Question

Calculus 1
Question is attached.

Answer 1

Sorry, my internet connection suddenly slows down when i try to upload a picture.

Answer 2

4d?

Answer 3

sorry number 5

Answer 4

lol oh that is a choice
\[\lim_{h \rightarrow 0}\frac{1}{k}\ln(\frac{2+h}{2})\]

Answer 5

I didn't realize it was a multiple choice thingy

Answer 6

\[F'(x)=f(x) \\ \int\limits_a^b f(x) dx=?\]
What does F'=f mean?
I will give you a hint that F is the ____-derivative of f.

Answer 7

I will also give you another hint:
fundamental theorem of calculus

Answer 8

for example:
how do you evaluate this:
\[\int\limits_{1}^{2}x^2 dx\]

Answer 9

Since i know that the derivative of F(x) is itself i can just switch them around in the integral equation right?

Answer 10

recall:
\[\frac{d}{dx}(\frac{x^3}{3})=x^2 \text{ for all } x \\ \ \text{ so } \int\limits_1^2 x^2 dx=[\frac{x^3}{3}]_1^2 =\frac{2^3}{3}-\frac{1^3}{3}\]


you are given
\[\frac{d}{dx}(F)=f \text{ for all } x \\ \int\limits_a^b f dx=[ ? ]_a^b\]
and that f is continuous which is another important thing

Answer 11

but wouldnt, f(x) in your example be the same function as its derivative?

Answer 12

are you saying f=f'?

Answer 13

math is case sensitive
so when they say F they don't mean f

Answer 14

so no we aren't given f'=f

Answer 15

do you know usually to integrate you need to find the antiderivative of the expression that is the integrand ?

Answer 16

so if we are given F'=f
that means the antiderivative of f is F
since F'=f

Answer 17

They do not give is the functions.
but lets assume the function is e^x

Answer 18

did you not understand the example I gave above?

Answer 19

i did not understand it

Answer 20

\[\frac{d}{dx}(\frac{x^3}{3})=x^2 \text{ for all } x \\ \ \text{ so } \int\limits_1^2 x^2 dx=[\frac{x^3}{3}]_1^2 =\frac{2^3}{3}-\frac{1^3}{3}\]
I started off exactly as your question did

Answer 21

replace the x^2 with f
and replace the x^3/3 with F
you can do this since (x^3/3)'=x^2 and (F)'=f

Answer 22

the problem with that is the fact that F and f are not the same for all values of x like the question stated.

Answer 23

No it is saying F'(x)=f(x) for all x

Answer 24

Also why do F and f have to be the same?
You are definitely not given that.

Answer 25

F' and f have to be the same for all x

Answer 26

which they are because when you differentiate (x^3/3) you do get x^2

Answer 27

x^2=x^2 for all x

Answer 28

http://mathworld.wolfram.com/FundamentalTheoremsofCalculus.html
This is just the fundamental theorem of calculus

Answer 29

Okay, I think I have a hard time understanding this because we never learned the fundamental principle of calculus and I am currently studying for the final, so that means my teacher never intended to teach that concept.

Answer 30

so you guys never cover definite integrals?

Answer 31

covered*

Answer 32

We did, but we were given the rules.

Answer 33

so have you ever done the one or know how to do the one I mentioned before:
\[\int\limits_1^2 x^2 dx\]

Answer 34

We were not taught how to evaluate an integral using the definition.

Answer 35

like how would you tackle that one then?

Answer 36

I can do definite integrals. Why did you choose x^2? as one of the functions?

Answer 37

I can choose 1 or x is doesn't matter
it is just an example

Answer 38

would e^x work then?

Answer 39

\[\int\limits_1^2 1 dx=?\]
sure we can use whatever function that is continuous and has a continuous derivative

Answer 40

I just want to see what you do to evaluate something like that if you never been taught the fundamental theorem of calculus

Answer 41

Like do you not normally find the antiderivative of the integrand ?

Answer 42

answer is 1

Answer 43

I know but I want to know how you get there

Answer 44

like what steps do you take

Answer 45

ill draw what i did