Differentiation and Integration Examples with Solutions

This article provides solved examples of differentiation and integration with step-by-step solutions. It covers topics like quotient rule, chain rule, product rule, finding maximum volume, area between curves, and population growth. Desklib's study material helps improve your calculus skills.

Solution 1:
(a)
Consider the expression, ------- (1)
Note that the quotient rule of differentiation state that if and are the functions
where and , then differentiation of with respect to x is
Use this rule to equation (1) and we have,
Hence
(b)
Consider the expression, ------- (1)
Note that the chain rule of differentiation state that if and are the functions
and , then differentiation of with respect to x is
Use this rule to equation (1) and we have,
Hence,
(c)
Consider the expression, ------- (1)
Note that the product rule of differentiation state that if and are the functions
and , then differentiation of with respect to x is
Use this rule to equation (1) and we have,
Hence,

Solution 2: Given the angular displacement radians as a function of time t is
(a)
Note that the angular velocity is,
Consider the expression
Now
After time
Hence after time , the angular velocity is
(b)
Note that the angular acceleration is
Since
So,
After , the angular acceleration is
Hence after time , the angular acceleration is
(c)
Suppose at time t, the angular acceleration is zero that is
So,
Hence at time , the acceleration will be zero.
Solution 3:
(a)
The simple diagram of the rectangular sheet of metal and the dimensions including the
squares of side is shown below.

(b)
Since from the above diagram, it is observed that the dimensions of the base of the box
are and , whose height is . So the volume of the box is
Hence volume is ----- (1)
(c)
Differentiate equation (1) with respect to we get
Form maximum volume, equate and obtain,
Simplify further,
That is
Note that is not possible since after removing a square of length from
both corners along the edge measuring ,there would not be sufficient material
remaining to remove the other square.
Hence the volume will be maximum when
Solution 4:
(a)
Consider the integral
Now,
Hence, where is constant of integration.
(b)

Consider the integral .
Now,
Hence, where is constant of
integration.
(c)
Consider the definite integral . Suppose then that is
and when we get and when we get so the integral
becomes,
Hence
Solution 5: Given the curve and
(a)
The area bounded by the curve between is
Simplify further,
Hence area between curves is square units.
(b)
The graph of the curve between and is shown below.

Now the area bounded by the curve between and is
Hence area bounded by the curves between and is
square units.
Solution 6:
Given the instantaneous rate of change of population
With initial population
(a)
Since this gives,
Integrate and obtain
Since that is
Hence population after t years is
(b)
The population after is
Hence population after years is approximately .
Solution 7:

Consider the indefinite integral . Let and
, then formula of integral by part is . So
Hence ,
Where C is constant of integration.