Locating Roots (2) Flashcards
The graph of the function f(x) = (2xe^x) - 3 crosses the x-axis at the point P (p,0)
Show that 0.7 < p < 0.8
f(0.7) = -0.1807 f(0.8) = 0.5608
f(x) is continuous and there’s a change of sign
The graph of the function f(x) = (2xe^x) - 3 crosses the x-axis at the point P (p,0)
Show that p=0.726 to 3 d.p.
f(0.7255) = -0.0025 f(0.7265) = 0.0045
f(x) is continuous and there’s a change of sign
g(x) = cosecx - 2
Show that x=0.5 is an approximation to one root of g(x)=0 to 1 d.p.
g(0.45)= 0.2990 g(0.55) = -0.0868
g(x) is continuous and there’s a change of sign
g(x) = cosecx - 2
Solve g(x)=0 to find an exact value for the first root in the interval [0.pi]
cosecx - 2 = 0
sinx = 1/2
sin^-1(1/2) = pi/6
x = pi/6
pi/6 is the only solution in the interval [0,pi]
Show that the equation sin3x + 3x = 1 can be written as 1/3(1-sin3x)
sin3x + 3x = 1
3x = 1 - sin3x
x = (1/3)(1 - sin3x)
What is a convergent staircase diagram?
The next iterations are getting closer to the points where the graphs intersect
The next iterations are getting closer to the points where the graphs intersect
What is a convergent staircase diagram?
What is a convergent cobweb diagram?
The iterations alternate between being below and above the root, but are getting closer every time
The iterations alternate between being below and above the root, but are getting closer every time
What is a convergent cobweb diagram?
What is a divergent staircase diagram?
The iterations are getting further away from the root
The iterations are getting further away from the root
What is a divergent staircase diagram?
Explain why the Newton-Raphson method fails with x_0=-2
Sub -2 into the method and explain the failure
Convex curves
An over-estimate
An over-estimate
Convex curves
Concave curves
An under-estimate
