#### Overview

#### Topics Covered

**Preview of Modeling and Calculus**

**1 Modeling with Functions**

1.1 Real Numbers and Functions

1.2 Data Fitting with Linear and Periodic Functions

1.3 Power Functions and Scaling Laws

1.4 Exponential Growth

1.5 Function Building

1.6 Inverse Functions and Logarithms

1.7 Sequences and Difference Equations

**2 Limits and Derivatives**

2.1 Rates of Change and Tangent Lines

2.2 Limits

2.3 Limit Laws and Continuity

2.4 Asymptotes and Infinity

2.5 Sequential Limits

2.6 Derivative at a Point

2.7 Derivatives as Functions

Group Projects

**3 Derivative Rules and Tools**

3.1 Derivatives of Polynomials and Exponentials

3.2 Product and Quotient Rules

3.3 Chain Rule and Implicit Differentiation

3.4 Derivatives of Trigonometric Functions

3.5 Linear Approximation

3.6 Higher Derivatives and Approximations

3.7 l’Hoˆ pital’s Rule

Group Projects

**4 Applications of Differentiation**

4.1 Graphing Using Calculus

4.2 Getting Extreme

4.3 Optimization in Biology

4.4 Decisions and Optimization

4.5 Linearization and Difference Equations

Group Projects

**5 Integration**

5.1 Antiderivatives

5.2 Accumulated Change and Area under a Curve

5.3 The Definite Integral

5.4 The Fundamental Theorem of Calculus

5.5 Substitution

5.6 Integration by Parts and Partial Fractions

5.7 Numerical Integration

5.8 Applications of Integration

Group Projects

**6 Differential Equations**

6.1 A Modeling Introduction to Differential Equations

6.2 Solutions and Separable Equations

6.3 Linear Models in Biology

6.4 Slope Fields and Euler’s Method

6.5 Phase Lines and Classifying Equilibria

6.6 Bifurcations

Group Projects

**7 Probabilistic Applications of Integration**

7.1 Histograms, PDFs, and CDFs

7.2 Improper Integrals

7.3 Mean and Variance

7.4 Bell-Shaped Distributions

7.5 Life Tables

Group Projects

**8 Multivariable Extensions**

8.1 Multivariate Modeling

8.2 Matrices and Vectors

8.3 Eigenvalues and Eigenvectors

8.4 Systems of Linear Differential Equations

8.5 Nonlinear Systems

Group Projects