For Exercise 115-120, factor the expressions over the set of complex numbers. For assistance, consider these examples. · In Section R .3 we saw that some expressions factor over the set of integers . For example: x 2 − 4 = x + 2 x − 2 . · Some expressions factor over the set of irrational numbers . For example: x 2 − 5 = x + 5 x − 5 . · To factor an expression such as x 2 + 4 , we need to factor over the set of complex numbers . For example, verify that x 2 + 4 = x + 2 i x − 2 i . a . x 2 − 3 b . x 2 + 3
For Exercise 115-120, factor the expressions over the set of complex numbers. For assistance, consider these examples. · In Section R .3 we saw that some expressions factor over the set of integers . For example: x 2 − 4 = x + 2 x − 2 . · Some expressions factor over the set of irrational numbers . For example: x 2 − 5 = x + 5 x − 5 . · To factor an expression such as x 2 + 4 , we need to factor over the set of complex numbers . For example, verify that x 2 + 4 = x + 2 i x − 2 i . a . x 2 − 3 b . x 2 + 3
Solution Summary: The author explains the factors of the expressions given below over the set of complex numbers.
For Exercise 115-120, factor the expressions over the set of complex numbers. For assistance, consider these examples.
·
In Section R
.3 we saw that some expressions factor over the set of integers
. For example:
x
2
−
4
=
x
+
2
x
−
2
.
·
Some expressions factor over the set of irrational numbers
. For example:
x
2
−
5
=
x
+
5
x
−
5
.
·
To factor an expression such as
x
2
+
4
, we need to factor over the set of complex numbers
. For example,
verify that
x
2
+
4
=
x
+
2
i
x
−
2
i
.
a
.
x
2
−
3
b
.
x
2
+
3
Combination of a real number and an imaginary number. They are numbers of the form a + b , where a and b are real numbers and i is an imaginary unit. Complex numbers are an extended idea of one-dimensional number line to two-dimensional complex plane.
This question builds on an earlier problem. The randomized numbers may have changed, but have your work for the previous problem available to help with this one.
A 4-centimeter rod is attached at one end to a point A rotating counterclockwise on a wheel of radius 2 cm. The other end B is free to move back and forth along a horizontal bar that goes through the center of the wheel. At time t=0 the rod is situated as in the diagram at the left below. The
wheel rotates counterclockwise at 1.5 rev/sec. At some point, the rod will be tangent to the circle as shown in the third picture.
A
B
A
B
at some instant, the piston will be tangent to the circle
(a) Express the x and y coordinates of point A as functions of t:
x= 2 cos(3πt)
and y= 2 sin(3t)
(b) Write a formula for the slope of the tangent line to the circle at the point A at time t seconds:
-cot(3πt)
sin(3лt)
(c) Express the x-coordinate of the right end of the rod at point B as a function of t: 2 cos(3πt) +411-
4
-2 sin (3лt)
(d)…
5. [-/1 Points]
DETAILS
MY NOTES
SESSCALCET2 6.5.AE.003.
y
y= ex²
0
Video Example
x
EXAMPLE 3
(a) Use the Midpoint Rule with n = 10 to approximate the integral
कर
L'ex²
dx.
(b) Give an upper bound for the error involved in this approximation.
SOLUTION
8+2
1
L'ex² d
(a) Since a = 0, b = 1, and n = 10, the Midpoint Rule gives the following. (Round your answer to six decimal places.)
dx Ax[f(0.05) + f(0.15) + ... + f(0.85) + f(0.95)]
0.1 [0.0025 +0.0225
+
+ e0.0625 + 0.1225
e0.3025 + e0.4225
+ e0.2025
+
+ e0.5625 €0.7225 +0.9025]
The figure illustrates this approximation.
(b) Since f(x) = ex², we have f'(x)
=
0 ≤ f'(x) =
< 6e.
ASK YOUR TEACHER
and f'(x) =
Also, since 0 ≤ x ≤ 1 we have x² ≤
and so
Taking K = 6e, a = 0, b = 1, and n = 10 in the error estimate, we see that an upper bound for the error is as follows. (Round your final
answer to five decimal places.)
6e(1)3
e
24(
=
≈
2. [-/1 Points]
DETAILS
MY NOTES
SESSCALCET2 6.5.015.
Use the Trapezoidal Rule, the Midpoint Rule, and Simpson's Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.)
ASK YOUR TEACHER
3
1
3 +
dy, n = 6
(a) the Trapezoidal Rule
(b) the Midpoint Rule
(c) Simpson's Rule
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