Question

f(x)=definite integral 1/(4+e^t)dt, [0, ln x], with x>0.
find (a) f(1)
(b) f'(1)
(c) inverse of f(0)

Answer 1

\[f(x)=\int\limits_{t=0}^{\ln x} dt \qquad \dfrac{1}{4+e^t} \]

Answer 2

Hint for part (a)


\[\Large f(x)=\int\limits_{t=0}^{t=\ln x} \dfrac{1}{4+e^t}dt\]


\[\Large f({\color{red}{x}})=\int\limits_{t=0}^{t=\ln ({\color{red}{x}})} \dfrac{1}{4+e^t}dt\]


\[\Large f({\color{red}{1}})=\int\limits_{t=0}^{t=\ln ({\color{red}{1}})} \dfrac{1}{4+e^t}dt\]

Answer 3

so that mean that f(1) = 0

Answer 4

why is f(1) equal to 0?

Answer 5

you are correct, but I want to make sure you are using the right integral property

Answer 6

it is because The Second Fundamental Theorem of Calculus

If ƒ is continuous on an open interval I containing a, then for every x in the interval,



ƒ(t) dt = ƒ(x)

Answer 7

no, it's because of this property

\[\Large \int_{a}^{a}f(x)dx = 0\]

Answer 8

the area under the curve from x = a to x = a is going to be 0 because the width of this region is 0 units wide

Answer 9

think of an infinitely thin strip or line with 0 area

Answer 10

but it is going from x=o to x= ln(1)

Answer 11

and ln(1) = 0

Answer 12

\[\Large \log_b(1) = 0\] for any valid base b

Answer 13

yes , you're right

Answer 14

so
\[\Large f({\color{black}{1}})=\int\limits_{t=0}^{t=\ln ({\color{black}{1}})} \dfrac{1}{4+e^t}dt\]
turns into
\[\Large f({\color{black}{1}})=\int\limits_{t=0}^{t=0} \dfrac{1}{4+e^t}dt\]

Answer 15

for part (b), you'll need to use the fundamental theorem of calculus

http://aleph0.clarku.edu/~ma120/FTC.jpg

the second part of it. Don't forget to use the chain rule

Answer 16

for part b, is it also 0

Answer 17

tell me what you got for \(\Large f \ '(x)\)

Answer 18

this page may help when it comes to the chain rule:
http://www.sosmath.com/calculus/integ/integ03/integ03.html

Answer 19

Look at the part that has
\[\LARGE F(x) = \int_{a}^{u(x)}f(t)dt\]

Answer 20

notice just under that is
\[\LARGE F \ '(x) = f{\Huge(}u(x){\Huge)}u \ '(x)\]

Answer 21

`for part b, is it also 0`

no, the answer to part (b) is NOT zero

Answer 22

so part b is 1/5

Answer 23

how did you get 1/5

Answer 24

what f ' (x) function did you get?