Buisness calculus notes, Lecture notes of Calculus

Download Buisness calculus notes and more Lecture notes Calculus in PDF only on Docsity!

MATH 1743

Lecture Outlines

Class Lecture Outline accompaniment to

Calculus I for Business, Life & Social Sciences.

Fall 2023

Mr. Gary Barksdale

Department of Mathematics

University of Oklahoma

1743 Binder Contents:

Algebra Skills Necessary for Success

Exam One Materials

Pages i - iv:

Pages 1 – 34:

Pages 35 - 76: Exam Two Materials

Pages 77 – 1 1 2: Exam Three Materials

Pages 113 – 120: Optimization

Your Syllabus, Semester Schedule, and all other course materials are only

available on your Canvas Page

Recall what constitutes a relation that IS a function VS a relation that is NOT a function AND

what is an independent variable (INPUT) and what is dependent variable (OUTPUT).

What is a verbal representation of a function?

Graphical representation?

Numeric representation?

Algebraic representation?

You MUST know how to perform all the algebra of functions and composition:

UNIT conversion: When converting from a smaller unit to larger, you must divide.

When converting from a larger unit to smaller, you must multiply.

ii

You MUST know all your basic shapes of graphs and characteristics of these functions:

Linear: 𝑦𝑦 = 𝑚𝑚𝑥𝑥 + 𝑏𝑏

Quadratic: 𝑦𝑦 = 𝑎𝑎𝑥𝑥 2 + 𝑏𝑏𝑥𝑥 + 𝑐𝑐

Cubic: 𝑦𝑦 = 𝑎𝑎𝑥𝑥 3 + 𝑏𝑏𝑥𝑥 2 + 𝑐𝑐𝑥𝑥 + 𝑑𝑑

Exponential: 𝑦𝑦 = 𝑎𝑎𝑏𝑏 𝑥𝑥

Logarithmic: 𝑦𝑦 = 𝑎𝑎 + 𝑏𝑏𝑏𝑏𝑏𝑏(𝑥𝑥), 𝑥𝑥 > 0

iii

Math 1743 Lecture Guide for Exam I materials Sections 1.1 -1.5, 1.8, 1. pages 1- 34 of this guide

Here are the assigned homework problems from these sections :

1. 1 (pp. 8-11) 1, 4, 5, 7, 10, 15, 18, 19, 24, 33, 34, 37, 38, 40, 41, 46, 47

1 .2 (pp. 19-22) 3, 4, 7, 10, 13, 14, 15, 19, 20, 24, 25, 28

1.3 (pp. 30-31) 2–5, 8 (change all 4's to 2's), 9, 11 - 13, 18, 21, 23, 26, 27-

1.4 (pp 42-45) 2, 3, 4, 8, 9, 10, 15, 18, 21, 23- 26

1.5 (pp. 53-56) 1, 7, 8, 9, 12, 13, 15, 16, 18, 19a, 20, 21, 23

1.10 (pp. 98-102) 3, 10, 12, 13, 17, 18, 23

Please be sure you are completing the assigned homework as each section

is covered in class.

1. 8 (pp. 81-85) 1, 10 – 13, 16, 17

Math 1743 Section 1.1: FUNCTIONS - FOUR REPRESENTATIONS Relations, Functions & Models

• A relation (or rule) links a variable quantity called an input to another variable quantity called an output.
• A function is a relation that assigned exactly one output to each input. A function f with an input of x is written as f(x). read as “ f of x ”.
• Most relations / functions can be represented 4 ways: Verbally, Numerically, Graphically, Algebraically Verbal Representation:

Numerical Representation:

Graphical Representation:

Algebraic Representation:

Example : #2, pg 9. Table representing Military Basic Monthly Officer Pay in 2009 We are instructed to a.) identify the representation, b.) state descriptions of both input and output, c.) indicate whether it is a function or not, d.) draw input/output diagram, IF it is a function (if not a function describe why not)

Example : State the input variable and units, output variable and units,. AND describe what are the four elements of the following mathematical models?

a. P t ( ) = 1.3 t 3 + 5.712 t − 7.918thousand dollars of profit for a small business, t years after 1998.

b. S(t) = 25t 2 +75t+1050 gives the number of new International Starbucks stores targeted to open between 2009 and 2011, t years after 2009.

It is absolutely necessary to be able to distinguish an input from an output in any function representation. Recall, when given an input, you EVALUATE the function for given input, and the result is the output that depends on that input Example: P t ( ) = 80(1.013) t million people of a resident population, t years after 1900.

a.) What was the population in 1950? We can use the calculator, see pgs 5 -6 of Calculator Instruction Manual.

b.) when will the population reach 250,000,000?

We must be able to solve part b algebraically, by hand, BUT this can be time consuming, so we can solve this by one of three methods using the TI84 calculator. Let us refer to the Calculator Instruction Guide, pages 6 – 8.

Method of Intersection of graphs. Graph P(t) as Y1 and Y2=250. The challenge for this method is finding the right window.

Zero Finder. Since P(t) = 250, we can set equal to zero, P t ( ) − 250 = 0 and enter as Y1. Again,

finding the right window is the challenge with this method.

Equation Solver! This involves a process! MATH, up one click, ENTER, up one click, Clear, VARS, Y-VARS, Function, Y1, ENTER. Then subtract the output value from Y1, ENTER. Now you must give a “guess”. Make it a reasonable guess. For this function, we KNOW there is only one solution, since an exponential function has no turning points, as you will recall from our algebra review. As the result, we need only give one guess. Make it reasonable. ALPHA ENTER.

