Suppose that a rectangle is bounded by the x -axis and the graph of y = cos x . a. Write a function that represents the area A x of the rectangle for. 0 < x < π 2 b. Complete the table. a. Graph the function from part (a) on the viewing window: 0 , π 2 , π 6 by − 3 , 3 , 1 and approximate the values of x for which the area is 1 square unit Round to 2 decimal places. b. In calculus, we can show that the maximum value of the area of the rectangle will occur at values of x for which 2 cos x − 2 x sin x = 0 . Confirm this result by graphing y = 2 cos x − 2 x sin x and the function from part (a) on the same viewing window. What do you notice?
Suppose that a rectangle is bounded by the x -axis and the graph of y = cos x . a. Write a function that represents the area A x of the rectangle for. 0 < x < π 2 b. Complete the table. a. Graph the function from part (a) on the viewing window: 0 , π 2 , π 6 by − 3 , 3 , 1 and approximate the values of x for which the area is 1 square unit Round to 2 decimal places. b. In calculus, we can show that the maximum value of the area of the rectangle will occur at values of x for which 2 cos x − 2 x sin x = 0 . Confirm this result by graphing y = 2 cos x − 2 x sin x and the function from part (a) on the same viewing window. What do you notice?
Solution Summary: The author analyzes the function that represents the area of the rectangle for 0xpi 2.
Suppose that a rectangle is bounded by the x-axis and the graph of
y
=
cos
x
.
a. Write a function that represents the area
A
x
of the rectangle for.
0
<
x
<
π
2
b. Complete the table.
a. Graph the function from part (a) on the viewing window:
0
,
π
2
,
π
6
by
−
3
,
3
,
1
and approximate the values of
x
for which the area is
1
square unit Round to
2
decimal places.
b. In calculus, we can show that the maximum value of the area of the rectangle will occur at values of
x
for which
2
cos
x
−
2
x
sin
x
=
0
. Confirm this result by graphing
y
=
2
cos
x
−
2
x
sin
x
and the function from part (a) on the same viewing window. What do you notice?
Use the properties of logarithms, given that In(2) = 0.6931 and In(3) = 1.0986, to approximate the logarithm. Use a calculator to confirm your approximations. (Round your answers to four decimal places.)
(a) In(0.75)
(b) In(24)
(c) In(18)
1
(d) In
≈
2
72
Find the indefinite integral. (Remember the constant of integration.)
√tan(8x)
tan(8x) sec²(8x) dx
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