Determine antiderivative of f. So I have a function: f ( t ) = { <mt

Pattab 2022-07-01 Answered
Determine antiderivative of f.
So I have a function:
f ( t ) = { 1 , if  2 t 0 2 t 2 6 t + 9 , if  0 < t < 6 1 , if  6 t 8
I need to find the antiderivative of f(t) on the interval: 2 < t < 8.
And I need to investigate whether the antiderivative has a maximum value on the interval.
So far I've figured out that the second function 2 t 2 6 t + 9 can be described as an absolute value: 2 | t 3 |.
That makes it 4 different functions in total that describes f(t)
However, I'm not sure exactly how to formulate a function for the antiderivative on the given interval.
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Answers (1)

Leslie Rollins
Answered 2022-07-02 Author has 25 answers
Step 1
First let's be explicit about what you say you already figured out:
f ( t ) = { 1 , if  2 t 0 t 1 , if  0 < t 3 5 t , if  3 < t < 6 1 , if  6 t 8
Step 2
F ( x ) = { 2 x f ( t ) d t if  2 x 0 , 2 0 f ( t ) d t + 0 x f ( t ) d t if  0 x 3 , 2 0 f ( t ) d t + 0 3 f ( t ) d t + 3 x f ( t ) d t if  3 x 6 , and so on.
Note that here I've written '' rather than < '' since altering the value of f at isolated points does not affect the integral.

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