P.S. This is one of the Calculator labs that the department makes up
#1. f(x) = x^4-17x^2+18
What is the relationship between the sign of f'(x) and the graph f(x) ?
What is the relationship between the roots of f''(x) and the graph of f(x) ?
What is the relationship between the sign of f''(x) and the graph of f(x) ?
#2. Given that f(x) = sin(x) use Newton's Method to find the first 2 positive zeros showing the first few approximations in each case.
So...
pi
___
x1: 3
x2: 3.142546543
x3: 3.141592653
x4: 3.14159255359
also...
2pi
____
x1: 6
x2: 6.29100619138
x3: 6.28310514772
x4: 6.28318530718
The first positive zero is: 3.14159
The second positive zero: 6.28319
What happens when you took pi/2 as first approximation? Explain
I got 1 but else is there to say
#3. if f(x) = x^4-17x^2+18
f''(x) = 12x^2-34
Therefore f''(x) = 0 if x = -sqrt(102)/6 and sqrt(102)/6 and these are the x-coordinates of the points of __ ___ of f(x)?
Thanks for the Help!
the sign of f'(x) let's you know when f(x) is increasing or decreasing. if f'(x) is positive, f(x) has a positive slope and is hence increasing. if f'(x) is negative, f(x) has a negative slope and is hence decreasing
the roots of f''(x) give the inflection points of f(x), that is the points where f(x) changes concavity from concave up to concave down or vice versa.
the sign of f''(x) indicates the concavity (and hence the nature of critical points) of f(x). if f''(x) is positive for some critical point in f(x), then f(x) is concave up at that point and it is a local min. if f''(x) is negative at some critical point of f(x), then f(x) is concave down at that point and it is a local max
points of inflection of f(x)...usually.
ok so these questions are weird, i don't know exactly what answer they are looking for, but anyway, we can see a few patterns. note that Newton's method is used to approximate the roots of a function. for each approximation we begin with, using Newton's method puts us closer to the root that is closest to that value.
not that when we begin with x = 3, each new approximation is bigger than the last and they keep getting closer to pi, 3.1415... since pi is the closest root to 3
when we start with x = 6, each new approximation is again bigger than the last, that is because we are increasing to 2pi, the next closest root to 6
when we begin with pi/2, each new approximation begins to decrease, this is because we are moving towards the closest root, which is zero. i find this interesting though, since pi is as close to pi/2 as 0 is