Download Exponential Growth and Decay

Section 4.1
Section 4.2
Section 4.3
Applications of Derivatives
Calculus for the Biological Sciences
February 24, 2006
Calculus for the Biological Sciences
Applications of Derivatives
Section 4.1
Section 4.2
Section 4.3
Analysis of Curves
Definition: A function f (x) is said to be an increasing function
over a certain interval of values of x if f (x) increases with
increases of x. That is, if x1 and x2 are any two values in the given
interval with x2 > x1 , then f (x2 ) > f (x1 ).
Definition: A function f (x) is said to be a decreasing function
over a certain interval of values of x if f (x) decreases with
increases of x. That is, if x1 and x2 are any two values in the given
interval with x2 > x1 , then f (x2 ) < f (x1 ).
Calculus for the Biological Sciences
Applications of Derivatives
Section 4.1
Section 4.2
Section 4.3
Analysis of Curves
Theorem 4.1.1:
(a) If f (x) is an increasing function that is differentiable, then
f 0 (x) ≥ 0.
(b) If f (x) is a decreasing function that is differentiable, then
f 0 (x) ≤ 0.
Calculus for the Biological Sciences
Applications of Derivatives
Section 4.1
Section 4.2
Section 4.3
Analysis of Curves
Theorem 4.1.2:
(a) If f 0 (x) > 0 for all x in some interval, then f (x) is an
increasing function of x over that interval.
(b) If f 0 (x) < 0 for all x in some interval, then f (x) is a decreasing
function of x over that interval.
Calculus for the Biological Sciences
Applications of Derivatives
Section 4.1
Section 4.2
Section 4.3
Analysis of Curves
Example: Find the values of x for which the function
f (x) = x 2 − 2x + 1
is increasing or decreasing.
Calculus for the Biological Sciences
Applications of Derivatives
Section 4.1
Section 4.2
Section 4.3
Analysis of Curves
Definition: A function f (x) is said to be convex(concave upwards)
00
over a certain interval of values of x if f > 0.
Definition: A function f (x) is said to be concave(concave
00
downwards) over a certain interval of values of x if f (x) ≤ 0.
A point of inflection on a curve is a point where the curve changes
its convexity.
Calculus for the Biological Sciences
Applications of Derivatives
Section 4.1
Section 4.2
Section 4.3
Analysis of Curves
Example: Find the values of x for which the function
1
f (x) = x 4 − x 3 + 2x 2
6
is concave upwards or concave downwards.increasing or decreasing.
Calculus for the Biological Sciences
Applications of Derivatives
Section 4.1
Section 4.2
Section 4.3
Analysis of Curves
Definition: A point of inflection on a curve is a point where the
curve changes from concave upwards to concave downwards, or
vice versa.
Calculus for the Biological Sciences
Applications of Derivatives
Section 4.1
Section 4.2
Section 4.3
Analysis of Curves
Example: Find the points of inflection of y = x 1/3 .
Calculus for the Biological Sciences
Applications of Derivatives
Section 4.1
Section 4.2
Section 4.3
Analysis of Curves
Theorem 4.1.3 (Rolle’s Theorem): Let the function f (x) be
defined and continuous at all x in the closed interval a ≤ x ≤ b
and be differentiable at all x in the open interval a < x < b. Let
f (a) = f (b). There exists at least one number c satisfying
a < c < b at which f 0 (c) = 0.
Calculus for the Biological Sciences
Applications of Derivatives
Section 4.1
Section 4.2
Section 4.3
Analysis of Curves
Theorem 4.1.4 (Mean Value Theorem): Let f (x) be
continuous on the closed interval a ≤ x ≤ b and differentiable on
the open interval a < x < b. Then there exists at least one
number c satisfying a < c < b such that
f 0 (c) =
f (b) − f (a)
.
b−a
Calculus for the Biological Sciences
Applications of Derivatives
Section 4.1
Section 4.2
Section 4.3
Maxima and Minima
Definition:
i. A function f (x) is said to have a local maximum at x = c if
f (c) > f (x) for all x sufficiently near c.
ii. A function f (x) is said to have a local minimum at x = c if
f (c) < f (x) for all x sufficiently close to c.
iii. The term extremum is used to denote either local maximum or
a local minimum.
iv. The value x = c is called a critical value for a continuous
function f if
1. either f 0 (c) = 0 or f 0 (x) fails to exist at x = c and if
2. f (c) is well-defined.
