## Problem Statement### Given:The second derivative of a function $ f(t) $ is provided as follows:\[ f''(t) = 4t - 10\sqrt{t} \]### Tasks:a) Find the most general formula for $ f'(t) $.b) Based on answer a, find $ f'(t) $.c) Based on answer b, find $ f(t) $.### Additional Notes:- The notation in the top right-hand corner indicates an "arbitrary constant" denoted by "C".## Solution Steps### a) Finding the General Formula for $ f'(t) $Start by integrating $ f''(t) $ to find $ f'(t) $. Use an arbitrary constant $ C $ since it's an indefinite integral.### b) Calculating $ f'(t) $Substitute specific values or conditions given (if any) to determine the constant from part a.### c) Solving for $ f(t) $Integrate $ f'(t) $ to find $ f(t) $, adding another constant $ C $ if further general solutions are needed.## ExplanationIn these steps, you will utilize methods from calculus, predominantly integration, to find the general functions and their particular solutions. The results will often depend on the conditions or initial values provided for the problem set.

It is given that, f'''t=4t-10t ......1 we have to find a f''t ,b f't…

Answered: arbitrary.constant f"" (+) = 4t-105E

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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## Problem Statement

### Given:
The second derivative of a function $ f(t) $ is provided as follows:
\[ f''(t) = 4t - 10\sqrt{t} \]

### Tasks:
a) Find the most general formula for $ f'(t) $.

b) Based on answer a, find $ f'(t) $.

c) Based on answer b, find $ f(t) $.

### Additional Notes:
- The notation in the top right-hand corner indicates an "arbitrary constant" denoted by "C".

## Solution Steps

### a) Finding the General Formula for $ f'(t) $

Start by integrating $ f''(t) $ to find $ f'(t) $. Use an arbitrary constant $ C $ since it's an indefinite integral.

### b) Calculating $ f'(t) $

Substitute specific values or conditions given (if any) to determine the constant from part a.

### c) Solving for $ f(t) $

Integrate $ f'(t) $ to find $ f(t) $, adding another constant $ C $ if further general solutions are needed.

## Explanation

In these steps, you will utilize methods from calculus, predominantly integration, to find the general functions and their particular solutions. The results will often depend on the conditions or initial values provided for the problem set.

Expert Solution

Step 1

$I t i s g i v e n t h a t, f''' (t) = 4 t - 10 \sqrt{t} . . . . . . (1) w e h a v e t o f i n d (a) f'' (t), (b) f' (t) a n d (c) f (t)$

$(a) O n i n t e g r a t i n g, e q u a t i o n (1) w . r . t' t' \int f''' (t) d t = \int (4 t - 10 \sqrt{t}) d t f'' (t) = (\frac{4 t^{2}}{2} - 10 \times \frac{2 (t) \frac{3}{2}}{3}) + C_{1}$

$f'' (t) = (2 t^{2} - \frac{20 (t) \frac{3}{2}}{3}) + C_{1} . . . . . . . . (2)$

$(b) A g a i n, i n t e g r a t i n g e q u a t i o n (2) w . r . t' t' . \int f'' (t) d t = \int (2 t^{2} - \frac{20 {(t)}^{\frac{3}{2}}}{3} + C_{1}) d t f' (t) = (\frac{2 t^{3}}{3} - \frac{8 (t) \frac{5}{2}}{3} + C_{1} t) + C_{2} . . . . . . . . . . (3)$

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