How is antiderivative the inverse of differentiation?What is the intuitive reason that "accumulating area" and "taking slopes" are inverses? What is the proof?

Question

How is antiderivative the inverse of differentiation?What is the intuitive reason that "accumulating area" and "taking slopes" are inverses?What is the proof?

Answer 1

start with area function F(x)=∫f(t)dt from a to x,,,,,for some continuous f(x),,,,a is a constant

can prove F'(x)=f(x),,,,,,F'(x) is slope of area function as it grows or reduces at x,,,,Fundamental Theorem of Calculus first part --prove with Mean Value Theorem for Integrals.

otherwise it is that

F(x+dx)-F(x)=f(x)dx,,,,,definition of derivative where dx=h and h->0,,,,,F(x+dx)-F(x)=dF= differential of F(x)

integration both sidesˆ

F(x)=∫f(t)dt from a to x,,,,,,same as ∫f(t)dt +C integrated from a to x

The Intuitive part of your query-----draw f(x)=x³ and mark area under curve to x-axis

from x=1 to x=3. The increment in area to be added is 27*dx,,,

x³ at x=3 is 27.

x⁴/4 is the area function(which can be determined with sums of tiny rectangles and formulas for sums) & the growth rate of area function is the value of f(x) at x=3, which 27 per unit x. That is the derivative of x⁴/4 is x³.

It is true F(x) is integral of f(x), and f(x) is the derivative of F(x) ,,for example

If it is known the integral of sin(x) = -cos(x) then derivative of -cos(x)=sin(x),,,, same as derivative of cos(x)=(-)sin(x).

If it is known the derivative of -sin(x)=--cos(x) then integral of -cos(x)=-sin(x),,,,,,,same as integral of cos(x)=sin(x).

Which means the one operation reverses the other, but this is not an inverse relationship between functions

If F(x) and f(x) are inverse functions then F[f(x)]=x and f[F(x)]=x

Answer 2

maybe the best intuition could come from looking at a simple example

such as a triangle formed by intersections of y=0, x=2, y=x

it has a slope of 1, an area of 2(2)/2 = 2

or the area of a rectangle formed by x=0, y=0, y=2, x=2

area = 2(2) = 4, slope =0

other things being equal, the greater the slope, the greater the area for a quadrant I area with a vertex at the origin

area is 2 dimensional in square units, slope is non-dimensional or a unitless pure number

or the line with the slope is 1 dimensional and is one side of the triangle, rectangle or polygon.

try a physics example

acceleration is the integral of velocity

velocity is the integral of distance

the derivative of distance = velocity

derivative of velocity = acceleration

derivative of acceleration = jerk (if you increase rate of acceleration you feel the car "jerk")

differentiate enough times and you get zero, if distance is finite

distance is in one dimensional units, such as feet

velociy is length per time, per second or per hour, ft/sec

acceleration is length per time per time, ft/sec^2

jerk is length per time per time per time ft/sec^3

rate of change of jerk is ft/sec^4 (that's Dom pouring it all on, trying to finish 1st in a close race in the Fast & Furious movies)

the denominator increases from 0 to 3 dimensions, as you integrate 0 to 3 times

integrate or differentiate and the dimensions change, slope to area is a dimensional change

integrate and get a higher dimension

integrate distance and go from 2 feet to 2x + c feet squared, an area

integrate area and get volume

integrate volume and get a 4th dimensional hyper-volume

in string theory there are 10 dimensions +

it could go on forever to some super hyper volume infinite dimensions

derivative is a rate of change

derivative of a derivative is a rate of change of the rate of change

rate of change is how fast something changes

everything changes, including what seems to stay the same, as it changes very slowly

the only constant in nature is change

except in the Twilight TV show episode where time froze. If time doesn't change, nothing changes, unless you somehow escape the constraints of time, and do time travel which is possible, in the future, but then if it's possible in the future, the future time travelers may be here among us now. sort of like space aliens disguised as us people

maybe like Prof. Sheldon Cooper explaining to Raj about "Flatland" with geometrical two dimensional figures acting like 3 D's

Okay I didn't answer the question at all, and probably didn't reach anything near intuition, but who knows?

Interesting question though. Ask Prof. Sheldon Cooper. He'll nail it and put it in a coffin, after driving a stake throught the heart. Or ask Howard Wallowitz, the engineer with "just" a masters degree, he is more practical, more intuitive, maybe.

Answer 3

That's a good question. Let me try to explain it using the function y=x. If we integrate it we get y=x^2/2+c and if we then proceed to take the derivative we go back to y=x. Hence they can transform one function into another and back again.

Your second question of how taking slopes and accumulating areas are inverses. They are not inverses they are finding different things. The derivative of a function will tell you the instantaneous slope of the original function at a given x value. The integral will tell you the area under the curve of the original function at a given x value.