For Exercises 1-16, identify which functions shown here ( f , g , h , and so on) have the given characteristics f x = sin π 2 x + 3 g x = − 3 cos 1 2 x − π 4 h x = 3 sin − 1 2 x − π 5 k x = − 3 sec 2 x + π m x = 2 csc 2 x − π 2 − 3 n x = 3 tan x − π 2 p x = − 2 cot 1 2 x + π t x = − 3 + 2 cos x Has a phase shift of − π 2
For Exercises 1-16, identify which functions shown here ( f , g , h , and so on) have the given characteristics f x = sin π 2 x + 3 g x = − 3 cos 1 2 x − π 4 h x = 3 sin − 1 2 x − π 5 k x = − 3 sec 2 x + π m x = 2 csc 2 x − π 2 − 3 n x = 3 tan x − π 2 p x = − 2 cot 1 2 x + π t x = − 3 + 2 cos x Has a phase shift of − π 2
Solution Summary: The author explains that the given functions have a phase shift of -pi 2.
For Exercises 1-16, identify which functions shown here (
f
,
g
,
h
,
and so on) have the given characteristics
f
x
=
sin
π
2
x
+
3
g
x
=
−
3
cos
1
2
x
−
π
4
h
x
=
3
sin
−
1
2
x
−
π
5
k
x
=
−
3
sec
2
x
+
π
m
x
=
2
csc
2
x
−
π
2
−
3
n
x
=
3
tan
x
−
π
2
p
x
=
−
2
cot
1
2
x
+
π
t
x
=
−
3
+
2
cos
x
Given lim x-4 f (x) = 1,limx-49 (x) = 10, and lim→-4 h (x) = -7 use the limit properties
to find lim→-4
1
[2h (x) — h(x) + 7 f(x)] :
-
h(x)+7f(x)
3
O DNE
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)
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