Practise using graph intersections or sign changes to locate roots and applying convergent fixed-point iterations with enough displayed values to justify the requested decimal…
By sketching a suitable pair of graphs, show that the equation
has only one root in the interval 0<x<21π.
Sketch a relevant graph, e.g. y= 2 x
May be sketching 2 x and -e^-x
For y= 2 x correct shape, asymptote in correct position. Cuts at (0,1)
For -e^x correct shape, cuts at (0,1)
For y= 2 x correct shape, correct max/min, cuts axis at /4
For -e^-x correct shape, cuts at (0,-1)
B1
Complete scale not needed, but should have enough
to imply the key points for both graphs.
Sketch a second relevant graph, e.g. y=-e^x and justify the given statement
B1
Needs to mark intersection with a dot, a cross, or say roots at points of intersection, OE.
Show by calculation that this root lies between 0.9 and 1 .
Calculate the values of a relevant expression or pair of expressions at
x=0.9 and x=1
M1
E.g. aligned & -4.401<-2.460
& -2.403>-2.718aligned or aligned & -1.94<0
& 0.315>0aligned
(or gathered0.179>0
-0.048<0gathered for the reciprocal graphs).
Complete the argument correctly with correct calculated values
A1
Show that if a sequence of values given by the iterative formula
converges, then it converges to the root of the equation in part (a).
State the equation x=1/2 ^-1(-e^-x) and rearrange to obtain 2 x=-e^x
Or rearrange 2 x=-e^x as x=1/2 ^-1(-e^-x) and state x_n+1=1/2 ^-1(-e^-x_n)
B1
Use the iterative formula given in part (c) to calculate x correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
Use the iterative process correctly at least once
M1
Obtain final answer 0.978
A1
Show sufficient iterations to at least 5 d. p. to justify 0.978 to 3 d.p. or show there
is a sign change in the interval (0.9775,0.9785)
A1
E.g.
1, 0.97376, 0.97903,0.97796, 0.97813
0.95, 0.98395, 0.97697, 0.97838, 0.97809
0.9, 0.99475, 0.97480, 0.97882, 0.97800, 0.97817