Calculus Problems with solutions

Calculus is the mathematics of change, of calculating problems that are continually evolving. There are mainly two branches of Calculus, Differential calculus or Differentiation and Integral Calculus or Integration which deals with areas and volumes of complex figures. Let us solve some of the Calculus Problems using the various rules or formulas stated in Calculus.

Calculus question on Limits: Find the limit of lim(x->5) [x^2- 25]/[x^2+x-30]
 First we need to factor both the numerator and the denominator.
That gives us, lim(x->5)[(x+5)(x-5)]/[(x+6)(x-5)].
 Next we cancel the common factors, which is (x-5).
We get, lim(x->5)(x+5)/(x+6). Now, we substitute x = 5 which gives us, 10/11 which is the required limit.
Differentiate the following calculus question, f(x) =x^2/(3x-1)
Using the Quotient rule of differentiation, we get
f’(x)= [(3x-1)d/dx[x^2]- (x^2)[d/dx(3x-1)]/(3x-1)^2
        =[(3x-1)(2x) – (x^2)(3)]/(3x-1)^2  combining the like terms, we get
        = [6x^2-2x-3x^2]/(3x-1)^2 on further simplification
        = [3x^2-2x]/(3x-1)^2  
        = x[3x-2]/(3x-1)^2
   
Calculus Problem on Differentiation of Trigonometry Functions: Differentiate y=2sin(x)-3cos(x)
We use the factor rule of differentiation, d/dx[cf(x) = cd/dx[f(x)].
So, we get, y’= 2 d/dx[sin(x)] – 3d/dx[cos(x)]
 which gives us 2cos(x)-3[-sin(x)]=2cos(x)+3sin(x)

Let us work on a Calculus question on differentiation of a logarithmic function: y = x^x
First let us apply natural logarithm on both sides, log y = log xx = xlog x.
Using the product rule on the right hand side, 1/y[y’]=x(1/x)+(1) ln x= 1+ln(x)
Multiplying y on both sides, y’ = x^x[1+ ln x]

Evaluate arccos [-sqrt(2)/2]. In this calculus question, we first put y =arccos[-sqrt(2)/2]
that gives us cos(y)=- sqrt(2)/2
From the unit circle, we get the angle y as 3(pi)/4

Find the fourth derivative of the given higher order derivatives calculus question, y = cos x
First derivative, y’ = -sin(x)
Second derivative, y’’= -cos(x)
Third derivative, y’’’=-[-sin(x)] = sin(x)
Fourth derivative, y^(4)= cos(x)

Calculus Problems using summation notation: Find the value of Summation(i=0to2)[5+sqrt(4i)] which gives us, =[5+sqrt(4^0)]+ [5+sqrt(4^1)]+ [5+sqrt(4^2)]
  = [5+sqrt(1)]+ [5+sqrt(4)]+ [5+sqrt(4^2)]
=[5+1] +[5+2]+[5+4]
=22

Integrate integral (2+3e^x) is a Calculus question on Integration.
The steps involved are as follows:
Integral(2+3e^x)= integral(2)dx – integral(3e^x)
= integral(2)dx- 3 integral(e^x)
         = 2x- 3e^x+ C

Integration by parts: Let us solve a Calculus question, Integrate integral (xsinx)dx
In this problem, let  u = x and dv = sin x dx
So, we get, du/dx = 1;
du = (1)dx and v = -cos x
Therefore, integral(xsinx)dx= x(-cos x) – integral (-cos x) dx = -x cos(x) + sin(x) + C

A problem solvers solution guide is a Calculus Problem Solver which gives a stepwise explanation of many of the calculus problems. We can even get online calculus question solvers.