MAT1193

Calculus for the Biosciences

MAT 1193. Calculus for the Biosciences. (3-0) 3 Credit Hours. (TCCN = MATH 2313)

Prerequisite: MAT 1093 or an equivalent course or satisfactory performance on a placement examination. An introduction to calculus is presented using discrete-time dynamical systems and differential equations to model fundamental processes important in biological and biomedical applications. Specific topics to be covered are limits, continuity, differentiation, antiderivatives, definite and indefinite integrals, the fundamental theorem of calculus, differential equations, and the phase-plane. (Formerly MAT 1194. Credit can be earned for only one of the following: MAT 1193, MAT 1194, or MAT 1214.) May apply toward the Core Curriculum requirement in Mathematics. Generally offered: Fall, Spring, Summer. Course Fees: DL01 $72; LRC1 $12; LRS1 $45; STSI $21.

Date	Section	Topic	Pre-requisite	Student Learning Outcome
Week 1	Example	Review of functions and change	Basic graphing skills The idea of a function Graphs of elementary functions (lines, parabola) Understanding of slope Periodic functions	Define a function and connect to a real-world dynamical model Estimate instantaneous rate of change by both visualization of average rate of change and calculations of the formula Understand formulas for distance, velocity and speed and make connection with slope formula Understand exponential functions and their graphs in terms of exponential growth/decay Understand logarithmic functions, graph and solve equations with log properties Analyze graphs of the sine and cosine by understanding amplitude and period
Week 2	Example	Instantaneous Rate of Change	Evaluating functions Tangent lines Average rate of change Equations of a line (slope-intercept, point-slope)	Comparing and contrasting the average rate of change (ARC) with instantaneous rate of change (IRC) Defining velocity using the idea of a limit Visualizing the limit with tangent lines Recognize graphs of derivatives from original function Estimate the derivative of a function given table data and graphically Interpret the derivative with units and alternative notations (Leibniz) Use derivative to estimate value of a function
Week 3	Example	Limits	Example	Use the limit definition to define the derivative at a particular point and to define the derivative function Understand the definition of continuity Apply derivatives to biological functions
Week 4	Example	Derivative Formulas (Derivatives for powers and polynomials)	Equations of lines Composite functions	Use constant formula and power formula to differentiate functions along with the sum and difference rule Use differentiation to find the equation of a tangent line to make predictions using tangent line approximation Differentiate exponential and logarithmic functions Differentiate composite functions using the chain rule Differentiate products and quotients
Week 4	Example	Derivative Formulas (Derivatives for trigonometric functions)	Exponential Logarithmic Trigonometric Applications	Differentiate trigonometric functions Applications of trigonometric function derivatives
Week 6	Example	Applications	Local & Global Maxima & Minima Concavity	Detecting a local maximum or minimum from graph and function values Test for both local and global maxima and minima using first derivative test (finding critical points) Test for both local and global maxima and minima using second derivative test (testing concavity) Using concavity for finding inflection points Apply max and min techniques in real world applications in the field of Biology (logistic growth)
Week 7	Example	Accumulated Change & the Definite Integral	Distance formula Summation formulas	Approximate total change from rate of change Computing area with Riemann Sums Apply concepts of finding total change with Riemann Sums
Week 8	Example	Antiderivatives	Basics in graphing	Be able to analyze area under the curve with antiderivatives graphically and numerically Use formulas for finding antiderivatives of constants and powers Use formulas for finding antiderivatives of trigonometric functions
Week 9	Example	Definite Integral and the Fundamental Theorem of Calculus	Average formula	Use the limit formula to compute a definite integral Compute area with the fundamental theorem of calculus (FTC) Interpreting the definite integral as area above and below the graph Use the definite integral to compute average value
Week 10	Example	Integration Applications	Example	Solve various biology applications using the fundamental theorem of calculus
Week 10	Example	Substitution Method	Example	Applying integration by substitution formulas
Week 11	Example	Integration by Parts and further applications	Example	Applying integration by integration by parts formulas Recognize which integration formulas to use
Week 12	Example	Differential Equations (Mathematical Modeling)	Word problem setup and understanding of mathematical models	Understand how to take information to set up a mathematical model Examine the basic parts of differential equations
Week 13	Example	Graphs of differential equations and separation of variables for solving differential equations	Graphing and factoring	Examine differential equations graphically with slope fields Use separation of variables for solving differential equations
Week 14	Example	Exponential growth and decay & surge function	Exponential functions	Apply differential equations to exponential growth & decay functions for population models Apply differential equations to surge functions for drug models