(a) Determine whether the following limits exist, and if so, find them: lim x → + ∞ e x cos x , lim x → − ∞ e x cos x (b) Sketch the graphs of the equations y = e x , y = − e x , and y = e x cos x in the same coordinate system , and label any points of intersection. (c) Use a graphing utility to generate some members of the family y = e a x cos b x ( a > 0 and b > 0 ), and discuss the effect of varying a and b on the shape of the curve.
(a) Determine whether the following limits exist, and if so, find them: lim x → + ∞ e x cos x , lim x → − ∞ e x cos x (b) Sketch the graphs of the equations y = e x , y = − e x , and y = e x cos x in the same coordinate system , and label any points of intersection. (c) Use a graphing utility to generate some members of the family y = e a x cos b x ( a > 0 and b > 0 ), and discuss the effect of varying a and b on the shape of the curve.
(a) Determine whether the following limits exist, and if so, find them:
lim
x
→
+
∞
e
x
cos
x
,
lim
x
→
−
∞
e
x
cos
x
(b) Sketch the graphs of the equations
y
=
e
x
,
y
=
−
e
x
,
and
y
=
e
x
cos
x
in the same coordinate system, and label any points of intersection.
(c) Use a graphing utility to generate some members of the family
y
=
e
a
x
cos
b
x
(
a
>
0
and
b
>
0
), and discuss the effect of varying a and b on the shape of the curve.
System that uses coordinates to uniquely determine the position of points. The most common coordinate system is the Cartesian system, where points are given by distance along a horizontal x-axis and vertical y-axis from the origin. A polar coordinate system locates a point by its direction relative to a reference direction and its distance from a given point. In three dimensions, it leads to cylindrical and spherical coordinates.
Sketch the graph of a function, f(x), that has each of the following properties:
• Domain: x € [-2, 1) U (1,7) U (7,00)
• Range: y € (-00, 10]
lim f(x) = -3
X-00
lim f(x) = -co and lim f(x) = 0
.
lim f(x) = 5
X→-2+
lim f(x) does not exist
X-9
The maximum output of f(x) occurs when x = 3
-10-
9
8
7
6
5
4
3
2
-1-
-10-9-8-7 -6 -5 -4 -3 -2 -1 0
-1
-2
-3
-4
--5-
--6-
-7-
-8
-9-
-10-
2 3 4 5 6 7 8 9 10
a) Sketch the graph of the function
f (x) = {2x – 3 if x 2
%3D
then evaluate lim f(x),
lim f(x) at a = 2.
X --> a+
X --> a-
b) Find the derivative of the following function
f (x) = 3x5 – 2x4 + 3x2 - 2x + 1
%3D
c) Find the derivative of the following function g (x) = (1+x) e
d) Find the derivative of the following function h (x) = (x + )
%3D
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