Test: Theorems of Integral Calculus- 2

 f has frequency components at 0 and  

f has frequency components at 0 and 

f has frequency components at  

f has frequency components at 

0    

1.5    

2.0    

3.0

∞    

2    

0

1

Continuous and differentiable  

 Continuous on the integral but not differentiable at all points  

 Neither continuous nor differentiable  

Differentiable but not continuous 

y is discontinuous at x = 0 

y is not defend at x = 0 

1  

-1  

0  

Does not exist 

1 

∞    

0   

-1

0    

1/6    

1/3    

1 

is continuous ∀ x∈ R and differentiable ∀ x ∈ R

is continuous ∀ x∈ R and differentiable ∀ x ∈ R except at x = 3/2

is continuous ∀ x∈ R  and differentiable ∀ x ∈ R except at x = 2/3

is continuous ∀ x∈ R except at x = 3 and differentiable ∀ x ∈ R

Indeter minate  

0  

1    

2 

a maxima at x= 1 and a minima at x = 3

a maxima at x = 3 and a minima at x = 1

no maxima, but a minima at x = 1

a maxima at x = 1, but no minima

1      

–1  

∞      

– ∞

1      

π  

2      

2π 

1.0      

2.5  

4.0      

5.0

 − π

− π/2

π/2

π

0.27  

0.67  

1  

 1.22 

-1/3

1/4

1/2

1

only when x(t) is bounded

only when x(t) is non-negative

only or t ≥ 0 if x (t) is bounded for t ≥ 0

even when x(t) is not bounded 

0

1

In 2

0    

1    

e    

1/e