Let f(x)=0 if x is less than -4 3 if x is greater than or equal to -4 and less than -1 -4 if x is greater than or equal to -1 and less than 4 0 if x is greater than or equal to 4 and g(x)=...

Let f(x)=

0 if x is less than -4

3 if x is greater than or equal to -4 and less than -1

-4 if x is greater than or equal to -1 and less than 4

0 if x is greater than or equal to 4

and g(x)= integrate from -4 to x of f(t)dt


Determine the value of each of the following:
g(5)

Asked on by rugby4life

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jeew-m | College Teacher | (Level 1) Educator Emeritus

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f(x) = 0 when x<-4

f(x) = 3 when -4<=x<-1

f(x) = -4 when -1<=x<4

f(x) = 0 when x>=4

 

`g(x) = int^x_(-4)f(t) dt`

Since f(x) changes with x we have to break the integral to several parts as x chages.

 

When `-4lt=xlt-1`

g(x)

`= int^x_(-4)f(t) dt`

`= int^x_(-4)3dt`

`= [3x]^x_(-4)`

`= 3x+12`

 

 

When `-1lt=xlt4`

g(x)

`= int^x_(-4)f(t) dt`

`= int^x_(-4)(-4) dt`

`= [-4x]^x_(-4)`

`= -4x-16`

 

 

When x>4

g(x)

`= int^x_(-4)f(t) dt`

`= int^x_(-4)(0) dt`

`= 0`

 

From `-4lt=xlt-1`

g(x) = 3x+12

When` -1lt=xlt4`

g(x) = -4x-16

When `xgt=4`

g(x) = 0

 

g(5) will occur when x>5 related to x>=4

g(5) = 0

Sources:

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