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In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

Read in the blogs more about the unique learning experience at nubtrek.continue

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Fundamental Theorem of Calculus

For functions `f(x), h(x), g(x)` as in`h(x) = d/(dx) f(x)`

`f(x) = int_a^x g(x)dx`

→ First part states that `h(x) = g(x)`; that is `d/(dx) int_a^x g(x) dx = g(x)`

→ First part implies that to find `f(x) = int_a^x g(x)dx`, find the inverse relationship `d/(dx) f(x) = g(x)` and thus, anti-derivative is defined.

→ Second part states that `int_a^b h(x) dx = f(b)-f(a)`

That is `int_a^b d/(dx) f(x) dx = f(b)-f(a)`.

→ Since the second part is true for any value `x` in the interval `[a,b]`, it implies that `int_a^x d/(dx) f(x) dx = f(x) +c`.

*plain and simple summary*

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Derivative and Integral are inverse operations.

• Derivative of integral of a function is the function.

• Integral of derivative of a function is the function or equivalent.

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In this page, the fundamental theorem of calculus is stated and proven. Anti-derivatives are introduced as the function that when differentiated, gives the integrand.

Starting on learning "". In this page, the fundamental theorem of calculus is stated and proven. Anti-derivatives are introduced as the function that when differentiated, gives the integrand.

This topic is going to be little mathematically involved and firmly anchors some of the concepts in integrals. To prove Fundamental Theorem of Calculus, let us first understand "mean-value-theorem" and "squeeze theorem".

Mean-Value-Theorem for Derivatives: For a continuous and differentiable function in the closed interval `[a,b]`, there exists `c` such that `a<=c<=b`

`d/(dx)f(x)|_(x=c)` `= (f(b)-f(a)) //(b-a)`

The same can be written as

`f′(c)` `= (f(b)-f(a)) //(b-a)`. The figure graphically explains the mean value theorem :

• ` (f(b)-f(a)) //(b-a)` is illustrated by the slope of hypotenuse of red triangle. In this, numeration and denominator are the two arms of right angle.

• `d/(dx)f(x)|_(x=c)` is illustrated by the slope of tangent given by orange line

Which of the following is correct?

- the tangent at `c` and the hypotenuse shown between `a` and `b` have same slope
- there is at-least one point `c` in which slope of tangent equals slope of the hypotenuse of red triangle
- both the above

The answer is "both the above"

Mean-Value-Theorem for integrals: For a continuous and integrable function in the closed interval `[a,b]`, there exists `c` such that `a<=c<=b`

`int_a^b f(x)dx ``= f(c) xx (b-a)`

The same can be written as

`F(b)-F(a) ``= f(c) xx (b-a)`

where `F(x)` is the integral of `f(x)`, given by `F(x) = int_a^x f(x)dx`. The figure graphically explains the mean value theorem:

• `F(b)-F(a)` is area under the curve shown in red.

• `f(c) xx (b-a)` is the area of rectangle shown in green.

Note: There is substantial overlap between red and green area in the figure.

Which of the following is correct?

- area under the curve in red and the area of the rectangle in green are equal
- there is at-least one point `c` in which area of the rectangle equals the area under the curve
- both the above

The answer is "both the above"

Does the name "mean-value-theorem" provide any clues as to what the theorem is about?

- Yes, one of the values in an interval is the mean for the interval.
- No, this does not have any meaning.

The answer is "Yes, one of the values in an interval is the mean for the interval."

Squeeze theorem: A value `c` is given such that `a<=c<=b`. If `a-b` tends to `0`, then `c=a=b`.

Does the name of the theorem provide any clues as to what the theorem is about?

- Yes, When an interval `a` and `b` is squeezed to `a=b`, then a value between these two `c` is squeezed to be equal to the two values.
- No, this does not have any meaning.

The answer is "Yes, When an interval `a` and `b` is squeezed to `a=b`, then a value between these two, `c` is squeezed to be equal to the two values."

Given a continuous function `f(x)` in a closed interval `[a,b]` and differentiable in open interval `(a,b)`.

`h(x) = d/(dx) f(x)`

`f(x) = int_a^x g(x)dx`

First part of Fundamental Theorem of Calculus states that `h(x) = g(x)`; that is `d/(dx) int_a^x g(x) dx = g(x)`

Note: *Derivative of integral* of a function is the function.

