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Introduction to Differential Calculus

Introduction to Differential Calculus

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Differentiation : Graphical Meaning


 »  Graphical Meaning of Derivative: slope of the tangent

    →  at a point `x=a`, the rate of change of the curve is `f′(a)`

    →  at a point `x=a`, the slope of the tangent is `f′(a)`.

    →  at the maxima of the curve `f(a_1)`, the derivative `f′(a_1)` crosses `0` from positive rate of change to negative rate of change.

    →  at the minima of the curve `f(a_2)`, the derivative `f′(a_2)` crosses `0` from negative rate of change to positive rate of change.

Differentiation: Graphical Meaning

plain and simple summary

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plain and simple summary

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The slope of the tangent on curve is the derivative evaluated at that point.

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In this page, graphical meaning of differentiation is discussed with examples.


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Stating on learning "Graphical Meaning of differentiation". In this page, graphical meaning of differentiation is discussed with examples.

Summary of differentiation: Given `y=f(x)` a function of variable `x`, the derivative of `y` is

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

Which of the following is correct?

  • The derivative is another function of variable `x`
  • The derivative is a numerical value

The answer is "The derivative is another function of variable `x`".

For some functions, the derivatives are numerical values, but it is not a numerical value for all functions.

Let us consider `y=(x^2)/2` and `(dy)/(dx) = x`.plot of x^2 and 2x The figure depicts both the functions. Blue color curve is `y=(x^2)/2` and orange color line is `(dy)/(dx) = x`.
note: The plot is not to the scale on x and y axes.

How are these two plots related?

  • the rate of change of blue curve is plotted as orange line
  • the two plots are not related

The answer is "the rate of change of blue curve is plotted as orange line".

The rate of change is a function of `x`

Let us consider `y=(x^2)/2` and `(dy)/(dx) = x` given in the figure.plot of x^2 and 2x The figure zooms in a small part of the plots. What do the two values at `x=a` signify?

  • `y(a)` is the value function evaluates to at `x=a`
  • `y′(a)` is the rate of change of `y` at `x=a`
  • both the above

The answer is "both the above". This result is known from the algebraic derivations. Let us see what this means in the given curve.

Considering `y=(x^2)/2` and `(dy)/(dx) = x` given in the figure.plot of x^2 and x The derivative in first principles is given as
`y′(x) `
`= d/(dx) y`
`= lim_(delta x->0) (delta y)/(delta x)`

`delta x` at `x=a` is shown in the figure.
`delta y` for the `delta x` is shown in the figure.
How is the derivative `y′` related to `delta y` and `delta x`?

  • derivative is not related to `delta x`
  • the derivative is derived using limit `delta x` tending to `0`

The answer is "the derivative is derived using limit `delta x` tending to `0`"

Considering `y=(x^2)/2` and `(dy)/(dx) = x` given in the figure.plot of x^2 and x The figure shows non-zero `delta x` and corresponding `delta y` before limit is applied.

The points `P` and `Q` are on the curve separated by `delta x` on `x=a`. Consider the line passing through the points `P` and `Q`. Is the line a secant or a tangent?

  • secant
  • tangent

The answer is "Secant", as the line passes through two points on the curve.

Considering `y=(x^2)/2` and `(dy)/(dx) = x` given in the figure.plot of x^2 and x It shows non-zero `delta x` and corresponding `delta y` before limit is applied. What is the slope of the line `bar(PQ)`?

  • `(delta y)/(delta x)`
  • slope cannot be found.

The answer is "`(delta y)/(delta x)`". The line passing through the points `P(x_1, y_1)` and `Q(x_1+delta x, y_1+delta y)`
Slope of the secant
`=(y_2-y_1)/(x_2-x_1)`
`=(delta y)/(delta x)`

Considering `y=(x^2)/2` and `(dy)/(dx) = x` given in the figure.plot of x^2 and x The derivative in first principles is given as
`y′(x) `
`= lim_(delta x->0) (delta y)/(delta x)`

Applying the limit, the `delta x` and `delta y` reduces "close to `0`", which is shown in red.
The points `P` and `Q` at the two ends of `dx` moves towards each other. When limit `delta x` is close to zero, what happens to the two points `P` and `Q`?

  • merge to become a single point
  • cross over and move away

The answer is "merge to become a single point". The points `P` and `Q` move towards each other and merge.

Considering `y=(x^2)/2` and `(dy)/(dx) = x` given in the figure.plot of x^2 and x Applying the limit, the `delta x` and `delta y` reduces close to `0`, which is shown in red.
The points `P` and `Q` merge to a single point. The line passing through `P` and `Q` is shown in red. As the limit is applied, the line touches the curve in only one point

Is the line a secant or a tangent?

