Unit Dates

Topics

College Board Standards

California Content Standards

Chapter P

9 Days

Preparation for Calculus

• Graphs and Models
• Linear Models and Rates of Change
• Functions and their Graphs

Unit Dates

Topics

College Board Standards

California Content Standards

Chapter 1

12 Days

Limits and Their Properties

• A Preview of Calculus
• Finding Limits Graphically and Numerically
• Evaluating Limits Analytically
• Continuity and One-Side Limits
• Infinite Limits

•  An intuitive understanding of the limiting process.
• Calculating limits using algebra.
• Estimating limits from graphs or tables of data.

• An intuitive understanding of continuity. (Close values of the domain lead to close values of the range.)
• Understanding continuity in terms of limits.

1.1     Students prove and use theorems evaluating the limits of sums, products, quotients, and composition of functions.

1.2     Students use graphical calculators to verify and estimate limits.

1.3     Students prove and use special limits, such as the limits of sinx/x and (1-cosx)/x as x tends to 0.

2.0     Students demonstrate knowledge of both the formal definition and the graphical interpretation of continuity of a function.

Unit Dates

Topics

College Board Standards

California Content Standards

Chapter 2

19 Days

Differentiation

• The Derivative and the Tangent Line Problem
• Basic Differentiation Rules and Rates of Change
• The Product and Quotient Rules and Higher Order Derivatives
• The Chain Rule
• Implicit Differentiation
• Related Rates

• Derivative presented geometrically, numerically, and analytically.
• Derivative interpreted as an instantaneous rate of change.
• Derivative defined as the limit of the difference quotient.
• Relationship between differentiability and continuity.
• Slope of a curve at a point. Examples are emphasized, including points at which there are vertical tangents and points at which there are no tangents.
• Instantaneous rate of change as the limit of average rate of change.
• Approximate rate of change from graphs and tables of values.
• Use of implicit differentiation to find the derivative of an inverse function.

• Knowledge of derivatives of basic functions, including power, exponential, trigonometric, and inverse trigonometric functions.
• Basic rules for the derivative of sums, products, and quotients of functions.
• Chain Rule and implicit differentiation.
• Geometric interpretation of differential equations via slope fields and the relationship between slope fields and derivatives of implicitly defined functions.

4.1 students demonstrate and understand the derivative of a function as the slope of the tangent line to the graph of the function.

4.2 Students demonstrate an understanding of the interpretation of the derivative as an instantaneous rate of change.  Students can use derivatives to solve a variety of problems from physics, chemistry, economics, and so forth that involve the rate of change of a function.

4.3 Students understand the relation between differentiability and continuity.

4.4 Students derive derivative formulas and use them to find the derivatives of algebraic, trigonometric, inverse trigonometric, exponential, and logarithmic functions.

5.0 Students know the chain rule and its proof and applications to the calculation of the derivative of variety of composite functions.

6.0 Students find the derivatives of parametrically defined functions and use implicit differentiation in a wide variety of problem in physics, chemistry, economics, and so forth.

7.0 Students compute derivatives of higher orders.

12.0 Students use differentiation to solve related rate problems in a variety of pure and applied contexts.

Unit Dates

Topics

College Board Standards

California Content Standards

Chapter 3

23 Days

Application of Differentiation

• Extrema on an Interval
• Rolle’s Theorem and the Mean Value Theorem
• Increasing and Decreasing Functions and the First Derivative Test
• Concavity and the Second Derivative Test
• Limits at Infinity
• A Summary of Curve Sketching
• Optimization Problems
• Newton’s Method
• Differentials

• Corresponding characteristics of the graphs of f, f', and f".
• Relationship between the concavity of f and the sign of f".
• Points of inflection as places where concavity changes.
• Analysis of curves, including the notions of monotonicity and concavity.
• Optimization, both absolute (global) and relative (local) extrema.
• Modeling rates of change, including related rates problems.
• Interpretation of derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration.
• Tangent line to a curve at a point and local linear approximation.

• Geometric understanding of graphs of continuous functions (Intermediate Value Theorem and Extreme Value Theorem).

• Understanding asymptotes in terms of graphical behavior.
• Describing asymptotic behavior in terms of limits involving infinity.

3.0     Students demonstrate an understanding and the application of the intermediate value theorem and the extreme value theorem.

9.0 Students use differentiation to sketch, by hand, graphs of functions.  They can identify maxima, minima, inflection points, and intervals in which the function is increasing and decreasing.

10.0Students know Newton’s method for approximating the zeros of a function.

11.0 Students use differentiation to solve optimization (maximum-minimum problems) in a variety of pure and applied contexts.

Unit Dates

Topics

College Board Standards

California Content Standards

Chapter 4

19 Days

Integration

• Antiderivatives and Indefinite Integration
• Area
• Riemann Sums and Definite Integrals
• The Fundamental Theorem of Calculus
• Integration by Substitution
• Numerical Integration

• Antiderivatives following directly from derivatives of basic functions.
• Antiderivatives by substitution of variables (including change of limits for definite integrals).

• Finding specific antiderivatives using initial conditions, including applications to motion along a line.

• Definite integral as a limit of Riemann sums over equal subdivisions.
• Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval:
• Use of the Fundamental Theorem to evaluate definite integrals.
• Use of the Fundamental Theorem to represent a particular antiderivative, and the analytical and graphical analysis of functions so defined.
• Numerical approximations to definite integrals. Use of Riemann and trapezoidal sums to approximate definite integrals of functions represented algebraically, geometrically, and by tables of values.
• Computation of Riemann sums using left, right and midpoint evaluation points.

13.0 Students know the definition of the definite integral by using Riemann sums.  They use this definition to approximate integrals.

14. 0 Students apply the definition of the integral to model problems in physics, economics, and so forth, obtaining results in terms of integrals.

15. 0 Students demonstrate knowledge and proof of the fundamental theorem of calculus and use it to interpret integrals as antiderivatives.

16. 0 Students use definite integrals in problems involving area, velocity, acceleration, volume of a solid, area of a surface of revolution, length of a curve and work.

17.0 Students compute by hand, the integrals of a wide variety of functions by using techniques of integration, such as substitution, integration by parts, and trigonometric substitution.  They can also combine these techniques when appropriate.

20.0 Students compute the integrals of trigonometric functions.

Unit Dates

Topics

College Board Standards

California Content Standards

Chapter 5

16 Days

Logarithmic and Exponential Functions

• The Natural Logarithmic Function: Differentiation
• The Natural Logarithmic Function:  Integration
• Inverse Functions
• Exponential Functions:  Differentiation and Integration
• Differential Equations:  Growth and Decay
• Differential Equations:  Separation of Variables

• Comparing relative magnitudes of functions and their rates of change. (For example, contrasting exponential growth, polynomial growth, and logarithmic growth.)

• Solving separable differential equations and using them in modeling. In particular, studying the equation y' = ky and exponential growth.

4.4 Students derive derivative formulas and use them to find the derivatives of algebraic, trigonometric, inverse trigonometric, exponential, and logarithmic functions.

17.0  Students compute by hand, the integrals of a wide variety of functions by using techniques of integration, such as substitution, integration by parts, and trigonometric substitution.  They can also combine these techniques when appropriate.

Chapter 6

17 Days

Application of Integration

• Area of Region Between Two Curves
• Volume:  The Disk Method

16.0  Students use definite integrals in problems involving area, velocity, acceleration, volume of a solid, area of a surface of revolution, length of a curve and work.