Lecture Seven

Overview One of the most important applications of the derivative is its use as a tool for
finding the optimal (best) solutions to problems. Optimization problems abound in mathematics,
physical science and engineering, business and economics, and biology and
medicine. For example, what are the height and diameter of the cylinder of largest volume
that can be inscribed in a given sphere? What are the dimensions of the strongest rectangular
wooden beam that can be cut from a cylindrical log of given diameter? Based on production
costs and sales revenue, how many items should a manufacturer produce to maximize
profit? How much does the trachea (windpipe) contract to expel air at the maximum
speed during a cough? What is the branching angle at which blood vessels minimize the
energy loss due to friction as blood flows through the branches?
In this chapter we use derivatives to find extreme values of functions, to determine
and analyze the shapes of graphs, and to solve equations numerically. We also introduce
the idea of recovering a function from its derivative. The key to many of these applications
is the Mean Value Theorem, which paves the way to integral calculus.
4.1 E xtreme Values of Functions
This section shows how to locate and identify extreme (maximum or minimum) values of
a function from its derivative. Once we can do this, we can solve a variety of optimization
problems (see Section 4.6). The domains of the functions we consider are intervals or
unions of separate intervals.
Applications of
Derivatives
4
Definitions Let ƒ be a function with domain D. Then ƒ has an absolute
maximum value on D at a point c if
ƒ(x) … ƒ(c) for all x in D
and an absolute minimum value on D at c if
ƒ(x) ? ƒ(c) for all x in D.
Maximum and minimum values are called extreme values of the function ƒ. Absolute
maxima or minima are also referred to as global maxima or minima.
For example, on the closed interval 3-p>2, p>24 the function ƒ(x) = cos x takes on
an absolute maximum value of 1 (once) and an absolute minimum value of 0 (twice). On
the same interval, the function g(x) = sin x takes on a maximum value of 1 and a minimum
value of -1 (Figure 4.1).
Functions with the same defining rule or formula can have different extrema (maximum
or minimum values), depending on the domain. We see th
Overview One of the most important applications of the derivative is its use as a tool for
finding the optimal (best) solutions to problems. Optimization problems abound in mathematics,
physical science and engineering, business and economics, and biology and
medicine. For example, what are the height and diameter of the cylinder of largest volume
that can be inscribed in a given sphere? What are the dimensions of the strongest rectangular
wooden beam that can be cut from a cylindrical log of given diameter? Based on production
costs and sales revenue, how many items should a manufacturer produce to maximize
profit? How much does the trachea (windpipe) contract to expel air at the maximum
speed during a cough? What is the branching angle at which blood vessels minimize the
energy loss due to friction as blood flows through the branches?
In this chapter we use derivatives to find extreme values of functions, to determine
and analyze the shapes of graphs, and to solve equations numerically. We also introduce
the idea of recovering a function from its derivative. The key to many of these applications
is the Mean Value Theorem, which paves the way to integral calculus.
4.1 E xtreme Values of Functions
This section shows how to locate and identify extreme (maximum or minimum) values of
a function from its derivative. Once we can do this, we can solve a variety of optimization
problems (see Section 4.6). The domains of the functions we consider are intervals or
unions of separate intervals.
Applications of
Derivatives
4
Definitions Let ƒ be a function with domain D. Then ƒ has an absolute
maximum value on D at a point c if
ƒ(x) … ƒ(c) for all x in D
and an absolute minimum value on D at c if
ƒ(x) ? ƒ(c) for all x in D.
Maximum and minimum values are called extreme values of the function ƒ. Absolute
maxima or minima are also referred to as global maxima or minima.
For example, on the closed interval 3-p>2, p>24 the function ƒ(x) = cos x takes on
an absolute maximum value of 1 (once) and an absolute minimum value of 0 (twice). On
the same interval, the function g(x) = sin x takes on a maximum value of 1 and a minimum
value of -1 (Figure 4.1).
