Math 229 Calculus I
Computing Square Roots in Your
Head
Professor Richard Blecksmith
[email protected]
Dept. of Mathematical Sciences
Northern Illinois University
http://math.niu.edu/∼richard/Math229
– p. 1
Reiterating Square Root Button
On a TI calculator, if you want to iterate the square
root process, that is take the square root of the square
root of the square root ..., use the following trick:
Reiterating Square Root Button
On a TI calculator, if you want to iterate the square
root process, that is take the square root of the square
root of the square root ..., use the following trick:
Initially, store a number, such as 7 in variable x:
7 STO → X
Reiterating Square Root Button
On a TI calculator, if you want to iterate the square
root process, that is take the square root of the square
root of the square root ..., use the following trick:
Initially, store a number, such as 7 in variable x:
7 STO → X
Now Calculate the square root of X and store the new
value in X:
√
X STO → X
Reiterating Square Root Button
On a TI calculator, if you want to iterate the square
root process, that is take the square root of the square
root of the square root ..., use the following trick:
Initially, store a number, such as 7 in variable x:
7 STO → X
Now Calculate the square root of X and store the new
value in X:
√
X STO → X
To reiterate, press the Return Key repeatedly.
Reiterating Square Root Button
On a TI calculator, if you want to iterate the square
root process, that is take the square root of the square
root of the square root ..., use the following trick:
Initially, store a number, such as 7 in variable x:
7 STO → X
Now Calculate the square root of X and store the new
value in X:
√
X STO → X
To reiterate, press the Return Key repeatedly.
Start with X = 7 and compute the next 15 successive
square roots.
Reiterating Square Root Button
On a TI calculator, if you want to iterate the square
root process, that is take the square root of the square
root of the square root ..., use the following trick:
Initially, store a number, such as 7 in variable x:
7 STO → X
Now Calculate the square root of X and store the new
value in X:
√
X STO → X
To reiterate, press the Return Key repeatedly.
Start with X = 7 and compute the next 15 successive
square roots.
Do you see a pattern?
Reiterating Square Root Button
On a TI calculator, if you want to iterate the square
root process, that is take the square root of the square
root of the square root ..., use the following trick:
Initially, store a number, such as 7 in variable x:
7 STO → X
Now Calculate the square root of X and store the new
value in X:
√
X STO → X
To reiterate, press the Return Key repeatedly.
Start with X = 7 and compute the next 15 successive
square roots.
Do you see a pattern?
Can you predict the value of the next square root in
your head?
The Square Root Trick
The following trick let’s you evaluated 10 digit square
roots in you head, to impress your friends and family,
to win fame and fortune.
The Square Root Trick
The following trick let’s you evaluated 10 digit square
roots in you head, to impress your friends and family,
to win fame and fortune.
The idea is that if x is near 1, then
The Square Root Trick
The following trick let’s you evaluated 10 digit square
roots 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
The Square Root Trick
The following trick let’s you evaluated 10 digit square
roots in you head, to impress your friends and family,
to win fame and fortune.
The idea is that if x is near 1, then
√
For example
√
x−1
x≈1+
2
1.000026452 ≈ 1.000013226
The Square Root Trick
The following trick let’s you evaluated 10 digit square
roots in you head, to impress your friends and family,
to win fame and fortune.
The idea is that if x is near 1, then
√
For example
√
x−1
x≈1+
2
1.000026452 ≈ 1.000013226
Why does this trick work?
Calculus to the Rescue
We are working with the function
√
y = f (x) = x.
Calculus to the Rescue
We are working with the function
√
y = f (x) = x.
By the power rule, the derivative is
1 −1/2
f (x) = x
.
2
′
Calculus to the Rescue
We are working with the function
√
y = f (x) = x.
By the power rule, the derivative is
1 −1/2
f (x) = x
.
2
′
When x = 1 the value of this derivative is
1 −1/2 1
= .
f (1) = (1)
2
2
′
Calculus to the Rescue
We are working with the function
√
y = f (x) = x.
By the power rule, the derivative is
1 −1/2
f (x) = x
.
2
′
When x = 1 the value of this derivative is
1 −1/2 1
= .
f (1) = (1)
2
2
′
The tangent line to the curve y = f (x) =
through the point (1, 1) and has slope
√
x goes
Calculus to the Rescue
We are working with the function
√
y = f (x) = x.
By the power rule, the derivative is
1 −1/2
f (x) = x
.
