Differential Calculus for the Life Sciences

Calculus arose as a tool for solving practical scientific problems through the centuries. However, it is often taught as a technical subject with rules and formulas (and occasionally theorems), devoid of its connection to applications. In this course, the applications form an important focal point, with emphasis on life sciences. This places the techniques and concepts into practical context, as well as motivating quantitative approaches to biology taught to undergraduates.

Calculus Mathematics

Tác giả: Leah Edelstein-Keshet
Nhà xuất bản: University of British Columbia
Nguồn: LibreTexts
Giấy phép: CC BY-NC-SA
Ngôn ngữ: EN
Danh mục: Khoa học cơ bản

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Mục lục· 111 mục

01: Power functions as building blocks
1.01: Power Functions
1.02: How big can a cell be A model for nutrient balance
1.03: Sustainability and energy balance on Earth
1.04: First steps in graph sketching
1.05: Rate of an enzyme-catalyzed reaction
1.06: Predator Response
1.07: Summary
1.08: Exercises
02: Average rates of change average velocity and the secant line
2.01: Time-dependent data and rates of change
2.02: The slope of a straight line is a rate of change
2.03: The slope of a secant line is the average rate of change
2.04: From average to instantaneous rate of change
2.05: Introduction to the derivative
2.06: Summary
2.07: Exercises
03: Three Faces of the Derivative - Geometric Analytic and Computational
3.01: The Geometric View - Zooming into the Graph of a Function
3.02: The analytic view- calculating the derivative
3.03: The computational view- software to the rescue
3.04: Summary
3.05: Exercises
04: Differentiation rules simple antiderivatives and applications
4.01: Rules of Differentiation
4.02: Application of the Second Derivative to Acceleration
4.03: Sketching the First and Second Derivative and the Anti-Derivatives
4.04: Summary
4.05: Exercises
05: Tangent lines Linear Approximation and Newtons Method
5.01: The Equation of a Tangent Line
5.02: Generic Tangent line Equation and Properties
5.03: Approximating a Function by its Tangent Line
5.04: Tangent Lines for Finding Zeros of a Function - Newtons Method
5.05: Aphids and Ladybugs revisited
5.06: Harder problems- finding the point of tangency
5.07: Summary
5.08: Exercises
06: Sketching the Graph of a Function using Calculus Tools
6.01: Overall Shape of the Graph of a Function
6.02: Special Points on the Graph of a Function
6.03: Sketching the Graph of a Function
6.04: Summary
6.05: Exercises
07: Optimization
7.01: Simple Biological Optimization Problems
7.02: Optimization with a constraint
7.03: Checking Endpoints
7.04: Optimal Foraging
7.05: Summary
7.06: Exercises
08: Introducing the Chain Rule
8.01: The Chain Rule
8.02: The chain rule applied to optimization problems
8.03: Summary
8.04: Exercises
09: Chain rule applied to related rates and implicit differentiation
9.01: Applications of the chain rule to related rates
9.02: Implicit Differentiation
9.03: The Power Rule for Fractional Powers
9.04: Summary
9.05: Exercises
10: Exponential Functions
10.01: Unlimited Growth and Doubling
10.02: Derivatives of Exponential Functions
10.03: Inverse Functions and Logarithms
10.04: Applications of the Logarithm
10.05: Summary
10.06: Exercises
11: Differential equations for exponential growth and decay
11.01: Introducing a new kind of equation
11.02: Differential equation for unlimited population growth
11.03: Radioactivity
11.04: Deriving a differential equation for the growth of cell mass
11.05: Summary
11.06: Exercises
12: Solving Differential Equations
12.01: Verifying that a Function is a Solution
12.02: Equations of the Form- y'(t) aby
12.03: Eulers Method and Numerical Solutions
12.04: Summary
12.05: Exercises
13: Qualitative Methods for Differential Equations
13.01: Linear and Nonlinear Differential Equations
13.02: The Geometry of Change
13.03: Applying qualitative analysis to biological models
13.04: Summary
13.05: Exercises
14: Periodic and Trigonometric Functions
14.01: Basic Trigonometry
14.02: Periodic Functions
14.03: Inverse trigonometric functions
14.04: Summary
14.05: Exercises
15: Cycles Periods and Rates of Change
15.01: Derivatives of Trigonometric Functions
15.02: Changing Angles and Related Rates
15.03: The Zebra Danios Escape Responses
15.04: Summary
15.05: Exercises
16: Additional Exercises
16.01: Exercises
17: Appendices
17.01: Appendix A- A review of Straight Lines
17.02: Appendix B- A precalculus review
17.03: Appendix C- A Review of Simple Functions
17.04: Appendix D- Limits
17.05: Appendix E- Proofs
17.06: Appendix F- Trigonometry Review
17.07: Appendix G- For Further Study
17.08: Appendix H- Short Answers to Problems