Computing Square Roots in Your Head - Northern …richard/Math229/square_roots.pdfMath 229 Calculus I Computing Square Roots in Your Head Professor Richard Blecksmith [email protected]

Math 229 Calculus I

Computing Square Roots in Your

Head

Professor Richard [email protected]

Dept. of Mathematical SciencesNorthern Illinois University

http://math.niu.edu/∼richard/Math229

– p. 1

Reiterating Square Root Button

On a TI calculator, if you want to iterate the squareroot process, that is take the square root of the squareroot of the square root ..., use the following trick:


On a TI calculator, if you want to iterate the squareroot process, that is take the square root of the squareroot of the square root ..., use the following trick:Initially, store a number, such as 7 in variable x:

7 STO → X



7 STO → X

Now Calculate the square root of X and store the newvalue in X: √

X STO → X



7 STO → X


X STO → X

To reiterate, press the Return Key repeatedly.



7 STO → X


X STO → X

To reiterate, press the Return Key repeatedly.Start with X = 7 and compute the next 15 successivesquare roots.



7 STO → X


X STO → X

To reiterate, press the Return Key repeatedly.Start with X = 7 and compute the next 15 successivesquare roots.Do you see a pattern?



7 STO → X


X STO → X

To reiterate, press the Return Key repeatedly.Start with X = 7 and compute the next 15 successivesquare roots.Do you see a pattern?Can you predict the value of the next square root inyour head?

The Square Root Trick

The following trick let’s you evaluated 10 digit squareroots in you head, to impress your friends and family,to win fame and fortune.


The following trick let’s you evaluated 10 digit squareroots in you head, to impress your friends and family,to win fame and fortune.The idea is that if x is near 1, then



√x ≈ 1 +

x− 1

2



√x ≈ 1 +

x− 1

2

For example

√1.000026452 ≈ 1.000013226



√x ≈ 1 +

x− 1

2

For example

√1.000026452 ≈ 1.000013226

Why does this trick work?

Calculus to the RescueWe are working with the function

y = f(x) =√x.


y = f(x) =√x.

By the power rule, the derivative is

f ′(x) =1

2x−1/2.


y = f(x) =√x.


f ′(x) =1

2x−1/2.

When x = 1 the value of this derivative is

f ′(1) =1

2(1)−1/2 =

1

2.


y = f(x) =√x.


f ′(x) =1

2x−1/2.


f ′(1) =1

2(1)−1/2 =

1

2.

The tangent line to the curve y = f(x) =√x goes

through the point (1, 1) and has slope


y = f(x) =√x.


f ′(x) =1

2x−1/2.


f ′(1) =1

2(1)−1/2 =

1

2.

The tangent line to the curve y = f(x) =√x goes

through the point (1, 1) and has slope m = f ′(1) = 1

2.

Tangent Line

The equation for the tangent line is just

Tangent Line


y − y0 = m(x− x0)

or

Tangent Line


y − y0 = m(x− x0)

or

y − 1 =1

2(x− 1)

or

Tangent Line


y − y0 = m(x− x0)

or

y − 1 =1

2(x− 1)

or

y = 1 +1

2(x− 1)

GeneralizationGiven any function f(x) and fixed x-value a


use the tangent line to approximate values of f(x).


use the tangent line to approximate values of f(x).We know the tangent line has slope


use the tangent line to approximate values of f(x).We know the tangent line has slope ?


use the tangent line to approximate values of f(x).We know the tangent line has slope f ′(a)


use the tangent line to approximate values of f(x).We know the tangent line has slope f ′(a) and goesthrough the point


use the tangent line to approximate values of f(x).We know the tangent line has slope f ′(a) and goesthrough the point ?


use the tangent line to approximate values of f(x).We know the tangent line has slope f ′(a) and goes

through the point (a, f(a))




y − y0 = m(x− x0)

or




y − y0 = m(x− x0)

or

y − f(a) = f ′(a)(x− a)

or




y − y0 = m(x− x0)

or

y − f(a) = f ′(a)(x− a)

or

y = f(a) + f ′(a)(x− a)




y − y0 = m(x− x0)

or

y − f(a) = f ′(a)(x− a)

or

y = f(a) + f ′(a)(x− a)

The function L(x) = f(a) + f ′(a)(x− a) is called the

linearization of f(x) at x = a.

Some Algebra

∆x = a small change in x

Some Algebra

∆x = a small change in xThis is the “h” in the limit definition of derivative.

Some Algebra

∆x = a small change in xThis is the “h” in the limit definition of derivative.∆y = f(x+∆x)− f(x).

Some Algebra

∆x = a small change in xThis is the “h” in the limit definition of derivative.∆y = f(x+∆x)− f(x).This is the numerator in the limit definition ofderivative.

Some Algebra

∆x = a small change in xThis is the “h” in the limit definition of derivative.∆y = f(x+∆x)− f(x).This is the numerator in the limit definition ofderivative.Now define

Some Algebra


dxdef= ∆x

Some Algebra


dxdef= ∆x

dydef= f ′(x)dx = f ′(x)∆x

Some Algebra


dxdef= ∆x


Note that this turns dydx into a genuine fraction.

Some Algebra


dxdef= ∆x



Text: Section 2.9, # 13.

Some Algebra


dxdef= ∆x



Text: Section 2.9, # 13.Text: Section 2.9, # 18.

Documents

Computing Square Roots in Your Head - Northern …richard/Math229/square_roots.pdfMath 229 Calculus I Computing Square Roots in Your Head Professor Richard Blecksmith [email protected]