- #1

Tush19

- 3

- 0

how does the first step use mean value theorem? I don't get it , can anyone explain , thanks.

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- A
- Thread starter Tush19
- Start date

- #1

Tush19

- 3

- 0

how does the first step use mean value theorem? I don't get it , can anyone explain , thanks.

- #2

- 20,004

- 10,658

$$

\int_x^{x+\delta x} f(s) ds = \delta x\, f(x^*)

$$

where ##x \leq x^* \leq x + \delta x##. Since ##f## is continuous, ##f(x^*) \to f(x)## for small ##\delta x##.

- #3

Tush19

- 3

- 0

thanks but I couldn't find that mean value theorem statement anywhere ,all it shows that mean value theorem is the following

$$

\int_x^{x+\delta x} f(s) ds = \delta x\, f(x^*)

$$

where ##x \leq x^* \leq x + \delta x##. Since ##f## is continuous, ##f(x^*) \to f(x)## for small ##\delta x##.

- #4

DrClaude

Mentor

- 8,120

- 4,935

https://www.kristakingmath.com/blog/mean-value-theorem-for-integralsthanks but I couldn't find that mean value theorem statement anywhere ,all it shows that mean value theorem is the following

- #5

- 20,004

- 10,658

Wrong mean value theorem:thanks but I couldn't find that mean value theorem statement anywhere ,all it shows that mean value theorem is the following

View attachment 295180

https://en.wikipedia.org/wiki/Mean_value_theorem#Mean_value_theorems_for_definite_integrals

- #6

- 20,004

- 10,658

$$

(b-a) f’(c) = f(b) - f(a) = \int_a^b f’(x) dx

$$

and let ##g(x) = f’(x)##. You now have

$$

\int_a^b g(x) dx = (b-a) g(c)

$$

for some ##c## such that ##a\leq c\leq b##.

- #7

Tush19

- 3

- 0

thank you so much

$$

(b-a) f’(c) = f(b) - f(a) = \int_a^b f’(x) dx

$$

and let ##g(x) = f’(x)##. You now have

$$

\int_a^b g(x) dx = (b-a) g(c)

$$

for some ##c## such that ##a\leq c\leq b##.

Share:

- Last Post

- Replies
- 5

- Views
- 336

- Last Post

- Replies
- 23

- Views
- 503

- Replies
- 3

- Views
- 531

- Last Post

- Replies
- 7

- Views
- 747

- Replies
- 3

- Views
- 369

- Replies
- 0

- Views
- 433

- Replies
- 9

- Views
- 519

- Replies
- 8

- Views
- 454

- Replies
- 2

- Views
- 405

- Replies
- 3

- Views
- 508