The gas mileage m x (in mpg) for a certain vehicle can be approximated m x = − 0.028 x 2 + 2.688 x − 35.012 , where x is the speed of the vehicle in mph. a. Determine the speed at which the car gets its maximum gas mileage. b. Determine the maximum gas mileage.
The gas mileage m x (in mpg) for a certain vehicle can be approximated m x = − 0.028 x 2 + 2.688 x − 35.012 , where x is the speed of the vehicle in mph. a. Determine the speed at which the car gets its maximum gas mileage. b. Determine the maximum gas mileage.
Solution Summary: The author calculates the speed at which the car gets its maximum gas mileage approximated by the quadratic equation.
The gas mileage
m
x
(in mpg) for a certain vehicle can be approximated
m
x
=
−
0.028
x
2
+
2.688
x
−
35.012
,
where x is the speed of the vehicle in mph.
a. Determine the speed at which the car gets its maximum gas mileage.
17. Suppose we know that the graph below is the graph of a solution to dy/dt = f(t).
(a) How much of the slope field can
you sketch from this information?
[Hint: Note that the differential
equation depends only on t.]
(b) What can you say about the solu-
tion with y(0) = 2? (For example,
can you sketch the graph of this so-
lution?)
y(0) = 1
y
AN
(b) Find the (instantaneous) rate of change of y at x = 5.
In the previous part, we found the average rate of change for several intervals of decreasing size starting at x = 5. The instantaneous rate of
change of fat x = 5 is the limit of the average rate of change over the interval [x, x + h] as h approaches 0. This is given by the derivative in the
following limit.
lim
h→0
-
f(x + h) − f(x)
h
The first step to find this limit is to compute f(x + h). Recall that this means replacing the input variable x with the expression x + h in the rule
defining f.
f(x + h) = (x + h)² - 5(x+ h)
=
2xh+h2_
x² + 2xh + h² 5✔
-
5
)x - 5h
Step 4
-
The second step for finding the derivative of fat x is to find the difference f(x + h) − f(x).
-
f(x + h) f(x) =
= (x²
x² + 2xh + h² -
])-
=
2x
+ h² - 5h
])x-5h) - (x² - 5x)
=
]) (2x + h - 5)
Macbook Pro
Evaluate the integral using integration by parts.
Sx² cos
(9x) dx
A Problem Solving Approach To Mathematics For Elementary School Teachers (13th Edition)
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