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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.

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mathsDifferential CalculusIntroduction to Differential Calculus

### Differentiation: Graphical Meaning

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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. 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. 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. 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. 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. 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. 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. 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. 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.

For a function f(x), the slope of tangent at x=a is d/(dx) f(x)|_(x=a). 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). 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. 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. 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. 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. •  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. 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