Suppose that the differentiable function f x , y has the property that f 2 + s 3 2 , 1 + s 2 = 3 s e s The directional derivative of f in the direction of u = 3 2 i + 1 2 j at (2, 1) is ______ .
Suppose that the differentiable function f x , y has the property that f 2 + s 3 2 , 1 + s 2 = 3 s e s The directional derivative of f in the direction of u = 3 2 i + 1 2 j at (2, 1) is ______ .
Suppose that the differentiable function
f
x
,
y
has the property that
f
2
+
s
3
2
,
1
+
s
2
=
3
s
e
s
The directional derivative of
f
in the direction of
u
=
3
2
i
+
1
2
j
at (2, 1) is
______
.
With integration, one of the major concepts of calculus. Differentiation is the derivative or rate of change of a function with respect to the independent variable.
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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Velocity of a Ball Thrown into the Air The position function of an object moving along a straight line is given by s = f(t). The average velocity of
the object over the time interval [a, b] is the average rate of change of f over [a, b]; its (instantaneous) velocity at t = a is the rate of change of f at a.
A ball is thrown straight up with an initial velocity of 128 ft/sec, so that its height (in feet) after t sec is given by s = f(t) = 128t - 16t².
(a) What is the average velocity of the ball over the following time intervals?
[3,4]
[3, 3.5]
[3, 3.1]
ft/sec
ft/sec
ft/sec
(b) What is the instantaneous velocity at time t = 3?
ft/sec
(c) What is the instantaneous velocity at time t = 7?
ft/sec
Is the ball rising or falling at this time?
O rising
falling
(d) When will the ball hit the ground?
t =
sec
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