What does the above mean?

- derivative is the inverse of integral
- derivative and integral are not related

The answer is "derivative is the inverse of integral".

*Given a continuous function `f(x)` in a closed interval `[a,b]` and differentiable in open interval `(a,b)`. `h(x) = d/(dx) f(x)` `f(x) = int_a^x g(x)dx` First part of Fundamental Theorem of Calculus states that `h(x) = g(x)`; that is `d/(dx) int_a^x g(x) dx = g(x)`* First part also implies the abstraction "indefinite integral" or anti-derivatives.

`int g(x)dx = c+ int_0^x g(x)dx`.

In indefinite integral, `f(x) = int_a^x g(x)dx`, the function `f(x)` is derived from aggregate of change or, equivalently, by limit of summation.

In anti-derivative, `f(x) +c = int g(x)dx`, the function `f(x)` is derived from the inverse operation of derivative. That is, derive `f(x)` such that, derivative of `f(x)` is `g(x)`.

In other words, `f(x) +c = int g(x)dx` means the anti-derivative of `g(x)` is `f(x)` (even though the representation uses integral symbol). The name anti-derivatives is coined to specify this.

What is the other name for indefinite integral?

- anti-derivatives
- algebraic integral

The answer is "anti-derivatives"

Given a continuous function `f(x)` in a closed interval `[a,b]` and differentiable in open interval `(a,b)`.

`h(x) = d/(dx) f(x)`

`f(x) = int_a^x g(x)dx`

Second part of Fundamental Theorem of Calculus states that `int_a^b h(x) dx = f(b)-f(a)`

That is `int_a^b d/(dx) f(x) dx = f(b)-f(a)`.*Integral of derivative* of a function is the function evaluated at the limits of integral.* (stated for definite integral.)*

What does the above mean?

- integral is the inverse of derivative
- integration of a derivative is not possible

The answer is "integral is the inverse of derivative"

*Given a continuous function `f(x)` in a closed interval `[a,b]` and differentiable in open interval `(a,b)`. `h(x) = d/(dx) f(x)` `f(x) = int_a^x g(x)dx` Second part of Fundamental Theorem of Calculus states that `int_a^b h(x) dx = f(b)-f(a)` That is `int_a^b d/(dx) f(x) dx = f(b)-f(a)`.* Since the second part is true for any value `x` in the interval `[a,b]`, it can be given as `int_a^x d/(dx) f(x) dx = f(x) +c`.

What does the above mean?

- integral is the inverse of derivative
- integration of a derivative is not possible

The answer is "integral is the inverse of derivative"

Proof for First part of Fundamental Theorem: *Given a continuous function `f(x)` in a closed interval `[a,b]` and differentiable in open interval `(a,b)`. `h(x) = d/(dx) f(x)` `f(x) = int_a^x g(x)dx` First part of Fundamental Theorem of Calculus states that `h(x) = g(x)`; that is `d/(dx) int_a^x g(x) dx = g(x)`* `h(x)`

`=d/(dx) f(x)`

`= lim_(delta->0) (f(x+delta) - f(x)) / delta`

`= lim_(delta->0) (int_a^(x+delta) g(x)dx ``- int_a^(x) g(x)dx) // delta`

`= lim_(delta->0) (int_a^x g(x)dx ``+ int_x^(x+delta) g(x)dx ``- int_a^(x) g(x)dx) // delta`

`= lim_(delta->0) (int_x^(x+delta) ``g(x)dx) // delta`

`= lim_(delta->0) (g(c)delta) // delta`

`= lim_(delta->0) g(c)`

`= g(x)`

What does the above prove?

- `d/(dx) int_a^x g(x) dx = g(x)`
- `int_a^x d/(dx) f(x) dx = f(x)+c`

The answer is "`d/(dx) int_a^x g(x) dx = g(x)`"

Proof for Second part of Fundamental Theorem: *Given a continuous function `f(x)` in a closed interval `[a,b]` and differentiable in open interval `(a,b)`. `h(x) = d/(dx) f(x)` `f(x) = int_a^x g(x)dx` Second part of Fundamental Theorem of Calculus states that `int_a^b h(x) dx = f(b)-f(a)` That is `int_a^b d/(dx) f(x) dx = f(b)-f(a)`.* To simplify the statements, take `a=0`

`f(b)-f(0)`

`= sum_(i=1)^n (f((b i)/n)-f((b(i-1))/n)`

`= sum_(i=1)^n b/n d/(dx) f(x)`

`= sum_(i=1)^n h(c) b/n`

`= lim_(n->oo) sum_(i=1)^n h(c) b/n`

`= lim_(n->oo) sum_(i=1)^n h((bi)/n) b/n`

`= int_0^b h(x)dx `

What does the above prove?

