In Problems 79-84, use the following discussion. The formula D = 24 [ 1 − cos − 1 ( tan i tan θ ) π ] can be used to approximate the number of hours of daylight D when the declination of the Sun is i ∘ at a location θ ∘ north latitude for any date between the vernal equinox and autumnal equinox. The declination of the Sun is defined as the angle i between the equatorial plane and any ray of light from the Sun. The latitude of a location is the angle θ between the Equator and the location on the surface of Earth, with the vertex of the angle located at the center of Earth. See the figure. To use the formula, cos − 1 ( tan i tan θ ) must be expressed in radians. Approximate the number of hours of daylight at the Equator ( 0 ∘ north latitude) for the following dates: Summer solstice ( i = 23.5 ∘ ) Vernal equinox i = 0 ∘ July 4 i = 0 ∘ ( i = 22 ∘ 48 ' ) What do you conclude about the number of hours of daylight throughout the year for a location at the Equator?
In Problems 79-84, use the following discussion. The formula D = 24 [ 1 − cos − 1 ( tan i tan θ ) π ] can be used to approximate the number of hours of daylight D when the declination of the Sun is i ∘ at a location θ ∘ north latitude for any date between the vernal equinox and autumnal equinox. The declination of the Sun is defined as the angle i between the equatorial plane and any ray of light from the Sun. The latitude of a location is the angle θ between the Equator and the location on the surface of Earth, with the vertex of the angle located at the center of Earth. See the figure. To use the formula, cos − 1 ( tan i tan θ ) must be expressed in radians. Approximate the number of hours of daylight at the Equator ( 0 ∘ north latitude) for the following dates: Summer solstice ( i = 23.5 ∘ ) Vernal equinox i = 0 ∘ July 4 i = 0 ∘ ( i = 22 ∘ 48 ' ) What do you conclude about the number of hours of daylight throughout the year for a location at the Equator?
In Problems 79-84, use the following discussion. The formula
can be used to approximate the number of hours of daylight D when the declination of the Sun is
at a location
north latitude for any date between the vernal equinox and autumnal equinox. The declination of the Sun is defined as the angle i between the equatorial plane and any ray of light from the Sun. The latitude of a location is the angle
between the Equator and the location on the surface of Earth, with the vertex of the angle located at the center of Earth. See the figure. To use the formula,
must be expressed in radians.
Approximate the number of hours of daylight at the Equator (
north latitude) for the following dates:
Summer solstice
Vernal equinox
July 4
What do you conclude about the number of hours of daylight throughout the year for a location at the Equator?
Consider the function f(x) = x²-1.
(a) Find the instantaneous rate of change of f(x) at x=1 using the definition of the derivative.
Show all your steps clearly.
(b) Sketch the graph of f(x) around x = 1. Draw the secant line passing through the points on the
graph where x 1 and x->
1+h (for a small positive value of h, illustrate conceptually). Then,
draw the tangent line to the graph at x=1. Explain how the slope of the tangent line relates to the
value you found in part (a).
(c) In a few sentences, explain what the instantaneous rate of change of f(x) at x = 1 represents in
the context of the graph of f(x). How does the rate of change of this function vary at different
points?
1. The graph of ƒ is given. Use the graph to evaluate each of the following values. If a value does not exist,
state that fact.
и
(a) f'(-5)
(b) f'(-3)
(c) f'(0)
(d) f'(5)
2. Find an equation of the tangent line to the graph of y = g(x) at x = 5 if g(5) = −3 and g'(5)
=
4.
-
3. If an equation of the tangent line to the graph of y = f(x) at the point where x 2 is y = 4x — 5, find ƒ(2)
and f'(2).
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