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Thought-Process to Discover Knowledge

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Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

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
mathsDifferential CalculusIntroduction to Differential Calculus

Differentiation: Graphical Meaning

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



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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 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 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
  • 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 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
  • 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)`
  • `(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
  • 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
  • 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)`
  • `(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
  • 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
  • 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 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
  • 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
  • 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(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.

Solved Exercise Problem:

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
  • all the above

The answer is "all the above"

                            
slide-show version coming soon