Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 57 doc
Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 57 doc
... equivalent to the previous answer, because
V(P)=
KT
P
.
(b) Here we are thinking of P as the independent variable and looking for
dT
dP
.NowV
is treated as a constant and T as a function of P . We can ... a constant.
10
(a) Suppose temperature is kept constant. Express the rate of change of volume with respect
to pressure.
(b) Suppose volume is kept constant. Express the rate...
Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 4 doc
... tax functions in states
1 and 2, respectively. We can describe the input-output relationship of the functions f and
g using formulas.
f(x)= 500 g(x) = 0.04x
The graphs of the functions f and ... Representations of Functions 19
stocks, and the rest on bonds. Let s be the amount he puts into slow-growth stocks, 2s be
the amount he puts into riskier stocks, and b be the a...
Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 6 doc
... length of the sides of the
base and let h denote the height of the box, s and h in inches.
(a) Express the volume of the box in terms of s and h.
(b) Express the surface area of the box in terms of ... For instance, suppose you were to calculate the value, V ,of
the change in your pocket. V is a function of q, d, n, and p, where q, d, n, and p are the
number of...
Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 11 doc
... position and velocity, and, more generally, between amount functions and their
corresponding rate functions.
EXERCISE 2.10 Oil is leaking from a point and spreading evenly in a thin, expanding ... thin, expanding disk. We can measure
the radius of the disk and want to know the rate of change of the area of the disk with respect
to the radius. The area is a function...
Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 15 docx
... peaks and valleys?
(c) How does the graph of y = cf(x)+k (where c and k are
constants) relate to that of y = f(x)?
In particular, let f(x)=x
2
.Onthe same set of axes sketch
the graphs of f(x )and ... decompose the function and find its rate of change from our knowledge
of the rates of change of these simpler functions.
8
PROBLEMS FOR SECTION 3.3
1. Let h(x) =f(g(x...
Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 17 doc
... graph of
y = mx + b. We want to show that the rate of change of y with respect to x is m, regardless
of our choice of points.
Since y = mx + b, the points (x
1
, y
1
) and (x
2
, y
2
) can be ... carefully and clearly,
it is easy to lose track of the main features of the equation:
the two variables,
the constants that are given, and
the unknown constants that you are...
Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 24 doc
... regardless of the values of a, b, and c. We can, for example,
set b = 0 and c = 0 and conclude that if f(x)=ax
2
then f
(x) = 2ax. Similarly, we
know that the derivative of bx is b, and the derivative ... taken out of the oven and put into the refrigerator. The refrigerator
is kept at a constant temperature. Newton’s Law of Cooling says that the difference
between the te...
Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 30 docx
... definition of derivative to find f
(x). Use your work to check your
answers to parts (a) and (b).
19. The domain of a function f is all real numbers. The zeros of f(x)are x =−1, x = 2,
and x = ... and
formulas for derivatives of
√
x and
1
x
found in Chapter 5 to arrive
at a formula for the derivative of x
n
for n = 0, 1, 2, 3, −1, and
1
2
.
Try your formula on anot...
Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 31 doc
... differentiating sums,
products, and quotients, and the ability to easily compute the derivative of any function of
the form x
n
, where n is an integer.
Sum Rule and Constant Multiple Rule
Section ... Revisited
Exploratory Problems for Chapter 8
Circles and Spheres
You know how to express the area and the circumference of a circle in
terms of its radius; you also know...
Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 32 docx
... growing a crop of tomatoes wants to know when he should harvest and sell them
in order to collect the most revenue. Both the weight of the crop and the price of tomatoes
are changing over time. ... for any exponent, but we will not be able to prove this until we know more
about exponential functions, their inverse functions, and taking the derivative of composite
func...
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