Solving Calculas Problems

Introduction to solving calculus problems:

Solving calculus problem is one of the great achievement in mathematics which concentrate on the functions, limits, integrals, derivatives, and infinite series. Solving calculus problem has two branches,the first one is differentiation and the second one is integration. The differential solving calculus problem helps us to locate out the rate of change of an amount wherever as the Integral solving calculus problem helps to find out the quantity where the rate of change is known.

Solving Calculus Problems-sample Problems:

Problem 1:

Solving the following calculus problem for the derivative of f(x) = 3x 3

Solution:

let c = 3

and g(x) = x 3,

then f '(x) = c g '(x)

= 3 (3x 2)

= 9 x 2 

Solving Calculus Problems-problem 2:

If F(x) is an anti derivative of f(x), then
(1/a) F(ax) is an anti-derivative of f(ax). True or false.

Solution:

Let u = a x and Differentiate (1/a) F(ax) w.r. to x

d/dx( (1 / a) F(a x) )

= (1 / a) d(u) / dx dF/dU

= (1/a) a f(u)

= f(a x)

Hence the given statement is True.

Problem 3:

The sum of two non negative numbers is 9 and so that the maximum number is the product of one number and the square of the other. Find those two non negative number.

Solution:

The sum of the two variables x and y is given to be 9 = x + y ,

so that, y = 9 - x

Maximize the product,

P = x y2 .

Substitute for y, we get

P = x y2

= x ( 9-x )2 .

On differentiating the above function, we get

P' = x (2) ( 9-x)(-1) + (1) ( 9-x)2

= ( 9-x) [ -2x + ( 9-x) ]

= ( 9-x) [ 9-3x ]

= ( 9-x) (3)[ 3-x ]

= 0 [for x=9 or x=3]

Note that since both x and y are non-negative numbers and their sum is 9, it follows the order that 0<= x <= 9.

If x=3 and y=6 ,

then P= 108

which is the largest possible product.