1. functions

Introduction

What Is Calculus?

• Advanced algebra and geometry. • Mathematics of change. • It  takes  a  problem  that  can’t  be  done  with  regular  

math because things are constantly changing — the changing quantities show up on a graph as curves — it zooms in on the curve till it becomes straight, and then it finishes off the problem with regular math.

The Big Guns of Calculus

1. Differentiation 2. Integration

Differentiation

• It’s  the  process  of  finding  a  derivative  of  a  curve.   • A derivative is just the fancy calculus term for a curve’s  slope  or  steepness.

• Slope is equal to the ratio of the rise to the run.

Integration

• Basically just fancy addition. • Process of cutting up an area into tiny sections,

figuring out their areas, and then adding them up to get the whole area.

Why Calculus Work

• Curves are straight at the microscopic level. • The earth is round, but to us it looks flat because we’re  sort  of  at  the  microscopic  level  when  compared to the size of the earth.

• Calculus works because when you zoom in and curves become straight, you can use regular algebra and geometry with them.

• This zooming-in process is achieved through the mathematics of limits.

Definition Example Numbers that can expressed as decimals

Real Numbers

Definition Example Rational Number: A number that may be written as a finite or infinite repeating decimal, in other words, a number that can be written in the form m/n such that m, n are integers

Irrational Number: A number that has an infinite decimal representation whose digits form no repeating pattern

Rational & Irrational Numbers

73205.13

285714.072

The Number Line

A geometric representation of the real numbers is shown below.

The Number Line

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

372

Open & Closed Intervals Definition Example

Open Interval: The set of numbers that lie between two given endpoints, not including the endpoints themselves

Closed Interval: The set of numbers that lie between two given endpoints, including the endpoints themselves

[-1, 4]

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 ,4

4x

41 x

Calculus and Functions

• Calculus is the mathematics of functions, which are relationships between sets consisting of objects called elements.

• The simplest type of function is a single-variable function, where the elements of two sets are paired off according to certain rules.

Functions

• is used to represent the dependence of one quantity upon another.

• is a rule that takes certain numbers as inputs and assigns to each a definite output number

• A function can be many-to-one, but never one-to-many. Sometimes, in order to emphasize the fact that no value of the independent variable maps into more than one value of the dependent variable,  we’ll  talk  about  this  type  of  relation  as  a  true function or a legitimate function.

Examples of Functions

• The outdoor air temperature is a function of the time of day.

• The number of daylight hours on June 21 is a function of latitude.

• The time required for a wet rag to dry is a function of the air temperature.

Functions in Application

(Response to a Muscle) When a solution of acetylcholine is introduced into the heart muscle of a frog, it diminishes the force with which the muscle contracts. The data from experiments of the biologist A. J. Clark are closely approximated by a function of the form

xbxxR

100

Functions in Application

where x is the concentration of acetylcholine (in appropriate units), b is a positive constant that depends on the particular frog, and R(x) is the response of the muscle to the acetylcholine, expressed as a percentage of the maximum possible effect of the drug.

(a) Suppose that b = 20. Find the response of the muscle when x = 60.

(b) Determine the value of b if R(50) = 60 – that is, if a concentration of x = 50 units produces a 60% response.

xbxxR

100

Functions in Application

SOLUTION

This is the given function. xbxxR

100(a)

Functions in Application

Replace b with 20 and x with 60.

60206010060

R

Simplify the numerator and denominator.

80

600060 R

Divide. 7560 R

Therefore, when b = 20 and x = 60, R (x) = 75%.

Functions in Application

This is the given function. xbxxR

100(b)

Replace x with 50. 505010050

b

R

Replace R(50) with 60.

505010060

b

Functions in Application

Simplify the numerator. 50500060

b

Multiply both sides by b + 50 and cancel.

5050

50006050

bb

b

Distribute on the left side. 5000300060 b

Functions in Application

Therefore, when R (50) = 60, b = 33.3.

Subtract 3000 from both sides. 200060 b

Divide both sides by 60. 3.33b

Functions

EXAMPLE

If , find f (a - 2). 342 xxxf

SOLUTION

This is the given function. 342 xxxf

Replace each occurrence of x with a – 2. 32422 2 aaaf

Functions

Evaluate (a – 2)2 = a2 – 4a + 4. 324442 2 aaaaf

Remove parentheses and distribute. 384442 2 aaaaf

Combine like terms. 12 2 aaf

Functions

• The set of all input numbers is called the domain of the function

• The set of resulting output numbers is called the range of the function.

• The input is called the independent variable • The output is called the dependent variable.

Domain

Definition Example Domain of a Function: The set of acceptable values for the variable x.