Calculus for the Biological Sciences
Applications of Derivatives
Section 4.1
Section 4.2
Section 4.3
Maxima and Minima
Theorem 4.2.1 (Second Derivative Test): Let f (x) be twice
differentiable at x = c. Then
1. x = c is a local maximum of f whenever
f 0 (c) = 0, f 00 (c) < 0;
2. x = c is a local minimum of f whenever
f 0 (c) = 0, f 00 (c) > 0.
Calculus for the Biological Sciences
Applications of Derivatives
Section 4.1
Section 4.2
Section 4.3
Maxima and Minima
Example: Determine the local maximum and minimum values of
x 3 + 2x 2 − 4x − 8.
Calculus for the Biological Sciences
Applications of Derivatives
Section 4.1
Section 4.2
Section 4.3
Maxima and Minima
Theorem 4.2.2 (First Derivative Test): If x = c is a critical
value for f (x), that is, either f 0 (c) = 0 or f 0 (x) fails to exist as
x → c:
1. x = c is a local maximum of f if f 0 (x) changes sign from
positive to negative as x changes from just below c to just
above c.
2. x = c is a local minimum of f if f 0 (x) changes sign from
negative to positive as x changes from just below c to just
above c.
3. x = c is not a local extremum if f 0 (x) does not change sign
as x changes from just below c to just above c. In this case
x = c will be either a point of inflection or a corner on the
graph of f (x).
Calculus for the Biological Sciences
Applications of Derivatives
Section 4.1
Section 4.2
Section 4.3
Maxima and Minima
Example: Find the local extrema for
f (x) =
Calculus for the Biological Sciences
x4
.
(x − 1)
Applications of Derivatives
Section 4.1
Section 4.2
Section 4.3
Maxima and Minima
Definition: The absolute maximum and absolute minimum is the
largest and smallest values of f (x) respectively taken over the
entire domain.
Calculus for the Biological Sciences
Applications of Derivatives
Section 4.1
Section 4.2
Section 4.3
Maxima and Minima
Example: Determine the absolute maximum and minimum values
of f (x) = 1 + 12x − x 3 over the interval 1 ≤ x ≤ 3.
Calculus for the Biological Sciences
Applications of Derivatives
Section 4.1
Section 4.2
Section 4.3
Maxima and Minima
Example: Find the absolute maximum and minimum values of
f (x) =
x2
x2 + 1
on the interval [−1, ∞).
Calculus for the Biological Sciences
Applications of Derivatives
Section 4.1
Section 4.2
Section 4.3
Applications of Maxima and Minima
Example: Find two numbers whose sum is 16 and whose product
is the largest possible.
Calculus for the Biological Sciences
Applications of Derivatives
Section 4.1
Section 4.2
Section 4.3
Applications of Maxima and Minima
Example: Construct a tank with a square base, rectangular sides
and no top. The tank must hold 4m3 of water. The material used
to make the tank costs $10/m2 . Determine the dimensions of the
tank which will minimize the cost.
Calculus for the Biological Sciences
Applications of Derivatives
Section 4.1
Section 4.2
Section 4.3
Applications of Maxima and Minima
Example: A grain silo is to be built in the form of a vertical
cylinder with a hemispherical roof (Fig. 4.34 page 189). The silo is
to be capable of storing 10, 000ft 3 of grain. No grain is stored in
the roof. The hemispherical roof costs twice as much per unit area
to manufacture as the cylindrical sides cost. What dimensions of
the silo will minimize the cost?
Calculus for the Biological Sciences
Applications of Derivatives