Calculus-Sec-1-1-Solution-Sage

<Calculus  (미적분학) with Sage>

CONTENTS

Part I  Single Variable Calculus

Chapter 1. Functions

1.1 Functions and Graph

1.2 Symmetry

1.3 Common Functions

1.4 Translation, Stretching and Rotation of 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?

(a)

(b)

(c)

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 ?

(a) .

Since .

Hence the required equation is .

(b) .

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 .

Hence .

(b) .

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?

(a)

(b)

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 .

(a)

(b)

(c) .

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 .

(a)

(b)

(c)

* Your tools to draw graphs (in Chrome or Firefox browser) :

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