Consider the function on the interval (0, 2π).
f(x) = sin(x) cos(x) + 2
(a) Find the open interval(s) on which the function is
increasing or decreasing. (Enter your answers using interval
notation.)
increasing
( )
decreasing
( )
(b) Apply the First Derivative Test to identify all relative
extrema.
relative maxima
(x, y) =
(smaller x-value)
(x, y) =
( )
(larger x-value)
relative minima
(x, y) =
(smaller x-value)
(x, y) =
...
Consider the function on the interval (0, 2π). f(x) = sin(x)/ 2
+ (cos(x))^2 (a) Find the open intervals on which the function is
increasing or decreasing. (Enter your answers using interval
notation.) increasing decreasing (b) Apply the First Derivative
Test to identify the relative extrema. relative maximum (x, y) =
relative minimum (x, y) =
Consider an initial value
problem
?′′ + 2? = ?(?) = cos? (0 ≤ ? < ?) , 0 (? ≥
?)
?(0) = 0 and ?′(0) = 0
(a) Express ?(?) in terms of the unit step function.
(b) Find the Laplace transform of ?(?).
(c) Find ?(?) by using the Laplace transform method.
Consider the vector function given below.
r(t) =
2t, 3 cos(t), 3 sin(t)
(a) Find the unit tangent and unit normal vectors T(t) and
N(t).
T(t) =
N(t) =
(b) Use this formula to find the curvature.
κ(t) =
The function F(x) = x2 - cos(π x) is defined on the
interval 0 ≤ x ≤ 1 radians. Explain how the Intermediate Value
Theorem shows that F(x) = 0 has a solution on the interval 0 < x
< .
Write a script that uses the function below to find root
brackets for ?(?) = cos(ex) + sin(?), between ? = 0 ???
? = ? with ns=100. Plot the output by first plotting the function
and then plotting ‘*’ at each bracket point (on the x-axis). You
may either give the plot() function two sets of inputs, or you can
use hold on ... hold off to add plots to your figure.
function xb = incsearchv(func,xmin,xmax,ns)