# 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 Topic Pre-requisite Student Learning Outcome
Week 1 Review of Functions and Change
• Define a function and connect to a real-world dynamical model
• Identify the parts of linear functions (slope, y-intercept).
• Demonstrate how to manipulate fractions.
• Identify power functions and polynomials.
• Identify exponential functions and their graphs in terms of exponential growth/decay.
• Identify logarithmic functions, graph and solve equations with log properties.
• Analyze graphs of the sine and cosine by recognizing amplitude and period.
• Identify and compute composite functions.
Week 2 Instantaneous Rate of Change
• Evaluating functions
• Tangent lines
• Average rate of change
• Equations of a line (slope-intercept, point-slope)
• Compute the average rate of change (ARC).
• Compute the instantaneous rate of change (IRC)
• Comparing and contrasting ARC with IRC
• Defining velocity using the idea of a limit
• Visualizing the limit with tangent lines
Week 3

Example

• 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
• Use the limit definition to define the derivative at a particular point and to define the derivative function
• Distinguish the definition of continuity of a function
• Apply derivatives to biological functions
Week 4 & 5 Derivative Formulas
• 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
• Differentiate trigonometric functions
• Applications of trigonometric function derivatives
Week 6 Applications of Derivatives
• 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 Accumulated Change
• 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 7 The Definite Integral
• 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 Antiderivatives Basics in graphing
• Use the limit formula to compute a definite integral
• 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 exponential and logarithm functions
• Use formulas for finding antiderivatives of trigonometric functions
Week 9 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)
Week 10 Applications of Integrals Example Solve various biology applications using the fundamental theorem of calculus
Week 11 Integration by Substitution Example Applying integration by substitution formulas
Week 12 Integration by Parts Example
• Applying integration by integration by parts formulas
• Recognize which integration formulas to use
Week 13 Differential Equations (Mathematical Modeling) Word problem setup and understanding of mathematical models
• Demonstrate how to take information to set up a mathematical model
• Examine the basic parts of differential equations
Week 14 Differential Equations Graphing and factoring
• Examine differential equations graphically with slope fields
• Use separation of variables for solving differential equations
Week 15 Differential Equations Applications Exponential functions
• Apply differential equations to exponential growth & decay functions for population models
• Apply differential equations to surge functions for drug models
• Modeling the spread of a disease