Question

Running out of calc problems for the night, get them while they're hot! Picture attached.

Answer 1

you are given that f'(x)=e^(8x)

1. integrate the f'(x) to get the f(x).
2. use the fact that f(0)=9/8 to solve for C.

Answer 2

first, what is \(\large\color{blue}{\displaystyle \int_{}^{}e^{8x}~dx}\) ?

Answer 3

Afraid I don't think we've learned integration. She said we should be able to do these intuitively but I'm struggling!

Answer 4

you can recognize a derivative.

Answer 5

Mhmm!

Answer 6

e^8x+C?

Answer 7

No not really....

lets do it together.

first, \(\large\color{blue}{\displaystyle \frac{d}{dx}e^{8x}=8e^{8x}}\). correct?

Answer 8

Thanks I would appreciate that! Yeah that's correct with the chain rule

Answer 9

yes, and what coefficient in front of the e^(8x) would make the derivative of e^(8x) not 8e^(8x), BUT rather just e^(8x) ?

Answer 10

1/8

Answer 11

yes, so your function f(x) is ?

Answer 12

1/8e^(8x)+C?

Answer 13

yes

Answer 14

Now, you know that f(0)=9/8, because that what having the point (0, 9/8) means.

Answer 15

Mhmm!

Answer 16

\(\large\color{blue}{\displaystyle f(x)=\frac{1}{8}e^{8x}+{\rm\color{red}{ C}} }\)

\(\large\color{blue}{\displaystyle \frac{9}{8}=\frac{1}{8}e^{8(0)}+{\rm\color{red}{ C}} }\)

Answer 17

that is how we are using our point to solve for C.

Answer 18

is this making sense ?

Answer 19

Yeah! :D

Answer 20

so C= ?

Answer 21

1? Or am I crazy

Answer 22

no no... 1 is correct.

Answer 23

e^(8x) becomes 1 when x=0,
and 1 times 1/8 is 1/8.

subtract 1/8 from both sides, and you get C=1.

Answer 24

Yep! Had to check what e^0 was

Answer 25

1/8e^(8x)+1

Answer 26

:D

Answer 27

Thanks a lot for your help!

Answer 28

\(\large\color{blue}{\displaystyle f(x)=\frac{1}{8}e^{8x}+{\rm\color{red}{ C}} \\[1.5em]}\)
\(\large\color{blue}{\displaystyle \frac{9}{8}=\frac{1}{8}e^{8(0)}+{\rm\color{red}{ C}} \\[1.5em]}\)
\(\large\color{blue}{\displaystyle \frac{9}{8}=\frac{1}{8}(1)+{\rm\color{red}{ C}} \\[1.5em]}\)
\(\large\color{blue}{\displaystyle \frac{9}{8}=\frac{1}{8}+{\rm\color{red}{ C}} \\[1.5em]}\)
\(\large\color{blue}{\displaystyle 1={\rm\color{red}{ C}} \\[1.5em]}\)

and yes,

\(\large\color{green}{\displaystyle f(x)=\frac{1}{8}e^{8(0)}+{\bf\color{darkgoldenrod}{ C}} \\[1.5em]}\)

Answer 29

sure anytime:)

Answer 30

I mean
\(\large\color{blue}{\displaystyle f(x)=\frac{1}{8}e^{8(0)}+{\rm\color{red}{ 1}} \\[1.5em]}\)

Answer 31

I'm about to post the last problem that has me stumped if you have an extra second! It's probably simple too the format is just confusing to me :)

Answer 32

wrote the C again, forgot to change my latex

Answer 33

yes, I got some time

Answer 34

(I think)

Answer 35

Okay I will open a new question and tag you!

Answer 36

sure