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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:
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: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 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
Now Calculate the square root of X and store the newvalue in X: √
X STO → X
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:Initially, store a number, such as 7 in variable x:
7 STO → X
Now Calculate the square root of X and store the newvalue 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 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
Now Calculate the square root of X and store the newvalue in X: √
X STO → X
To reiterate, press the Return Key repeatedly.Start with X = 7 and compute the next 15 successivesquare roots.
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:Initially, store a number, such as 7 in variable x:
7 STO → X
Now Calculate the square root of X and store the newvalue in 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?
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:Initially, store a number, such as 7 in variable x:
7 STO → X
Now Calculate the square root of X and store the newvalue in 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 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 idea is that if x is near 1, then
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 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 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
For example
√1.000026452 ≈ 1.000013226
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 idea is that if x is near 1, then
√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.
By the power rule, the derivative is
f ′(x) =1
2x−1/2.
Calculus to the RescueWe are working with the function
y = f(x) =√x.
By the power rule, the derivative is
f ′(x) =1
2x−1/2.
When x = 1 the value of this derivative is
f ′(1) =1
2(1)−1/2 =
1
2.
Calculus to the RescueWe are working with the function
y = f(x) =√x.
By the power rule, the derivative is
f ′(x) =1
2x−1/2.
When x = 1 the value of this derivative is
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
Calculus to the RescueWe are working with the function
y = f(x) =√x.
By the power rule, the derivative is
f ′(x) =1
2x−1/2.
When x = 1 the value of this derivative is
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
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).
GeneralizationGiven 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
GeneralizationGiven 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 ?
GeneralizationGiven 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)
GeneralizationGiven 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 goesthrough the point
GeneralizationGiven 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 goesthrough the point ?
GeneralizationGiven 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))
GeneralizationGiven 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))
y − y0 = m(x− x0)
or
GeneralizationGiven 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))
y − y0 = m(x− x0)
or
y − f(a) = f ′(a)(x− a)
or
GeneralizationGiven 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))
y − y0 = m(x− x0)
or
y − f(a) = f ′(a)(x− a)
or
y = f(a) + f ′(a)(x− a)
GeneralizationGiven 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))
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 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
∆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
dxdef= ∆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.Now define
dxdef= ∆x
dydef= f ′(x)dx = f ′(x)∆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.Now define
dxdef= ∆x
dydef= f ′(x)dx = f ′(x)∆x
Note that this turns dydx into a genuine fraction.
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
dxdef= ∆x
dydef= f ′(x)dx = f ′(x)∆x
Note that this turns dydx into a genuine fraction.
Text: Section 2.9, # 13.
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
dxdef= ∆x
dydef= f ′(x)dx = f ′(x)∆x
Note that this turns dydx into a genuine fraction.
Text: Section 2.9, # 13.Text: Section 2.9, # 18.