Ch. 5 – Applications of
Derivatives
5.4 – Modeling and Optimization
• Ex: Find two positive numbers whose sum is 10 and whose product
is as large as possible.
– First, identify some variables and set up equations to solve. Let’s use x and y for our two
numbers and P for our product.
x  y  10
xy  P
– Our goal here is to maximize P. Any problem that requires us to minimize and maximize is
called an optimization problem.
– Whenever we maximize/minimize a variable, set its derivative to zero!!!
– For this problem, we must first get P in terms of one variable...
y  10  x
P  x(10  x)
P  10 x  x 2
P '  10  2 x  0 @ x  5
– So x=5 is a critical point. What are the endpoints of the problem (think about what values of x
and y restrict the problem)?
– Since x and must be positive, 0<x<10, so (0, 0) and (10, 0) are the endpoints. Do a P’ number
line test...
10
5
0
– This number line shows that x=5 is a maximum.
– If x=5, then y=5...
– Answer: 5 and 5
+
-
• Ex: A rectangular plot of land will be bounded on one side by a river
and on the other 3 sides by a fence. If you have 800 m of fence,
what is the largest area you can enclose?
– Drawing a picture helps a lot!
b
h
– We want to maximize area. We need 2 equations...
A  bh
b  2h  800
river
– We want area in terms of one variable, so solve and substitute!
b  800  2h
2
A  (800  2h)h  800h  2h
– Now we maximize A by setting the derivative equal to zero...
A '  800  4h  0
800  4h
– So the maximum area will be...
b  2(200)  800
b  400
Amax  (400)(200)  80, 000 m 2
h
h  200
• Ex: A rectangle is inscribed between
one arch of a sine curve and the x-axis.
Find the largest area possible for such
a rectangle.
1
π
– Drawing a picture helps a lot!
– We want to maximize area (A = bh), but what are b and h?
– Consider the first intersection point of the curve and the rectangle, (x, sinx). Using this point
(and knowing the symmetry of the sine curve), we have what we need...
(x, sinx)
sinx
π
x
π-2x
A  (  2 x) sin x
– Let’s use our calculator to find the maximum of this function.
– Due to the restrictions of the problem, assume 0<x<π/2
– The maximum is at (.710, 1.122)…
2
max
A
 1.122 u
x