Geometric Interpretations

Introduction

Differentiation is an important concept in Mathematics that binds it with Physics. Every theorem and mathematical concept is incomplete without knowing their geometric interpretations. Thus, in this Geometric interpretations study material, we will discuss geometric interpretations of some fundamental theorems like Rolle’s Theorem and its extensions, i.e., Lagrange’s Mean Value Theorem, or simply Mean Value Theorem (MVT). We will also take up the Sandwich Theorem and the geometric interpretations of the derivative of a point. These theorems are very important for having a good hand on calculus. Their derivations also hold importance as it is the derivation which gives us a feel of the concept and makes us understand it at a deeper level.

Sandwich Theorem

The sandwich theorem is used to find the limit of some trigonometric functions by comparing two other functions whose limits are known. Sandwich Theorem is also known as the Squeeze theorem or Pinching theorem, as it squeezes one function with an unknown limit between the two other functions with the known limit. 

Statement: Let f, g and h be real functions such that f(x) ≤ g(x) ≤ h(x) for all x in the common domain of definition. For some real number a, if xaf(x) = xah(x) then xag(x) will also be equal to the other two.

Consider the inequality relating to trigonometric functions as:

cos x < sin x/x < 1 for 0 < |x| < π/2

Since,

sin (– x) = – sin x

cos( – x) = cos x

Hence, it is sufficient to prove the inequality for 0 < x < π/2 only.

Proof:

• Draw a unit circle with centre O such that ∠AOC is x radians and 0 < x < π/2.
• Line segments AB and are to OA. Join OC as shown.
• From the above figure, we can observe that;
• Area of ΔOAC < Area of sector OAC < Area of Δ OAB

⇒1/2×OA×CD<x/2π×π(OA)2<1/2×OA.AB

Here, since OA is common, cancel it out  to get;

CD < x . OA < AB.

In OCD,

sin x = CD/OC = CD/OA (since OA=OC ), therefore CD = OA sin x

Also,

tan x = AB/OA

⇒ AB = OA tan x

Thus,

OA sin x < OA . x < OA tan x

Since OA is positive, we can simplify the above inequality as;

sin x < x < tan x…(1)

From the given, 0 < x < π/2, we can interpret that sin x is positive (all trigonometric functions are positive in the first quadrant).

Dividing equation (1) by sin x,

Thus,

(sin x/sin x) < (x/sin x) < (tan x/sin x) 1 < (x/sin x) < (1/cos x) {since tan x = sin x/cos x}

Reciprocating  throughout, to get

cos x < (sin x/x) < 1

Derivative at a point

Proof:

Let y=f(x) be a continuous curve of a differentiable function f(x).

Let points  P (a,f(a)) and Q (a+h,f(a+h)) present on the curve. PQ is a chord formed by joining P&Q

Since, Slope = (y2-y1)/(x2-x1) 

Slope of PQ =   (f(a+h)-f(a))/(a+h-a)

( f(a+h)-f(a))/h

Let l be a tangent at point P; if Q approaches P, then the h will approach 0, and PQ will eventually move to l.

Thus, the slope of a point at the tangent l = slope of PQ when QP

Thus, we have;

h0(f(a+h)-f(a))/h= QPQR/PR= f’(c)

Therefore, the slope of the tangent at P = f’(c) at P = differentiation of f(x) at P.

Thus, tan=f'(a)

Thus, the derivative of f(x) at x=a is equal to the slope of the tangent on the y=f(x) curve at point with coordinates (a,f(a)).

Rolle’s Theorem

Theorem: Let f : [a, b] → R be continuous on [a, b] and differentiable on (a, b), such that f(a) = f(b), where a and b are some real numbers. Then, there exists some c in (a, b) such that f′(c) = 0.

As per the theorem, there is at least one point through which the tangent on the curve between A&B is parallel to the x-axis.

Lagrange’s Mean Value Theorem

It is an extension of Rolle’s Theorem and states that when f : [a, b] → R be a continuous function on [a, b] and differentiable on (a, b). Then, there exists some c in (a, b) such that f’(c)= (f(b)-f(a))/(b-a)

Conclusion

Rolle’s theorem geometrically signifies that on a continuous curve of y=f(x), and f(a)=f(b), there is at least one point between a and b where the slope is zero and a tangent through it is parallel to the x-axis. Lagrange’s Mean Value Theorem is an extension of Rolle’s theorem.

 Mean Value Theorem says that is a point on the continuous curve y=f(x) with f:[a,b], such that the slope of the tangent or point on the tangent is the same as secant between (b,f(b)) and (a,f(a)). The sandwich model helps to find the limit of a function using two functions with known limits. Discussing the derivative of a point, here we find the slope of the tangent at a point on the curve.