  • secant
  • tangent

The answer is "tangent", as the line touches at a single point on the curve.

Considering `y=(x^2)/2` and `(dy)/(dx) = x` given in the figure.plot of x^2 and x Applying the limit, the `delta x` and `delta y` reduces close to `0`, which is shown in red. What is the slope of the tangent at `x=a`?

  • `(dy)/(dx)|_(x=a)`
  • slope cannot be calculated

The answer is "`(dy)/(dx)|_(x=a)`". The rate of change is the slope at that point.

`(dy)/(dx)|_(x=a)` is the slope of the tangent on curve `y` at position `x=a`. plot of x squared with tangent.

For a function `f(x)`, the slope of tangent at `x=a` is `d/(dx) f(x)|_(x=a)`.plot of x squared and x The figure shows `f(x)` in blue and `d/(dx) f(x)` in orange. Three positions are identified with `3` vertical lines. Which of the following describe the slope of `f(x)` at the position `P`?

  • negative or decreasing rate of change
  • positive or increasing rate of change
  • zero or no change for small change in `x`

The answer is "negative or decreasing rate of change". This is also evident from the negative value at the same position in orange line.

For a function `f(x)`, the slope of tangent at `x=a` is `d/(dx) f(x)|_(x=a)`.plot of x squared and x The figure shows `f(x)` in blue and `d/(dx) f(x)` in orange. Which of the following describe the slope of `f(x)` at the position `Q`?

  • negative or decreasing rate of change
  • positive or increasing rate of change
  • zero or no change for small change in `x`

The answer is "positive or increasing rate of change". This is also evident from the positive value at the same position in orange line.

Figure shows `y=x+1` in blue and `(dy)/(dx) = 1` in orange. plot of x Which of the following is observed?

  • the function has constant rate of change
  • the rate of change of the function is 0

The answer is "the function has constant rate of change". This is evident from the flat orange line.

The figure shows `y=sin x` in blue and `(dy)/(dx) =cos x` in orange. plot of sin x Which of the following is observed?

  • when `y` reaches maximum, the rate of change is `0`
  • when `y` reaches minimum, the rate of change is `0`
  • both the above

The answer is "both the above". This is evident from the values on orange line, the derivative of the function, having value 0 at the positions of maxima and minima.

The figure shows `y` in blue and `(dy)/(dx)` in orange. Two positions are identified.plot of x^3-x^2 Which of the following is observed?

  • when `y` reaches maximum, the rate of change crosses `0` from positive to negative
  • when `y` reaches minimum, the rate of change crosses `0` from negative to positive
  • both the above

The answer is "both the above". This is evident from the values on orange line, the derivative of the function, crossing the x axis at the positions of maxima and minima.

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

Graphical Meaning of Derivative: For a function `f(x)`, the derivative `f′(x)` is another function of the variable.plot of x cube with tangent. •  at a point `x=a`, the rate of change of the curve is `f′(a)`

•  at a point `x=a`, the slope of the tangent is `f′(a)`.

•  at the maxima of the curve `f(a_1)`, the derivative `f′(a_1)` crosses `0` from positive rate of change to negative rate of change.

•  at the minima of the curve `f(a_2)`, the derivative `f′(a_2)` crosses `0` from negative rate of change to positive rate of change.



           

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

The figure shows `y` in blue and `(dy)/(dx)` in orange.plot of x squared big Which of the following observation is true for the given function?

  • for negative values of x, the rate of change is negative
  • for positive values of x, the rate of change is positive
  • at `x=0`, the rate of change crosses `0` from negative to positive
  • all the above

The answer is "all the above"

