Greatest Integer Functions MCQ Quiz in বাংলা - Objective Question with Answer for Greatest Integer Functions - বিনামূল্যে ডাউনলোড করুন [PDF]

Concept:

Greatest Integer Function and Definite Integral:

The greatest integer function, denoted by [x], gives the greatest integer less than or equal to x.

When integrating a function involving the greatest integer function, it is important to break the integral over intervals where the floor value is constant.

Given integral:

\(\displaystyle \int_0^3 \left[ \frac{1}{e^{x-1}} \right] dx = \int_0^3 \left[ e^{1-x} \right] dx = \alpha - \log_e 2 \)

Calculation

 \((f(x) = 2, e^{1-x} = 2 \Rightarrow x = 1 - \ln 2.\)

 \((f(x) = 1, (e^{1-x} = 1 \Rightarrow x = 1\)

Also 

f(0) = e^1-0 = e ≈2.718 > 2

f(1) = e¹ =0

f(2) = e^-1 = 0.367

f(3) = e^-2 = ≈ 0.1353

\[ \int_0^3 [f(x)] \, dx = \int_0^{1-\ln 2} 2 \, dx + \int_{1-\ln 2}^1 1 \, dx + \int_1^3 0 \, dx \]

\[ = 2(1-\ln 2) + (1 - (1-\ln 2)) + 0 = 2 - 2\ln 2 + \ln 2 = 2 - \ln 2 \]

Also

\[ \int_0^3 [f(x)] \, dx = \alpha - \log_e 2 \]

Comparing,

\[ 2 - \ln 2 = \alpha - \ln 2 \implies \alpha = 2 \]

Therefore,

\[ \alpha^3 = 2^3 = 8 \]

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Concept :

⇒ f(x) = [x] denotes a step function whose graph is as follows :

⇒ Thus by the graph we can depict any value, for example [2.93] = 2, [-0.5] = -1, ...

Calculation :

Given the function f(x) = [x + 1] - (sin\(\frac{π }{2}\)[x]).

It is given that l₁ = limx→0-f(x).

⇒ l1 = limx→0-f(x) =  limx→0- {[x + 1] - (sin\(\frac{π }{2}\)[x])} = {[0-+1] - (sin\(\frac{π }{2}\)[0-])} = {[1-] - (sin\(\frac{π }{2}\)[0-])}.

⇒ l1 = {[1_-] - (sin\(\frac{-π }{2}\))} = {0- (-1)} = 1.

It is given that l₂ = limx→0+f(x).

⇒ l₂ = limx→0+f(x) =  limx→0+ {[x + 1] - (sin\(\frac{π }{2}\)[x])} = {[0++1] - (sin\(\frac{π }{2}\)[0+])} = {[1+] - (sin\(\frac{π }{2}\)[0+])}.

⇒ l₂ = {[1+] - (sin(0))} = {1- 0} = 1.

Thus l1 = l2 = 1.

Mistake Points

Student often gets mistaken in two points when solving these type of problems :

1. Observation and representation of step bracket wherever necessary.
2. sin\(\frac{π }{2}\)(0-) = sin\(\frac{π }{2}\)(0+) = 0, but their Difference when a step function is used on them.

       ⇒ sin\(\frac{π }{2}\)[0-] = sin\(\frac{-π }{2}\) = -1 and sin\(\frac{π }{2}\)[0+] = sin0 = 0.

Concept:

Greatest Integer Function: (Floor function)

The function f (x) = [x] is called the greatest integer function and means greatest integer less than or equal to x i.e [x] ≤ x.

Domain of [x] is R and range is I.

If f :A → B and g : C → D. Then (fog) (x) will exist if and only if co-domain of g = domain of f i.e D = A and (gof) (x) will exist if and only if co-domain of f = domain of g i.e B = C.

Calculation:

Given: f(x) = (x)[x] where [.] denotes greatest integer function and g(x) = x2 

Here, we have to find the value of g o f(5/2) 

⇒ g o f(5/2)  = g( f(5/2))

∵ f(x) = (x)[x] where [.] denotes greatest integer function

⇒ f(5/2) = (5/2)[5/2]

As we know that [5/2] = 2

⇒ f(5/2) = (5/2)2 = 25/4

⇒ g o f(5/2)  = g(25/4)

∵ g(x) = x2 so, g(25/4) = 625/16

Hence, g o f(5/2) = 625/16