2
′
When x = 1 the value of this derivative is
1 −1/2 1
= .
f (1) = (1)
2
2
′
√
The tangent line to the curve y = f (x) = x goes
through the point (1, 1) and has slope m = f ′ (1) = 21 .
Tangent Line
The equation for the tangent line is just
Tangent Line
The equation for the tangent line is just
y − y0 = m(x − x0 )
or
Tangent Line
The equation for the tangent line is just
y − y0 = m(x − x0 )
or
or
1
y − 1 = (x − 1)
2
Tangent Line
The equation for the tangent line is just
y − y0 = m(x − x0 )
or
or
1
y − 1 = (x − 1)
2
1
y = 1 + (x − 1)
2
Generalization
Given any function f (x) and fixed x-value a
Generalization
Given any function f (x) and fixed x-value a
use the tangent line to approximate values of f (x).
Generalization
Given any function f (x) and fixed x-value a
use the tangent line to approximate values of f (x).
We know the tangent line has slope
Generalization
Given any function f (x) and fixed x-value a
use the tangent line to approximate values of f (x).
We know the tangent line has slope ?
Generalization
Given any function f (x) and fixed x-value a
use the tangent line to approximate values of f (x).
We know the tangent line has slope f ′ (a)
Generalization
Given any function f (x) and fixed x-value a
use the tangent line to approximate values of f (x).
We know the tangent line has slope f ′ (a) and goes
through the point
Generalization
Given any function f (x) and fixed x-value a
use the tangent line to approximate values of f (x).
We know the tangent line has slope f ′ (a) and goes
through the point ?
Generalization
Given any function f (x) and fixed x-value a
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))
Generalization
Given any function f (x) and fixed x-value a
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))
or
y − y0 = m(x − x0 )
Generalization
Given any function f (x) and fixed x-value a
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))
or
or
y − y0 = m(x − x0 )
y − f (a) = f ′ (a)(x − a)
Generalization
Given any function f (x) and fixed x-value a
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))
or
or
y − y0 = m(x − x0 )
y − f (a) = f ′ (a)(x − a)
y = f (a) + f ′ (a)(x − a)
Generalization
Given any function f (x) and fixed x-value a
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))
or
or
y − y0 = m(x − x0 )
y − f (a) = f ′ (a)(x − a)
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 x
This is the “h” in the limit definition of derivative.
Some Algebra
∆x = a small change in x
This is the “h” in the limit definition of derivative.
∆y = f (x + ∆x) − f (x).
Some Algebra
∆x = a small change in x
This is the “h” in the limit definition of derivative.
∆y = f (x + ∆x) − f (x).
This is the numerator in the limit definition of
derivative.
Some Algebra
∆x = a small change in x
This is the “h” in the limit definition of derivative.
∆y = f (x + ∆x) − f (x).
This is the numerator in the limit definition of
derivative.
Now define
Some Algebra
∆x = a small change in x
This is the “h” in the limit definition of derivative.
∆y = f (x + ∆x) − f (x).
This is the numerator in the limit definition of
derivative.
Now define
def
dx = ∆x
Some Algebra
∆x = a small change in x
This is the “h” in the limit definition of derivative.
∆y = f (x + ∆x) − f (x).
This is the numerator in the limit definition of
derivative.
Now define
def
dx = ∆x
def ′
dy = f (x)dx = f ′ (x)∆x
Some Algebra
∆x = a small change in x
This is the “h” in the limit definition of derivative.
∆y = f (x + ∆x) − f (x).
This is the numerator in the limit definition of
derivative.
Now define
def
dx = ∆x
def ′
dy = f (x)dx = f ′ (x)∆x
dy
Note that this turns dx
into a genuine fraction.
Some Algebra
∆x = a small change in x
This is the “h” in the limit definition of derivative.
∆y = f (x + ∆x) − f (x).
This is the numerator in the limit definition of
derivative.
Now define
def
dx = ∆x
def ′
dy = f (x)dx = f ′ (x)∆x
dy
Note that this turns dx
into a genuine fraction.
Text: Section 2.9, # 13.
Some Algebra
∆x = a small change in x
This is the “h” in the limit definition of derivative.
∆y = f (x + ∆x) − f (x).
This is the numerator in the limit definition of
derivative.
Now define
def
dx = ∆x
def ′
dy = f (x)dx = f ′ (x)∆x
dy
Note that this turns dx
into a genuine fraction.
Text: Section 2.9, # 13.
Text: Section 2.9, # 18.