Solar eclipse. On July 2, 2019, a total solar eclipse will occur over the South Pacific Ocean and parts of South America. ( Source : eclipse.gsfc.nasa.gov. ) The path of the eclipse is modeled by f ( t ) = 0.00259 t 2 − 0.457 t + 36.237 , Where f ( t ) is the latitude in degrees south of the equator at t minutes after the start of the total eclipse. What is the latitude closest to the equator, in degrees, at which the total eclipse will be visible?
Solar eclipse. On July 2, 2019, a total solar eclipse will occur over the South Pacific Ocean and parts of South America. ( Source : eclipse.gsfc.nasa.gov. ) The path of the eclipse is modeled by f ( t ) = 0.00259 t 2 − 0.457 t + 36.237 , Where f ( t ) is the latitude in degrees south of the equator at t minutes after the start of the total eclipse. What is the latitude closest to the equator, in degrees, at which the total eclipse will be visible?
Solar eclipse. On July 2, 2019, a total solar eclipse will occur over the South Pacific Ocean and parts of South America. (Source: eclipse.gsfc.nasa.gov.) The path of the eclipse is modeled by
f
(
t
)
=
0.00259
t
2
−
0.457
t
+
36.237
,
Where
f
(
t
)
is the latitude in degrees south of the equator at t minutes after the start of the total eclipse. What is the latitude closest to the equator, in degrees, at which the total eclipse will be visible?
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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