Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 55 potx
Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 55 potx
... instantaneous
rate of change of y with respect to x at x = 2.
Notation: y
, f
, f
(x), y
(x),
dy
dx
,
df
dx
all mean the same thing.
d
dx
is an operator that means
“take the derivative of ... They can get to the store by a
combination of boat and foot. Where should they build their dock
so the commute to the store takes the minimum amount of time?
home
6 miles
R...
Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 9 potx
... undefined and discontinuous at x = 0.
(e) j is 1 -to- 1.
Answers to Exercise 2.8
(a) g and h are even functions.
(b) f and j are odd functions.
(c) The graph of an odd function is said to have symmetry ... Characterizing Functions and Introducing Rates of Change
Definition
A is inversely proportional to B if
A =
k
B
for some constant k. As B increases A decreases,...
Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 20 potx
... f(a) )and (b, f(b)).
The instantaneous rate of change of f at x = a can be approximated by the average
rate of change of f on [a, a + h] for h very close to zero. As h gets closer and closer
to ... rate of change over the tiny interval approaches the instantaneous
rate of change at x = a.
The instantaneous rate of
change of f at x = a
≈
f(a +h) − f(a)
(a + h) − a
for h...
Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 26 potx
... equation of the parabola with the specifications
given.
17. x-intercepts of 3 and −2; maximum value of 1
18. x-intercepts of π and 3π; y-intercept of 6
19. x-intercepts of π and 3π; y-intercept of −2
20. ... (i) and three arbitrary points we can obtain a system of three
linear equations that we can solve for the constants a, b, and c. We can use the x-intercepts
r
1...
Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 35 potx
... Animals eat plants and so have the same ratio of C
14
to C
12
as do the plants
and the atmosphere. Thus, during the lifetime of a living organism the ratio of C
14
to C
12
in the organism is fixed; ... sketch the graphs of f(x)=2
x
,g(x) =
3
x
, and h(x) = 10
x
. Then, on another set of axes, draw a rough
sketch of the graphs of f
(x), g
(x), and h
(x). The grap...
Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 36 potx
... chances of winning?
(e) Sketch a graph of P (n). Use [0, 100] as the range of the graph. Explain the practical
significance of any asymptotes.
9.4 THE DERIVATIVE OF AN EXPONENTIAL FUNCTION
As an ... average or
at instantaneous rates of change, we can state the following.
Exponential functions grow at a rate proportional to themselves.
Exponential functions have a cons...
Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 49 potx
... Logarithmic and Exponential Functions
We are looking for the slope of the tangent line to ln x at x = b. Let L
1
be the tangent
line to e
x
at (c, b) and L
2
be the tangent line to ln x at (b, ... slope at any point on the graph of e
x
is given by the y-coordinate
of that point), the slope of the tangent line to e
x
at (c, b)isb.Since the roles of x and y are
interch...
Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 72 potx
... are the maximum and minimum values of f ?
12. If we ignore air resistance, a baseball thrown from shoulder level at an angle of θ radians
with the ground and at an initial velocity of v
0
meters ... 3π
radians
minute
.
dθ
dt
= 3π radians/minute
Strategy We know
dθ
dt
and we want to find
dx
dt
. Our strategy is to write an equation relating
θ and x.
11
We then differenti...
Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 85 potx
... estimating
the area of a plot of land or body of water.
Suppose our aim is to approximate
b
a
f(x)dx.Wecan collect values of f(x)at equally
spaced intervals and use Simpson’s rule to approximate ... 2y
4
+···+2y
2n−2
+4y
2n−1
+y
2n
)x.
The answer to part (c) is given at the end of the section.
Notice that because Simpson’s rule requires both the midpoint and trapezoid...
Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 112 potx
... between F and L.
The axis of symmetry runs
through F and perpendicular
to L.
F
V
L
d
1
d
1
d
2
d
2
d
3
d
3
Axis of symmetry
Figure E.4
An Ellipse
An ellipse is the set of points (x, y), the sum of whose ... they can be thought of as the intersection
of a plane with a pair of circular cones with vertices joined in an hourglass configuration.
Depending upon the orientat...
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