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
< .
Consider the following differential equation:
x2y"-2xy'+(x2+2)y=0
Given that y1=x cos x is a solution, find a second
linearly independent solution to the differential equation.
Write the general solution.