The speed v L (in m/sec) of an ocean wave in deep water is approximated by V L = 1.2 L , where L (in meters) is the wavelength of the wave. (The wavelength is the distance between two consecutive wave crests.) a. Find the average rate of change in speed between waves that are between 1 m and 4 m in length. b. Find the average rate of change in speed between waves that are between 4 m and 9 m in length. c. Use a graphing utility to graph the function. Using the graph and the results from parts (a) and (b), what does the difference in the rates of change mean?
The speed v L (in m/sec) of an ocean wave in deep water is approximated by V L = 1.2 L , where L (in meters) is the wavelength of the wave. (The wavelength is the distance between two consecutive wave crests.) a. Find the average rate of change in speed between waves that are between 1 m and 4 m in length. b. Find the average rate of change in speed between waves that are between 4 m and 9 m in length. c. Use a graphing utility to graph the function. Using the graph and the results from parts (a) and (b), what does the difference in the rates of change mean?
Solution Summary: The author calculates the average rate of change in speed between the ocean waves that are 1m and 4m in length.
The speed
v
L
(in m/sec) of an ocean wave in deep water is approximated by
V
L
=
1.2
L
,
where
L
(in meters) is the wavelength of the wave. (The wavelength is the distance between two consecutive wave crests.)
a. Find the average rate of change in speed between waves that are between 1 m and 4 m in length.
b. Find the average rate of change in speed between waves that are between 4 m and 9 m in length.
c. Use a graphing utility to graph the function. Using the graph and the results from parts (a) and (b), what does the difference in the rates of change mean?
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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