Math 1743 Section 1.2: FUNCTION BEHAVIOR, LIMITS & END BEHAVIOR Limits

Let us recall that a function is said to increase over an interval of ( a,b ) only if f ( a ) < f ( b ). Also, that a function decreases over ( a,b ) only if f ( a ) > f ( b ). Also, a function is constant over ( a,b ) only if f ( a ) = f ( b )

Over what interval does this function increase? Decrease? Is constant?

Concavity: A function, f , is concave up over an interval ( a, b) if it appears to ‘arc upward’ over the interval. ie: forms a ‘bowl’ shape. CCU A function, f , is concave down over an interval ( a, b) if it appears to ‘arc downward’ over the interval. ie: forms a ‘hill’ shape. CCD

Example : #5, pg 20.

over what interval is the function increasing? Decreasing? CCU or CCD?

Inflection point. The point at which a function changes concavity is called the Inflection Point, abbreviated as “ IP ”.

Algebra review time: Recall HA (horizontal asymptotes).

Horizontal Asymptote When a function approaches a numeric value, L, as the input increases (or decreases) without bound, the function has a limiting value of L. the horizontal line with the equation y=L is the horizontal asymptote of the function.

How does this relate to limits and end behavior?

Limits of infinity and End Behavior. Bounded End Behavior.

The notation, lim^ ( ) x

f x L →∞

= (^) means that as x increases without bound the function approaches L.

The notation, lim ( ) x f x L →−∞ = means that as x decreases without bound the function approaches L.

This means the end behavior of the function f(x) is asymptotic to the value L , or L is the limiting value of the function f(x).The function is said to be bounded by this value.

Unbounded End Behavior

The notation, lim ( ) x f x →∞ = ∞ means that as x increases without bound, f(x) increases without bound_._

The notation, lim ( ) x f x →∞ = −∞ means that as x increases without bound, f(x) decreases without bound_._

The notation, lim ( ) x f x →−∞ = ∞ means that as x decreases without bound, f(x) increases without bound_._

The notation, lim^ ( ) x

f x →−∞

= −∞ (^) means that as x decreases without bound, f(x) decreases without bound_._

(See page 17 of text)

#8, pg 20. A model for the number of deaths in the US Army in the first 3 months of the 1918 flu epidemic is

a.) Is A(t) increasing, decreasing, or constant?

b .) Discuss the concavity of A(t) and approximate the input value for the location of the IP.

c.) express, using limit notation, the behavior of this function as x increases without bound.

Example: #26, pg 21.

Example: #11, pg 20.

Math 1743 Section 1.3: LIMITS & CONTINUITY Limit of a function at a point, Numeric Method, Algebraic Method, Rules of Limits, Continuity.

We will start with a graphical demonstration:

What is the domain of this function? What is f (5)=?

As the x values approach, from the left, the value of 5 (since 5 does not exist in this function), what value does the function approach?

As the x values approach, from the right, the value of 5 (since 5 does not exist in this function), what value does the function approach?

Left Limit AND Right Limit So, THE Limit is

5 lim ( ) 3 x − f^ x →

5 lim ( ) 3 x

• f^ x →

5 lim ( ) 3 x f x →

If the Limit from the Left = Limit from the Right, then the limit exists.

lim ( ) x a − f^ x^ L →

= AND lim ( ) x a

• f^ x^ L →

THEN, lim ( ) x a

f x L →

Numerically estimate the limit at a point.

To numerically estimate a limit at a point, we must reference our General Guidelines for Numerical Estimates of Limits. Here we will need to use two tables, one showing the numerical process from the left and one from the right.

Example:

Numerically estimate 2 lim ( ) x f x →− for the function

x x f x x

So, the Left Limit = Right Limit, therefore the Limit Exists! But, if we attempt to evaluate f ( 2)− , the value DNE!

So, f ( x ) is NOT continuous at x = -2.

Algebraic Determination of the Limit.

Recall, all your factoring skills, cancellation of common factors, HA rules, etc…. We will need to learn and commit to memory all the rules of limits.

Constant Rule: The limit of a constant is that constant. If c and K are both constants, then lim x a

K K

→

Replacement Rule: If x = a is a VALID input to f ( x ), then lim ( ) ( ) x a f x f a → = usually f ( x ) is a polynomial function or a rational fnct with valid input.

x → − 2 − f^ ( ) x^ x^ → −^2 + f^ ( ) x

Sum and Difference Rules: The limit of the sum is the sum of the limits. The limit of the difference is the difference of the limits

lim x → a (^) [ f ( ) x ± g x ( ) (^) ] = lim x → a f ( ) x ±lim x → a g x ( )

Product Rule: The limit of the product is the product of the limits. lim x → a (^) [ f ( ) x ⋅ g x ( ) (^) ] = lim x → a f ( ) lim x ⋅ x → a g x ( )

Quotient Rule: The limit of the quotient is the quotient of the limits.

( )^ lim^ ( ) lim ( ) lim ( )

x a x a x a

f x f^ x g x g x

→ → →

{ g x ( )^^ ≠^0 }

Coefficient Rule (or as the text puts it CONSTANT MULTIPLIER RULE): If k is a numeric coefficient, constant multiplier, then, lim ( ) lim ( ) x a x a kf x k f x → →

What this says is that you can “factor out” the coefficient from the limit.

Cancellation Rule: If f ( x ) is a rational function that can be simplified by cancelling a common factor, to the function g ( x ), through factoring or other algebraic manipulation, and x = a is NOT a valid input to f ( x ), but IS a valid input to the simplified function, g ( x ), then, lim ( ) lim ( ) ( ) x a x a f x g x g a → →

Examples: Algebraically find following limits:

3 lim(2 3) x x →

lim p (^3 ) p → p p