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Summary of differentiation: Given y = f of x, a function of variable x, the derivative of y is, d y by d x = limit delta tending to 0, f of x +delta, minus f of x, divided by delta. ;; Which of the following is correct.
another;variable;x
The derivative is another function of variable x
numerical;value
The derivative is a numerical value
The answer is "The derivative is another function of variable x". For some functions, the derivatives are numerical values, but it is not a numerical value for all functions.
Let us consider y = x squared divided by 2 and d y by d x = x. The figure depicts both the functions. Blue color curve is the function and orange color curve is the derivative. ;; How are these two plots related.
rate;change;blue;orange;curve;line
the rate of change of blue curve is plotted as orange line
2;not
the two plots are not related
The answer is "the rate of change of blue curve is plotted as orange line". the range of change is a function of x
Let us consider y = x squared divided by 2 and d y by d x = x, given in the figure. The figure zooms in a small part of the plots. What are the two values at x=a signify.
value;function;evaluates
y of a is the value function evaluates to at x = a
prime;rate;change
y prime of a is the rate of change of y at x = a
both;above
both the above
The answer is "both the above". This result is known from the algebraic derivations. Let us see what this means in the given curve.
Let us consider y = x squared divided by 2 and d y by d x = x, given in the figure. The derivative in first principles is given. ;; How is the derivative y prime relate to delta y and delta x.
not;related
derivative is not related to delta x
derived;using;limit;becomes
the derivative is derived using limit delta x tending to 0
The answer is "the derivative is derived using limit delta x tending to 0 "
Considering y = x squared divided by 2 and d y by d x = x, given in the figure. The figure shows non-zero delta x and corresponding delta y, before limit is applied. The point P and Q are on the curve separated by delta x on x=a. ;; Consider the line passing through the points P and Q. Is the line a secant or a tangent.
secant
secant
tangent
tangent
The answer is "Secant", as the line passes through two points on the curve.
Considering y = x squared divided by 2 and d y by d x = x, given in the figure. It shows non-zero delta x and corresponding delta y before limit is applied. What is the slope of the line P Q.
1
2
The answer is "delta y by delta x". The derivation of slope is given.
Considering y = x squared divided by 2 and d y by d x = x, given in the figure. The derivative in first principles is given. ;; Applying the limit, the delta x and delta y reduces, close to 0, which is shown in red. The points P and Q at the two ends of d x moves towards each other. When limit delta x is close to zero, what happens to the two points p and q.
merge;become;single
merge to become a single point
cross;over;move;away
cross over and move away
The answer is "merge to become a single point". The points P and Q move towards each other and merge.
Considering y = x squared divided by 2 and d y by d x = x, given in the figure. Applying the limit, the delta x and delta y reduces close to 0, which is hsown in red. The points P and Q merge to a single point. The line passing through P and Q is shown in red. As the limit is applied, the line touches the curve in only one point. ;; Is the line a secant or a tangent.
secant
secant
tangent
tangent
The answer is "tangent", as the line touches at a single point on the curve.
Considering y = x squared divided by 2 and d y by d x = x, given in the figure. Applying the limit, the delta x and delta y reduces close to 0, which is shown in red. What is the slope of the tangent.
1
2
The answer is "d y by d x at x equals a". The rate of change is the slope at that point.
d y by d x at x = a is the slope of the tangent on curve y at position x=a.
For a function f of x, the slope of tangent at x=a, is d by d x f of x at x =a. The figure shows the function in blue and derivative in orange. Three positions are identified with 3 vertical lines. Which of the following describe the slope of f of x at the position P.
negative;decreasing
negative or decreasing rate of change
positive;increasing
positive or increasing rate of change
0;no;small
zero or no change for small change in x
The answer is "negative or decreasing rate of change". This is also evident from the negative value at the same position in orange line.
For a function f of x, the slope of tangent at x=a, is d by d x f of x at x=a. The figure shows the function in blue, and the derivative in orange. Which of the following describe the slope of f of x at the position Q.
negative;decreasing
negative or decreasing rate of change
positive;increasing
positive or increasing rate of change
0;no;small
zero or no change for small change in x
The answer is "positive or increasing rate of change". This is also evident from the positive value at the same position in orange line.
Figure shows y = x plus 1 in blue and d y by dx =1 in orange. Which of the following is observed.
1
2
The answer is "the function has constant rate of change". This is evident from the flat orange line.
The figure shows y = sine x in blue and d y by d x = cos x in orange. Which of the following is observed.
maximum
when y reaches maximum, the rate of change is 0
minimum
when y reaches minimum, the rate of change is 0
both;above
both the above
The answer is "both the above". This is evident from the values on orange line, the derivative of the function, having value 0 at the positions of maxima and minima.
The figure shows y in blue and d y by dx in orange. Two positions are identified. Which of the following is observed.
maximum
when y reaches maximum, the rate of change crosses 0 from positive to negative
minimum
when y reaches minimum, the rate of change crosses 0 from negative to positive
both;above
both the above
The answer is "both the above". This is evident from the values on orange line, the derivative of the function, crossing the x axis at the positions of maxima and minima.
The slope of the tangent on curve is the derivative evaluated at that point.
Graphical Meaning of Derivative: For a function f of x, the derivative f prime x is another function of the variable. ;; at a point x=a, the rate of change of the curve is f prime of a. ;; at a point x=a, the slope of the tangent is f prime of a.;; At the maxima of the curve f of a 1, the derivative f prime of a 1 crosses 0 from positive to negative rate of change. ;; At the minima of the curve f of a 2, the derivative f prime of a 2 crosses 0 from negative to positive rate of change.
The figure shows y in blue and d y by d x in orange. Which of the following observation is true for the given function.
negative
for negative values of x, the rate of change is negative
positive
for positive values of x, the rate of change is positive
0;2;to
at x=0 , the rate of change crosses 0 from negative to positive
all;above
all the above
The answer is "all the above"

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