A quarterback throws a football with an initial velocity of 72 ft/sec at an angle of 25 ° . The height of the ball can be modeled by h t = − 16 t 2 + 30.4 t + 5 , where h t is the height (in ft) and t is the time in seconds after release. a. Determine the time at which the ball will be at its maximum height. b. Determine the maximum height of the ball. c. Determine the amount of time required for the bail to reach the receiver's hands if the receiver catches the ball at a point 3 ft off the ground.
A quarterback throws a football with an initial velocity of 72 ft/sec at an angle of 25 ° . The height of the ball can be modeled by h t = − 16 t 2 + 30.4 t + 5 , where h t is the height (in ft) and t is the time in seconds after release. a. Determine the time at which the ball will be at its maximum height. b. Determine the maximum height of the ball. c. Determine the amount of time required for the bail to reach the receiver's hands if the receiver catches the ball at a point 3 ft off the ground.
Solution Summary: The author explains how the arc of the parabolic curve of a quadratic function, h(t), will open downwards and the vertex coordinates will provide the maximum values
A quarterback throws a football with an initial velocity of 72 ft/sec at an angle of
25
°
.
The height of the ball can be modeled by
h
t
=
−
16
t
2
+
30.4
t
+
5
, where
h
t
is the height (in ft) and
t
is the time in seconds after release.
a. Determine the time at which the ball will be at its maximum height.
b. Determine the maximum height of the ball.
c. Determine the amount of time required for the bail to reach the receiver's hands if the receiver catches the ball at a point 3 ft off the ground.
(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
Let f be defined as follows.
y = f(x) = x² - 5x
(a) Find the average rate of change of y with respect to x in the following intervals.
from x = 4 to x = 5
from x = 4 to x = 4.5
from x = 4 to x = 4.1
(b) Find the (instantaneous) rate of change of y at x = 4.
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