A function f is given by f ( x ) = 1 ( x − 5 ) 2 . This function takes a number x , subtracts 5 from it, squares the result, and takes the reciprocal of the square. Find f ( 3 ) , f ( − 1 ) , f ( k ) , and f ( x + h ) .
A function f is given by f ( x ) = 1 ( x − 5 ) 2 . This function takes a number x , subtracts 5 from it, squares the result, and takes the reciprocal of the square. Find f ( 3 ) , f ( − 1 ) , f ( k ) , and f ( x + h ) .
Solution Summary: The author explains how to calculate the values of f(3) and c.
A function
f
is given by
f
(
x
)
=
1
(
x
−
5
)
2
.
This function takes a number
x
, subtracts 5 from it, squares the result, and takes the reciprocal of the square. Find
f
(
3
)
,
f
(
−
1
)
,
f
(
k
)
, and
f
(
x
+
h
)
.
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
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