- `d/(dx) int_a^x g(x) dx = g(x)`
- `int_a^x d/(dx) f(x) dx = f(x)+c`

The answer is "`int_a^x d/(dx) f(x) dx = f(x)+c`"

The significance of Fundamental Theorem of Calculus:

The relationship between derivative and integral is understood as *inverse operations*.

Summary of what we have learned:

• Differentiation, by the first principles, is the instantaneous-rate-of-change.

• Integration, by the first principles, is the continuous-aggregate-of-change.

• Integration is addition of change to an initial value.

• Indefinite integral is defined as a function of variable, where integration is carried out between `0` and variable `x`, along with an initial constant.

• Definite integral is defined as the quantity added for a change in interval `a` and `b`. This results in a numerical value.

• First part of fundamental theorem of calculus states that derivative of integral of a function is the function itself.

• Second part of fundamental theorem of calculus states that integral of derivative of a function is the function or an equivalent numerical result.

*comprehensive information for quick review*

Jogger

*comprehensive information for quick review*

Jogger

dummy

**Fundamental Theorem of Calculus:** Given a continuous function `f(x)` in a closed interval `[a,b]` and differentiable in open interval `(a,b)`.

`h(x) = d/(dx) f(x)`

`f(x) = int_a^x g(x)dx`

First part states that `h(x) = g(x)`; that is `d/(dx) int_a^x g(x) dx = g(x)`

First part implies that to find `f(x) = int_a^x g(x)dx`, find the inverse relationship `d/(dx) f(x) = g(x)` and thus, anti-derivative is defined.

Second part states that `int_a^b h(x) dx = f(b)-f(a)`

That is `int_a^b d/(dx) f(x) dx = f(b)-f(a)`.

Since the second part is true for any value `x` in the interval `[a,b]`, it implies that `int_a^x d/(dx) f(x) dx = f(x) +c`.

*practice questions to master the knowledge*

Exercise

*practice questions to master the knowledge*

Exercise

*your progress details*

Progress

*About you*

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This topic is going to be little mathematically involved and firmly anchors some of the concepts in integrals. To prove Fundamental Theorem of Calculus, let us first understand "mean-value-theorem" and "squeeze theorem".

mean value theorem for derivatives: For a continuous and differential function in the closed interval a to b, there exists c such that, a, less than or equal, c, less than or equal b, d by d x, of f of x, at x = c = f of b minus f of a divided by b minus a. The same can be written as f prime c . The figure graphically explains the mean value theorem. Which of the following is correct?

slope;tangent;hypotenuse

the tangent at c and the hypotenuse shown between a and b have same slope

there;which;red;triangle

there is at-least one point c in which slope of tangent equals slope of the hypotenuse of red triangle

both;above

both the above

The answer is "both the above"

mean value theorem for integrals: for a continuous and integrable function in the closed interval a to b. There exists c such that, a, less than or equal, c, less than or equal b, integral a to b f of x d x, = f of c multiplied b minus a. The same can be written as capital F of b minus capital f of a equals f of c multiplied b minus a. Where capital f of x is the integral of f of x, given by capital f of x, = integral a to x f of x dx. The figure graphically explains the mean value theorem. which of the following is correct.

area;curve;red;green

area under the curve in red and the area of the rectangle in green are equal

at;least;1;point

there is at-least one point c in which area of the rectangle equals the area under the curve

both;above

both the above

The answer is "both the above"

Does the name "mean-value-theorem" provide any clues as to what the theorem is about?

yes;s;1;values;interval

Yes, one of the values in an interval is the mean for the interval.

no;not;any

No, this does not have any meaning.

The answer is "Yes, one of the values in an interval is the mean for the interval."

