^t + 1 v'(t) = log (1/3) *(1/3)^t The required first derivative of v(t) = ( 1 + 3^t) / 3^t...

Calculus

### Calculate the antiderivative of the function f(x) given by f(x)=2x/(x+1)*(x^2+1)?

We need to find the integral of f(x) = 2x/(x+1)*(x^2+1) f(x) = 2x/(x+1)*(x^2+1) finding the partial fractions => f(x) = (Ax + B)/(x^2 + 1) + C/(x + 1) => f(x) = [Ax^2 + Ax + Bx + B + Cx^2 +...

Calculus

### What is the maximum and minimum value of the function f(x) = 2x^3 - 4x + 3

To find the maximum and minimum values of the function f(x) = 2x^3 - 4x + 3, we need to find the first derivative. f(x) = 2x^3 - 4x + 3 f’(x) = 6x^2 - 4 Equate this to zero and solve for x 6x^2 - 4...

Calculus

### Find the derivative of y=2^(x^2+2x)

The function given is y = 2^(x^2 + 2x). Take the natural logarithm of both the sides ln y = ln ( 2^(x^2 + 2*x)) use the property log x^a = a*log x => ln y = (x^2 + 2*x)*ln 2 Differentiate both...

Calculus

### Evaluate the integral `int (x*e^(2x))/(2x+1)^2 dx` .

The 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`...

Calculus

### Evaluate the limit of the function y if x goes to infinite? y=(3x^2-4x+1)/(-8x^2+5)

We have to find lim x--> inf. [(3x^2-4x+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...

Calculus

### Find the limit of the function (x^2+2x-3)/|x-1| if x approaches 1 from the left.

We have to find the value of (x^2+2x-3)/|x-1| as x approaches from the left. As x approaches from the left x - 1 is always negative, so we have |x - 1| = (1 - x) lim x--> 1 [ (x^2+2x-3)/(1 -...

Calculus

### Derivative Consider an example of a square root function and find it's derivative using definition of derivative

Let f(x) = sqrt x f'(x) = lim h-->0 [ (sqrt (h + x) - sqrt x)/h] => lim h-->0 [ (sqrt (h + x) - sqrt x)/(h + x - x)] => lim h-->0 [ (sqrt (h + x) - sqrt x)/(sqrt (h + x))^2 - (sqrt...

Calculus

### Definite integral. How to find the value of definite integral of an exponential function, if the base is >1 and...

For f(x)=3^4x, to find the definite integral between x = 0 and x = 1, first find the indefinite integral. F(0) = Int [ f(x) ] = 3^(4x) / 4*ln 3 The definite integral is F(1) - F(0) => 3^(4*1) /...

Calculus

### Where is the function f(x) discontinuous? f(x) = (x^2-x-2)/(x-2), if x is not 2 f(x) = 1, if x=2

We have the function: f(x) = (x^2-x-2)/(x-2), if x is not 2 f(x) = 1, if x=2 At x = 2, f(x) = 1 For lim x-->2 [f(x)] => lim x--> 2 [(x^2-x-2)/(x-2)] => lim x--> 2 [(x^2-2x + x -...

Calculus

### What is the extreme point of the curve 3x-6x^2?

We have to find the extreme point of the curve y = 3x - 6x^2. To do that we find the first derivative of 3x - 6x^2 and equate it to zero. This is solved for x. Now y = 3x - 6x^2 y' = 3 - 12x 3 -...

Calculus

### What is derivative of the function f(x)=(x^2-4)^(1/(x-2))?

We have to differentiate f(x)=(x^2-4)^(1/(x-2)) We can make the differentiation easier by eliminating the exponent on the right. For this, we take the log to the base e of both the sides. ln f(x) =...

Calculus

### Absolute value equation x^2-4|x|+3=0

x^2-4|x|+3=0 We have two options fo the absolute value. lxl = x for x > 0 lxl = -x for x < 0 Then, we have two options for the equation. x^2 - 4x + 3 = 0 OR x^2 + 4x + 3 = 0 We will solve...

Calculus

### What is the first derivative of function given by f(x)=(x-square rootx)/(5x^2-3) ?

We have to find the derivative of (x - sqrt x) / (5x^2 - 3). As we have the expression as a quotient, we use the quotient rule. f(x) = u(x) / v(x) => f'(x) = (u'(x)*v(x) - u(x)*v'(x))/v(x)^2...

Calculus

### If f(x) = x^6+4x^4+3x, find the second derivative of f''(x).

We have the function f(x) = x^6 + 4x^4 + 3x and we have to find the second derivative. First let's find the first derivative f'(x) = 6x^5 + 16x^3 + 3 The second derivative is found by...

Calculus

### Differentiate the function: f(x) = (cos^ 2 x)* ln (x^2).

The derivative of f(x) = (cos^ 2 x)* ln (x^2) has to be found. f(x) = (cos^ 2 x)* ln (x^2) => f(x) = (cos x)^2 * ln (x^2) f'(x) = [(cos x)^2]' * ln (x^2) + (cos x)^2 * [ln (x^2)]' f'(x) = [2*cos...

Calculus

### solve the equation Integral sin^2xdx+Integral cos^2xdx=(x^3-4x)/(x-2)

We have to solve the equation: Int[(sin x)^2 dx]+Int[(cos x)^2 dx] = (x^3-4x)/(x-2) Now Int [ a dx ] + Int [ b dx] = Int [ (a + b) dx] Int[(sin x)^2 dx]+Int[(cos x)^2 dx] = (x^3 - 4x)/(x-2) =>...

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