The domain of the function is

x

xf

31

03 xx3

Concept Test

As a person hikes down from the top of a mountain, the variable T represents the time, in minutes, since the person left the top of the mountain, and the variable H represents the height, in feet, of the person above the base of the mountain. Table 1.1 gives values at several different times for these variables. Table 1.1

Time T 20 30 40 50 60 Height H 1000 810 730 810 580

Which of the following statements is true? a) T is a function of H b) H is a function of T c) Both statements are true: T is a function of

H and H is a function of T d) Neither statement is true: T is not a function

of H and H is not a function of T

ANSWER

(b) We see that H is a function of T since for every value of T, there is a unique value of H. As the person hikes down the mountain, the height at any given moment in time is uniquely determined. We know that T is not a function of H, since we see in Table 1.1 that, for example, the height H = 810 corresponds to at least two values of T.

As a person hikes down from the top of a mountain, the variable T represents the time, in minutes, since the person left the top of the mountain, and the variable H represents the height, in feet, of the person above the base of the mountain. We have H = f(T). The statement f(100) = 300 means:

a) The mountain rises 300 feet above its base, and it takes 100 minutes to descend from the top of the mountain.

b) The mountain rises 100 feet from its base and it takes 300 minutes to descend from the top of the mountain.

c) At a time of 100 minutes after leaving the top of the mountain, the person is 300 feet above the base of the mountain.

d) At a time of 300 minutes after leaving the top of the mountain, the person is 100 feet above the base of the mountain.

ANSWER

(c) The statement f(100) = 300 tells us that when T = 100, we have H = 300. Therefore, (c) is the correct answer.

As a person hikes down from the top of a mountain, the variable T represents the time, in minutes, since the person left the top of the mountain, and the variable H represents the height, in feet, of the person above the base of the mountain. We have H = f(T). The vertical intercept for the graph of this function represents:

a) The time it takes the person to descend from the top of the mountain to the base of the mountain.

b) The height of the person in feet above the base of the mountain when the person is at the top of the mountain.

c) The height of the person in feet above the base of the mountain, as the person hikes down the mountain.

d) The time when the person begins to descend down the mountain.

ANSWER

(b) Since H = f(T), the vertical intercept is the value of H when T = 0. Since T = 0 means the person is at the top of the mountain, we want the value of H when the person is at the top of the mountain, which is answer (b).

As a person hikes down from the top of a mountain, the variable T represents the time, in minutes, since the person left the top of the mountain, and the variable H represents the height, in feet, of the person above the base of the mountain. We have H = f(T). The horizontal intercept for the graph of this function represents:

a) The time it takes the person to descend from the top of the mountain to the base of the mountain.

b) The height of the person in feet above the base of the mountain when the person is at the top of the mountain.

c) The height of the person in feet above the base of the mountain, as the person hikes down the mountain.

d) The time when the person begins to descend down the mountain.

ANSWER

(a) Since H = f(T), the horizontal intercept is the value of T when H = 0. Since H = 0 means the person is at the base of the mountain, we want the value of T when the person reaches the base of the mountain, which is answer (a).

A  patient’s  heart  rate,  R, in beats per minute, is a function of the dose, D of a drug, in mg. We have R = f(D). The statement f(50) = 70 means: a) The  patient’s  heart  rate  goes  from  70  beats  per  

minute to 50 beats per minute when a dose is given.

b) When  a  dose  of  50  mg  is  given,  the  patient’s  heart rate is 70 beats per minute.

c) The dose ranges from 50 mg to 70 mg for this patient.

d) When  a  dose  of  70  mg  is  given,  the  patient’s  heart rate is 50 beats per minute.

ANSWER

(b) Since R = f(D), the statement f(50) = 70 means that when D = 50, we have R = 70, so the answer is (b).

A  patient’s  heart  rate,  R, in beats per minute, is a function of the dose, D of a drug, in mg. We have R = f(D). The vertical intercept for the graph of this function represents: a) The maximum dose of the drug. b) The maximum heart rate. c) The  dose  of  the  drug  at  which  the  patient’s  heart  

stops beating. d) The  patient’s  heart  rate  if  none  of  the  drug  is  

administered.

ANSWER

(d) Since R = f(D), the vertical intercept is the value of R when D = 0. This is the heart rate when the dose of the drug is 0, so the answer is (d).

Which of the following functions has its domain identical with its range? (a) f(x) = x2

(b) (c) h(x) = x3

(d) i(x) = |x|

( )g x x

ANSWER

COMMENT: It is worth considering the domain and range for all choices.

(b) and (c). For , the domain and range are all nonnegative numbers, and for h(x) = x3, the domain and range are all real numbers.

( )g x x