Study the roots of the nonlinear equation f(x) = cos(x) + (1 /(1
+ e^2x)) both theoretically and numerically. (a) Plot f(x) on the
interval x ∈ [−15, 15] and describe the overall behaviour of the
function as well as the number and location of its roots. Use the
“zoom” feature of Matlab’s plotting window (or change the axis
limits) in order to ensure that you are identifying all roots – you
may have to increase your plotting point density...
(a) Find the limit of the following functions:
-lim as x approaches 0 (1-cos3(x)/x)
-lim as x approaches 0 (sin(x)/2x)
-lim as theta approaches 0 (tan (5theta)/theta)
(b) Find the derivative of the following functions:
-f(x) = cos (3x2-2x)
-f(x) = cos3 (x2/1-x)
(c) Determine the period of the following functions:
-f(x) = 3 cos(x/2)
-f(x)= 21+ 7 sin(2x+3)
A graphing calculator is recommended.
For the limit
lim x → 2 (x3 −
2x + 4) = 8
illustrate the definition by finding the largest possible values
of δ that correspond to ε = 0.2 and ε =
0.1. (Round your answers to four decimal places.)
ε =
0.2
δ =
ε =
0.1
δ =