Business Calculus: Calculus for Life Sciences I – Syllabus and Core Concepts

Calculus for Life Sciences I (Business Calculus)

Course Overview

This course is designed to serve the needs of students in the life sciences and related fields, including business calculus. The curriculum covers fundamental calculus concepts with applications to real-world problems in the sciences. Key topics include functions, limits, derivatives, applications of derivatives, exponential and logarithmic functions, trigonometric functions, integration, and the Fundamental Theorem of Calculus.

• Course Code: MATH 2321

• Textbook: Calculus with Applications (12th edition) by Lial, Greenwell, & Ritchey

• Software: MyLab Math for homework and assignments

Core Topics and Subtopics

Functions and Their Properties

Understanding functions is foundational in calculus. Functions describe relationships between variables and are used to model real-world phenomena.

• Definition: A function is a rule that assigns to each input exactly one output.

• Types of Functions: Linear, quadratic, polynomial, rational, exponential, logarithmic, and trigonometric.

• Example: The function $f(x) = 2x + 3$ is linear; $f(x) = e^x$ is exponential.

Limits and Continuity

Limits are essential for defining derivatives and integrals. Continuity ensures that functions behave predictably without sudden jumps.

• Limit: The value that a function approaches as the input approaches a certain point.

• Continuity: A function is continuous at a point if the limit exists and equals the function's value at that point.

• Formula: $\lim_{x \to a} f(x) = L$

• Example: $\lim_{x \to 2} (3x + 1) = 7$

Rates of Change

Rates of change describe how a quantity changes with respect to another. In business and life sciences, this often models growth, decay, or cost changes.

• Average Rate of Change: $\frac{f(b) - f(a)}{b - a}$

• Instantaneous Rate of Change: Given by the derivative at a point.

• Example: The rate at which a population grows at a specific time.

Derivatives and Differentiation

The derivative measures the instantaneous rate of change of a function. Differentiation is the process of finding derivatives.

• Definition: The derivative of $f(x)$ at $x = a$ is $f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$

• Rules: Power rule, product rule, quotient rule, chain rule.

• Example: If $f(x) = x^2$, then $f'(x) = 2x$

Applications of Derivatives

Derivatives are used to find maxima and minima (optimization), rates of change in applied contexts, and to analyze the behavior of functions.

• Critical Points: Where $f'(x) = 0$ or $f'(x)$ is undefined.

• Optimization: Finding maximum profit or minimum cost in business applications.

• Concavity and Inflection Points: Determined by the second derivative $f''(x)$.

• Example: Maximizing revenue by setting the derivative of the revenue function to zero.

Exponential and Logarithmic Functions

These functions model growth and decay processes, such as population growth, radioactive decay, and compound interest.

• Exponential Function: $f(x) = a e^{bx}$

• Logarithmic Function: $f(x) = \ln(x)$

• Derivative of Exponential: $\frac{d}{dx} e^{x} = e^{x}$

• Derivative of Logarithm: $\frac{d}{dx} \ln(x) = \frac{1}{x}$

• Example: Modeling bacterial growth with $P(t) = P_0 e^{kt}$

Trigonometric Functions and Their Derivatives

Trigonometric functions are used in modeling periodic phenomena. Their derivatives are essential in calculus applications.

• Basic Functions: $\sin(x)$, $\cos(x)$, $\tan(x)$

• Derivatives: $\frac{d}{dx} \sin(x) = \cos(x)$, $\frac{d}{dx} \cos(x) = -\sin(x)$

• Example: Modeling seasonal changes in population.

Integration and the Fundamental Theorem of Calculus

Integration is the reverse process of differentiation and is used to find areas, accumulated quantities, and solve differential equations.

• Indefinite Integral: $\int f(x) dx$

• Definite Integral: $\int_a^b f(x) dx$

• Fundamental Theorem of Calculus: If $F$ is an antiderivative of $f$, then $\int_a^b f(x) dx = F(b) - F(a)$

• Example: Calculating total profit over a time interval.

Course Schedule (Sample)

The following table outlines the suggested schedule and topics for the semester, based on the syllabus:

Class #	Date	Section	Topic
1	8/26	2.1	Intro to course and review functions
2	8/29	2.1	Functions
3	9/1	3.2	Limits
4	9/4	3.3	Limits and Continuity
5	9/6	3.4	Rates of Change
6	9/8	3.4	Definition of Derivative, Quiz 1
7	9/11	4.1	Differentiation Rules
8	9/13	4.3	Chain Rule
9	9/15	4.4/5	Catch up/Review for Test 1
10	9/18		Test 1
11	9/20	2.1/2.2	Exponential and Logarithmic Functions
12	9/22	3.1	Derivatives of Exponential and Log Functions
13	9/27	10.7	Trigonometric Functions
14	9/29	13.1	Derivatives of Trigonometric Functions, Quiz 2
15	10/2	5.1	Exp/Log Functions
16	10/4	5.2	Exp/Log Functions
17	10/6	5.3	Relative Max/Min
18	10/9	5.4	Relative Max/Min
19	10/13	5.3	Higher Derivatives/Concavity
20	10/16	5.4	Test 2
21	10/18	6.1	Absolute extrema
22	10/20	6.2	Curve Sketching
23	10/23	6.3	Optimization/Application of derivatives, Quiz 3
24	11/1	7.1	Antiderivatives
25	11/6	7.2	Area and Definite Integrals
26	11/8	7.3	Substitution
27	11/15	7.3	THANKSGIVING Break-No Class
28	12/4	7.4	Fundamental Theorem of Calculus
29	12/7/11	ALL	Comprehensive Review
30	12/11	ALL	FINAL EXAM, 8:00-10:30 am

Assessment and Grading

• Quizzes: 15 points (5 points each, lowest dropped)

• MyLab Math Homework: 18 points

• Midterm Tests: 45 points (15 points each, best two out of three count)

• Final Exam: 22 points (comprehensive)

• Extra Credit: Syllabus Quiz and corrections of midterm exams

Academic Honesty and Policies

• All work must be your own; plagiarism and cheating are not tolerated.

• Electronic devices must be turned off during class and exams.

• Attendance is essential for success in this course.

Resources

• Student Learning Assistance Center (SLAC): 4th floor of Alkek Library

• Collaborative Learning Center (CLC): 4th floor of R5 Mitte Building

• Math CATS: Derrick Hall 238

Additional info:

• This syllabus is tailored for students in the life sciences, but the calculus content is directly relevant to business calculus as well, especially in the application of derivatives and integrals to real-world problems.