Bryan Lynn

University of Massachusetts Boston

Biography

Bryan is a PhD student at Oregon State University. His research is at the intersection of math and biology with a focus on the evolution of cooperation.

Education

BS Mathematics

University of Massachusetts Boston

Phd Integrative Biology

Oregon State University

Topics Covered

Animal Diversity

Invertebrates

The Chemistry of Life

Life - A Darwinian Approach

The History of Life

Bacteria and Archaea

The Cell

The Cell Cycle and Cellular Reproduction

Metabolism 101

The Central Dogma - Gene Expression

Genomics and Biotechnology

Animal Form and Function

Animal Nutrition

Circulation and Gas Exchange

The Immune System

The Nervous System

An Introduction to the Biosphere

Ecosystems and Restoration Ecology

Conservation Biology and Global Change

Phylogeny

Protists

Plant Diversity

Soil and Plant Nutrition

Speciation

Community Ecology

Cellular Respiration and Fermentation

Photosynthesis

Cell Signaling

Vertebrates

Vascular Plant Structure and Development

Hormones and the Endocrine System

Osmoregulation and Excretion

Equations and Inequalities

Series

Probability Topics

Introduction to Sequences and Series

Differentiation

Integration Techniques

Differential Equations

Second-Order Differential Equations

Derivatives

Vectors

Functions

Limits

Applications of the Derivative

Continuous Functions

Population Ecology

Sensory and Motor Mechanisms

Fungi

Introduction to Matrices

Polynomials

Mendelian Genetics

Vascular Plant Transport

Plant Responses to Signals

Partial Derivatives

Population Evolution

Viruses

Animal Reproduction and Development

Animal Behavior

Amino Acids, Peptides, and Proteins

Rational Functions

Trigonometry

Quadratic Functions

Integrals

Integration

Applications of Integration

Polygons

Area and Perimeter

Congruent Triangles

Relationships Within Triangles

Bryan's Textbook Answer Videos

Graph the surface and the tangent plane at the given point. (Choose the domain and viewpoint so that you get a good view of both the surface and the tangent plane.) Then zoom in until the surface and the tangent plane become indistinguishable.
$ z = \sqrt{9 + x^2y^2} $, $ (2, 2, 5) $

Chapter 14: Partial Derivatives

Section 4: Tangent Planes and Linear Approximations

Draw the graph of $ f $ and its tangent plane at the given point. (Use your computer algebra system both to compute the partial derivatives and to graph the surface and its tangent plane.) Then zoom in until the surface and the tangent plane become indistinguishable.
$ f(x, y) = \dfrac{1 + \cos^2(x - y)}{1 + \cos^2(x + y)} $, $ \biggl( \dfrac{\pi}{3}, \dfrac{\pi}{6}, \dfrac{7}{4} \biggr) $

Chapter 14: Partial Derivatives

Section 4: Tangent Planes and Linear Approximations

Find the linear approximation of the function $ f(x, y) = 1 - xy \cos \pi y $ at $ (1, 1) $ and use it to approximate $ f(1.02, 0.97) $. Illustrate by graphing $ f $ and the tangent plane.

Chapter 14: Partial Derivatives

Section 4: Tangent Planes and Linear Approximations

Use the table in Example 3 to find a linear approximation to the heat index function when the temperature is near $ 94 ^\circ F $ and the relative humidity is near 80%. Then estimate the heat index when the temperature is $ 95^\circ F $ and the relative humidity is 78%.

Chapter 14: Partial Derivatives

Section 4: Tangent Planes and Linear Approximations

The wind-chill index $ W $ is the perceived temperature when the actual temperature is $ T $ and the wind speed is $ v $, so we can write $ W = f(T, v) $. The following table of values is an excerpt from Table 1 in Section 14.1. Use the table to find a linear approximation to the wind-chill index function when $ T $ is near $ -15^\circ C $ and $ v $ is near 50 km/h. Then estimate the wind-chill index when the temperature is $ -17^\circ C $ and the wind speed is 55 km/h.

Chapter 14: Partial Derivatives

Section 4: Tangent Planes and Linear Approximations

A model for the surface area of a human body is given by $ S = 0.1091w^{0.425}h^{0.725} $, where $ w $ is the weight (in pounds), $ h $ is the height (in inches), and $ S $ is measured in square feet. If the errors in measurement of $ w $ and $ h $ are at most 2%, use differentials to estimate the maximum percentage error in the calculated surface area.

Chapter 14: Partial Derivatives

Section 4: Tangent Planes and Linear Approximations

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