   Chapter 3.9, Problem 41E

Chapter
Section
Textbook Problem

# 23-42 Find f. f ′ ′ ( t ) = t 3 − cos t ,   f ( 0 ) = 2 ,   f ( 1 ) = 2

To determine

To find:

The function f(t)

Explanation

1) Concept:

If F is an antiderivative of f on an interval I, then the most general antiderivative of f on I is Fx+c where c is an arbitrary constant.

2) Definition:

A function F  is called an antiderivative of f on an interval I if F'x=fx f for all x in I.

3) Given:

f''t=t3-cost, f0=2 and f1=2

4) Calculations:

Here f''t=t3-cost

f''t=t13-cost

The general antiderivative of f't using rules of antiderivative is

f't=t4343-sint+C , Where C is the arbitrary constant

f't=34t43-sint+C

Userules of antiderivative once more to find f(t)

ft=34×t7373+cost+Ct+D where,C & D are the arbitrary constants

ft=34×37t73+cost+Ct+D

ft=928t73+cost+Ct+D

It is given that f<

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