Problem of the Day No calculator! What is the instantaneous rate of change at x = 2 of f(x) = x2 - 2...
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Problem of the Day No calculator! What is the instantaneous rate of change at x = 2 of f(x) = x2 - 2 ? x - 1 A) -2 C) 1/2 E) 6 B) 1/6 D) 2
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Transcript of Problem of the Day No calculator! What is the instantaneous rate of change at x = 2 of f(x) = x2 - 2...
Problem of the Day No calculator!
What is the instantaneous rate of change at x = 2 of f(x) = x2 - 2 ? x - 1
A) -2 C) 1/2 E) 6
B) 1/6 D) 2
Problem of the Day No calculator!
What is the instantaneous rate of change at x = 2 of f(x) = x2 - 2 ? x - 1
A) -2 C) 1/2 E) 6
B) 1/6 D) 2
(take derivative and then substitute in)
Newton's Method
A technique for approximating the real zeroes of a function using tangent lines
Newton's Method
A technique for approximating the real zeroes of a function using tangent lines
If the function is continuous on [a, b] and differentiable on (a, b) and if f(a) and f(b) differ in sign then by the ___________________________ f must have at least one zero in (a, b)
a b
y
x
Newton's Method
A technique for approximating the real zeroes of a function using tangent lines
If the function is continuous on [a, b] and differentiable on (a, b) and if f(a) and f(b) differ in sign then by the Intermediate Value Theorem f must have at least one zero in (a, b)
a b
y
x
Newton's Method
A technique for approximating the real zeroes of a function using tangent lines
Visual Calculus Link
Newton's Method
A technique for approximating the real zeroes of a function using tangent lines
In summary, the x-intercept will be approximately
xn+1 = xn - f(xn) f '(xn)
Calculate 3 iterations of Newton's Method to approximate a zero of f(x) = x2 - 2 starting with x = 1.
Iteration xn f(xn) f '(xn)
f(xn)f '(xn)
xn - f(xn)f '(xn)
Calculate 3 iterations of Newton's Method to approximate a zero of f(x) = x2 - 2 starting with x = 1.
Iteration xn f(xn) f '(xn)
f(xn)f '(xn)
xn - f(xn)f '(xn)
123
11.51.416
-1.25.006945
232.83
-.5.083.002451
1.51.4161.414216
Calculate 3 iterations of Newton's Method to approximate a zero of f(x) = x2 - 2 starting with x = 1.
Iteration xnyour equation
nderiv(Y1,x,x)
x -
xn - f(xn)f '(xn)
Y1 =
Ti-84
Y2 =
Y3 =
Ti-Nspire
f1 = your equation
f2 =
f3 =
Newton's Method will not always produce an answer, such as when
1) the derivative within the interval is zero at any point2) functions similar to f(x) = x1/3
You can test for convergence to see if it will work with the following formula
f(x) f ''(x) [f '(x)]2 < 1
Another precaution
Do not round in intermediary steps. Let your calculator carry the numbers.
