<Calculus (미적분학) with Sage>
Part I Single Variable Calculus
Chapter 1. Functions
1.1 Functions and Graph
For Computer Algebra System(CAS) problems, we recommend that each student try to manipulate the problem values for additional experience and practice.
1.(Celsius and Fahrenheit). If the temperature is degrees Celsius, then the temperature is also degrees Fahrenheit, where
(a) Find , , and .
(b)Suppose the outside temperature is degrees Celsius. What is the temperature in degrees Fahrenheit?
(c)What temperature is the same in both degrees Fahrenheit and in degrees Celsius?
Let . Then .
That is . Hence .
Therefore the temperature that the same in both degrees Fahrenheit and in degrees Celsius is .
2. (Brain Weight Problem) The weight of a human's brain is directly proportional to his/her body weight .
(a) It is known that a person whose weight is has a brain that weighs .
Find an equation of variation expressing as a function of .
(b) Express the variation constant as a percent and interpret the resulting equation.
(c) What is the weight of the brain of a person who weighs ?
Since , .
Hence the required equation is .
That is, .
Hence the variation constant is and brain weight is of body weight.
(c) Since ,
3. (Muscle Weight) The weight of the muscles in a human is directly proportional to his/her body weight .
(a) It is known that a person who weighs has of muscles. Find an equation of variation expressing as a function of .
(b) Express the variation constant in percentage and interpret the resulting equation.
(a) Since , .
So, we know that .
That is, .
Hence the variation constant is and muscle weight is of body weight.
4. (Estimating Heights) An anthropologist can use certain linear functions to estimate the height of a male or female, given the length of certain bones.
The humerus is the bone from the elbow to the shoulder. Let be the length of the humerus, in centimeters. Then the height, in centimeters, of a male with a humerus of length is given by
The height in centimeters of a female with a humerus of length is given by .
A humerus was uncovered in some ruins.
(a) If we assume it was from a male, how tall was he?
(b) If we assume it was from a female, how tall was she?
5. (Urban Population) The population of a town is . After a growth of , its new population is .
(a) Assuming that is directly proportional to , find an equation of variation.
(b) Find when .
(c) Find when .
That is .
6. (Average Age of Women at First Marriage) In general, our society is marrying at a later age. The average age of women at first marriage can be approximated by the linear function ,
where is the average age of women at 1st marriage years after . Thus, is the average age of women at 1st marriage in the year , is the mean age in , and so on.
(a) Find , , , and .
(b) What will be the mean age of women at first marriage in ?
(c) Graph .
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Calculus-Grapher : http://matrix.skku.ac.kr/cal-lab/sage-grapher.html
Parametric Equation Grapher : http://matrix.skku.ac.kr/cal-lab/sage-grapher-para.html
Polar Equation Grapher : http://matrix.skku.ac.kr/cal-lab/sage-grapher-polar.html
Implicit Function Grapher : http://matrix.skku.ac.kr/cal-lab/sage-grapher-imp.html