Functions with the same defining rule or formula can have different extrema (maximum
or minimum values), depending on the domain. We see thOverview One of the most important applications of the derivative is its use as a tool for
finding the optimal (best) solutions to problems. Optimization problems abound in mathematics,
physical science and engineering, business and economics, and biology and
medicine. For example, what are the height and diameter of the cylinder of largest volume
that can be inscribed in a given sphere? What are the dimensions of the strongest rectangular
wooden beam that can be cut from a cylindrical log of given diameter? Based on production
costs and sales revenue, how many items should a manufacturer produce to maximize
profit? How much does the trachea (windpipe) contract to expel air at the maximum
speed during a cough? What is the branching angle at which blood vessels minimize the
energy loss due to friction as blood flows through the branches?
In this chapter we use derivatives to find extreme values of functions, to determine
and analyze the shapes of graphs, and to solve equations numerically. We also introduce
the idea of recovering a function from its derivative. The key to many of these applications
is the Mean Value Theorem, which paves the way to integral calculus.
4.1 E xtreme Values of Functions
This section shows how to locate and identify extreme (maximum or minimum) values of
a function from its derivative. Once we can do this, we can solve a variety of optimization
problems (see Section 4.6). The domains of the functions we consider are intervals or
unions of separate intervals.
Applications of
Derivatives
4
Definitions Let ƒ be a function with domain D. Then ƒ has an absolute
maximum value on D at a point c if
ƒ(x) … ƒ(c) for all x in D
and an absolute minimum value on D at c if
ƒ(x) ? ƒ(c) for all x in D.
Maximum and minimum values are called extreme values of the function ƒ. Absolute
maxima or minima are also referred to as global maxima or minima.
For example, on the closed interval 3-p>2, p>24 the function ƒ(x) = cos x takes on
an absolute maximum value of 1 (once) and an absolute minimum value of 0 (twice). On
the same interval, the function g(x) = sin x takes on a maximum value of 1 and a minimum
value of -1 (Figure 4.1).
Functions with the same defining rule or formula can have different extrema (maximum
or minimum values), depending on the domain. We see thOverview One of the most important applications of the derivative is its use as a tool for
finding the optimal (best) solutions to problems. Optimization problems abound in mathematics,
physical science and engineering, business and economics, and biology and
medicine. For example, what are the height and diameter of the cylinder of largest volume
that can be inscribed in a given sphere? What are the dimensions of the strongest rectangular
wooden beam that can be cut from a cylindrical log of given diameter? Based on production
costs and sales revenue, how many items should a manufacturer produce to maximize
profit? How much does the trachea (windpipe) contract to expel air at the maximum
speed during a cough? What is the branching angle at which blood vessels minimize the
energy loss due to friction as blood flows through the branches?
In this chapter we use derivatives to find extreme values of functions, to determine
and analyze the shapes of graphs, and to solve equations numerically. We also introduce
the idea of recovering a function from its derivative. The key to many of these applications
is the Mean Value Theorem, which paves the way to integral calculus.
4.1 E xtreme Values of Functions
This section shows how to locate and identify extreme (maximum or minimum) values of
a function from its derivative. Once we can do this, we can solve a variety of optimization
problems (see Section 4.6). The domains of the functions we consider are intervals or
unions of separate intervals.
Applications of
Derivatives
4
Definitions Let ƒ be a function with domain D. Then ƒ has an absolute
maximum value on D at a point c if
ƒ(x) … ƒ(c) for all x in D
and an absolute minimum value on D at c if
ƒ(x) ? ƒ(c) for all x in D.
Maximum and minimum values are called extreme values of the function ƒ. Absolute
maxima or minima are also referred to as global maxima or minima.
For example, on the closed interval 3-p>2, p>24 the function ƒ(x) = cos x takes on
an absolute maximum value of 1 (once) and an absolute minimum value of 0 (twice). On
the same interval, the function g(x) = sin x takes on a maximum value of 1 and a minimum
value of -1 (Figure 4.1).
Functions with the same defining rule or formula can have different extrema (maximum
or minimum values), depending on the domain. We see th