Practise locating roots with graph intersections or a sign change, calculating both endpoints and explaining why the stated interval contains a root or exactly one root.
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