General Information:
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- Instructor(s): Paul Tsopméné
- Instructor(s) Email: paul.tsopmene@ubc.ca
- Instructor(s) Office: SCI 259
- Duration: Term 2 Winter 2021 (Jan – Apr 2022)
- Delivery Modality: Online (until February 7, 2022 – the Zoom link is posted on Canvas) and In-Person for the rest of the term.
- Course Location: FIP 204
- Classroom Schedule: Tue and Thu: 2:00 PM – 3:30 PM
- Office Hours: Tue and Thu: 5:00 PM – 6:30 PM. I am also available by email or by appointment if these times do not work for you.
- Course Website: Course materials are available on Canvas. My primary method of communication is through Canvas messages/emails. Make sure you check this website regularly.
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Course Objective: This course aims to give students a broad range of mathematical concepts foundational to biology and data analysis in biology. Specific objectives include:
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- to understand the essential calculus concepts and techniques at the same level as the general calculus course.
- to gain some exposure to more advanced calculus concepts from a computational viewpoint as opposed to a full mathematically rigorous treatment.
- to appreciate the connection between biological phenomena and mathematical models by using linear algebra and differential equations.
- to understand some basic statistical methods to understand a set of data.
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Course Overview: This course will focus predominantly on single-variable integration and its applications. While the list of topics aligns well with a general calculus course, most of the concepts will be presented through applications in life sciences, including applications in Biology. The course will start off with integration. In this first chapter, we will learn how to integrate using the substitution rule, find the area between two curves, and use integration to calculate cumulative rates of change and average values. In the second chapter, we will learn how to integrate using integration by parts, integrate by using partial fractions, find improper integrals, and integrate numerically. In the third chapter, we will learn how to use integrals to solve separable differential equations and describe the behavior of solutions. In the fourth chapter, we will learn how to solve a system of linear equations (2 equations and 2 unknowns), perform algebraic operations on 2 x 2 matrices, and find the eigenvalues and eigenvectors of 2 x 2 matrices. In chapter 5, we will learn how to calculate the sum of a geometric series and find the Taylor series of a function. In the last chapter, we will learn how to calculate probabilities for continuous random variables, find the expected value, variance, and standard deviation of a continuous random variable, and find the linear regression line.
Content: Topics include
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- Integration: antiderivatives and indefinite integrals, area and the definite integral, the fundamental theorem of calculus, applications of integration (areas, net change, average values).
- Techniques of Integration and Computational Methods: integration by parts, integration using partial fractions, improper integrals, numerical integration (the trapezoidal rule and Simpson’s rule).
- Differential Equations: solving first-order separable differential equations, equilibria and their stability, integrating factors, applications (population models, one-compartment models, and more).
- Linear Algebra: systems of linear equations, matrix operations, eigenvectors and eigenvalues, an application: the Leslie matrix. (We restrict ourselves to 2 x 2 matrices.)
- Infinite Series: geometric series, Taylor series.
- Continuous Probability Distributions and Linear Regression: probability density functions, expected value and variance, normal distribution, linear regression
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Learning Outcomes: Upon successful completion of this course, students will be able to:
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- Understand the basic idea of integral calculus: finding the area under a curve.
- Approximate the area under a curve by using the left, right, or midpoint Riemann sums.
- Evaluate integrals by using basic antiderivative formulas.
- Evaluate integrals by using techniques such as substitution, integration by parts, and partial fraction decomposition.
- Use integrals to find the area between two curves, the cumulative/net change, and the average value of a function.
- Approximate the value of a definite integral using the midpoint rule and the trapezoidal rule.
- Determine whether an improper integral converges or diverges and evaluate it if it is convergent.
- Solve and analyze first-order separable differential equations. Specifically, students should be able to: (1) Use integrals to solve separable differential equations; (2) Find equilibria and determine their stability graphically and analytically; (3) Describe the behavior of solutions of differential equations starting from different initial conditions. (4) Construct a differential equation modelling a quantity described in a problem.
- Compute the eigenvalues and eigenvectors of 2 x 2 matrices, and apply this to the study of Leslie matrices, which are used extensively in ecology to model the changes in the population of organisms over a period of time.
- Find the sum of a geometric series or determine that a geometric series is divergent.
- Find the Taylor (or Maclaurin) series of a function.
- Calculate probabilities for continuous variables, find the expected values and the standard deviation, and find the linear regression line.
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Learning Resources:
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- Textbook (Optional): Calculus for Biology and Medicine, by Claudia Neuhauser, 3rd edition, Pearson. (If you have the 4th edition instead, that’s fine.) This is optional, as the lecture contents and the practice problems I will post on Canvas will be enough.
- Practice Problems: Practice problems and detailed solutions as well as reviews of the relevant theories will be posted on Canvas every week. While completion of these problems is essential for the success in this course, you do not hand them in for grading.
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Course Outline: More information about this course can be found in the course outline.