Class Notes
(806,429)

Canada
(492,247)

York University
(33,488)

MATH 1013
(72)

Anthony Anthony
(1)

Lecture

# Applications of Derivatives: Section 4.1

Unlock Document

York University

Mathematics and Statistics

MATH 1013

Anthony Anthony

Fall

Description

Unit 4: Applications of Differentiation
Now we know differentiation rules, we can pursue better applications.
Ex. How derivative affect the graph of f(x), to locate max and mins.
Ex. Maximize/Minimize a cost or area for optimization.
Recap: Intermediate Value Theorem: If a function f is continuous on a closed interval [a, b], and c is any number between f(a)
and f(b) inclusive, then there is at least one number x in the closed interval such that f(x) = c.
- If there is 2 points connected by a function, and it has one point above a horizontal line, and another below, then it MUST
cross that line!
4.1: Maximum and Minimum Values:
Optimization: To find the optimal/best way of doing something through the use of maximum/minimum
values.
o The shape of a can to minimize manufacturing costs?
o Maximum acceleration of a space shuttle?
o Angle blood vessels branch to minimize energy of pumping blood?
(Also known as extreme values of f).
Absolute Maximum: If f(d) >= f(x) for ALL x in the domain of f, where f(d) is called the maximum value of
f on the domain.
Absolute Minimum: If f(a) <= f(x) in ALL x in the domain of f, where f(a) is called the minimum value of f
on the domain.
Local Maximum: The largest values of a certain interval. Ex. For interval (a,c) - f(b) is the local maximum
in f(x), where x is sufficiently close to b.
Local Minimum: The smallest values of a certain interval. Ex. For interval (b,d) - f(c) is the local minimum
in f(x). Since f(c) <= f(x) where x is sufficiently close to c.
------------------------------------------------------------------------------------------------- Ex. For f(x) = cosx: Its absolute and local maximum is 1, and its absolute and local minimum is -1.
2
Ex. If f(x) = x , then f(x) >= f(0) for all x. Therefore f(0) = 0 is the absolute and local minimum of f.
There is no highest point on a parabola, so there is no max.
3
Ex. If f(x) = x , then the function neither has a max or min (abs and loc)
f(-1) = Absolute maximum (It occurs at an end point)
f(1) = Local maximum
f(0) = Local minimum
f(3) = Absolute & Local minimum
f(4) = Neither local or absolute maximum.
-------------------------------------------------------------------------------------------------
Extreme Value Theorem:
If f is continuous on a closed interval [a,b]:
o There is at least an absolute max value f(c) and absolute m

More
Less
Related notes for MATH 1013