Squeeze theorem: A value c is given such that, a, less than or equal, c, less than or equal, b. If a minus b tends to 0, then c equals a equals b. Does the name of the theorem provide ant clues as to what the theorem is about.

1

2

The answer is "Yes, When an interval a and b is squeezed to a=b , then a value between these two, c is squeezed to be equal to the two values."

Given a continuous function f of x in a closed interval a to b, and differentiable in the open interval a to b. ;; h of x = derivative of f of x ;; and f of x = integral of g of x ;; First part of fundamental theorem of calculus states that h of x equals g of x. That is derivative of integral of g of x = g of x. Note: Derivative of integral of a function is the function. ;; What does the above mean.

inverse

derivative is the inverse of integral

related

derivative and integral are not related

The answer is "derivative is the inverse of integral".

Repeating the first part of fundamental theorem of calculus. First part also implies the abstraction indefinite integral or anti-derivatives. integral g of x d x = c + integral 0 to x, g of x d x. Note that the left hand side is the indefinite integral and right hand side involves definite integral. ;; In indefinite integral, f of x = integral a to x, g of x d x, the function f of x is derived from aggregate of change or equivalently, by limit of summation. ;; In anti-derivative f of x + c = integral g of x d x, the function f of x is derived from the inverse operation of derivative. That is, derive f of x such that, derivative of f of x is g of x . In other words, f of x + c = integral g of x dx means the anti-derivative of g of x is f of x. The name anti-derivative is coined to specify this. What is the other name for indefinite integral.

anti-derivatives;derivatives;anti

anti-derivatives

algebraic;integral

algebraic integral

The answer is "anti-derivatives"

Given a continuous function f of x in a closed interval a to b, and differentiable in the open interval a to b. ;; h of x = derivative of f of x ;; and f of x = integral of g of x ;; Second part of the fundamental theorem of calculus states that integral a to b of h of x d x = f of b minus f of a. That is integral a to b, d by d x of f of x d x = f of b minus f of a. Integral of derivative of a function is the function evaluated at the limits of integral. What does the above mean.

inverse

integral is the inverse of derivative

possible;not

integration of a derivative is not possible

The answer is "integral is the inverse of derivative"

Repeating the second part of fundamental theorem of calculus. Since the second part is true for any value x in the interval a to b, it can be given as, integral a to x, d by d x f of x d x = f of x + c. Integral of derivative of a function is the function with a constant of integration. What does the above mean.

inverse

integral is the inverse of derivative

possible;not

integration of a derivative is not possible

The answer is "integral is the inverse of derivative"

Proof for First part of Fundamental Theorem is given. What does this prove?

1

2

The answer is "derivative of integral is the same function"

Proof for Second part of Fundamental Theorem is given. What does the above prove.

1

2

The answer is "integral of derivative is the same function"

The significance of Fundamental Theorem of Calculus: The relationship between derivative and integral is understood as inverse operations.

Summary of what we have learned: ;; Differentiation, by the first principles, is the instantaneous-rate-of-change. ;; Integration, by the first principles, is the continuous-aggregate-of-change. ;; Integration is addition of change to an initial value. ;; Indefinite integral is defined as a function of variable, where integration is carried out between 0 and variable x, along with an initial constant. ;; Definite integral is defined as the quantity added for a change in interval a and b. This results in a numerical value. ;; First part of fundamental theorem of calculus states that derivative of integral of a function is the function itself. ;; Second part of fundamental theorem of calculus states that integral of derivative of a function is the function or an equivalent numerical result.

Derivative and Integral are inverse operations. ;; Derivative of integral of a function is the function. ;; Integral of derivative of a function is the function or equivalent.

Fundamental Theorem of Calculus: Given a continuous function f of x in a closed interval a to b, and differentiable in the open interval a to b. h of x is the derivative of f of x. f of x is the indefinite integral of g of x ;; First part state that h of x = g of x ; that is derivative of integral of a function g of x is the function g of x itself. First part implies that to find f of x = integral a to x g of x dx, find the inverse relationship d by dx of f of x = g of x and thus, anti-derivative is defined. Second part states that integral a to b h of x d x = f of b minus f of a. That is integral a to b, d by d x of f of x, dx = f of b minus f of a. Since the second part is true for any value of x in the interval a to b, it implies that integral from a to x d by d x of f of x = f of x + c.