Derivative Functions
The derivative function gives the derivative of a function at each point in the domain of the original function for which the derivative is defined. We can formally define a derivative function as follows.
Definition:
let f be a function. The derivative function, denoted by f', is the function whose domain consists of those values of x such that the following limit exists:
Answer:
The artist should make 15 caricatures and 30 portraits.
Step-by-step explanation:
First, we identify the variables for the problem. In this case, the variables will be:
C=number of caricatures
P= number of portraits.
Next, we can use the variables to make the inequalities that constrain the problem.
We know the artist can't spend more than $360 for supplies and we also know that it costs $4 to make a caricature and $10 to make a portrait, so:
We also know the artist has a limit of time of 20 hours, which is the same as 20*60=1200 minutes to paint. We know it takes her 20 minutes to make a caricature and 30 minutes to make a portrait, so:
and we also know she must make a positive number of caricatures and a positive number of portraits so the other two constrains are:
So we can go ahead and graph them. (see attached picture)
Once we graphed the inequalities we can determine what the corners of the polygon created by the intersection of the shaded areas are, so we proceed and calculate them:
first point:
(0,0)
second point:
4(0)+10P=360
P=36
(0,36)
Third point:
20C+30(0)=1200
C=60
(60,0)
Fourth point:
We multiply the first equation by -5 so we get:
-20C-50P=-1800
20C+30P=1200
------------------------
-20P=-600
P=30
so:
20C+30(30)=1200
20C=1200-900
20C=300
C=15
(15,30)
Once we found the 4 points, we go ahead and evaluate them for the objective function. In this case, the objective function is the total amount of money she got from selling the pieces of art, so we get:
I=10C+20P
Point 1
I=10(0)+20(0)=0
Point 2
I=10(0)+20(36)=720
Point 3
I=10(60)+20(0)=600
Point 4
I=10(15)+20(30)=750
Next, we look for the greatest income which in this case will be 750 so that's why she must make 15 caricatures and 30 Portraits.
