Strand Analysis

Calculus

Calculus - 2004

Strand Course The learner will be able to…

The Rate of Change of a Function

1. Coordinates for the Plane Calculus identify and graph coordinates in the plane
2. The Slope of a Line Calculus determine slopes of lines and identify lines that are perpendicular and parallel
3. Equations for Lines Calculus write an equation for a line and find the distance from a point to a line
4. Functions and Graphs Calculus identify a relation as a function and state the domain and range, graph a function, and compose functions
5. Absolute Values, Greatest Integer Function Calculus understand and apply absolute value and its properties; understand and apply the greatest integer function
6. Tangent Lines and Slopes of Curves Calculus understand average rate of change, find the slope of a tangent line to a curve, and write the equation of the tangent line
7. Derivatives Calculus use the definition to find derivatives and find the slopes of curves
8. Velocity and Other Rates of Change Calculus find the average velocity of a moving body and use the derivative to find the instantaneous velocity of the body; understand average and instantaneous rates of change
9. Limits Calculus understand the definition of limit, evaluate the limits of a function; show that a limit exists by using the definition
10. Properties of Limits Calculus understand and apply the properties of limits
11. Continuity Calculus use the concept of limit to determine a function continuous or discontinuous

Derivatives

1. Derivatives of Polynomial Functions Calculus find the first and second derivatives of a polynomial function using the power rule and sum rule
2. Products, Powers, Quotients Calculus find derivatives for products and quotients of functions
3. Implicit Differentiation and Fractional Powers Calculus find derivatives of implicit functions and functions with fractional powers
4. Linear Approximations and Differentials Calculus find the linearization of a function to approximate its values; use differentials to find estimated change
5. The Chain Rule Calculus use the Chain Rule to differentiate composite functions
6. Derivatives of Trigonometric Functions Calculus find the derivatives of the six trigonometric functions
7. Parametric Equations Calculus find the first and second derivatives of parametric equations
8. Newton's Method Calculus use Newton's Method to approximate a solution of an equation
9. Differential Notation Calculus understand the relationship between derivatives and differentials

Applications of Derivatives

1. Sketching Curves with the First Derivative Calculus use the first derivative test to aid in sketching polynomial functions
2. Concavity and Points of Inflection Calculus use the second derivative to test a function for intervals of concavity and inflection points
3. Asymptotes and Symmetry Calculus graph rational functions by determining symmetries and asymptotes
4. Maxima and Minima Theory Calculus apply first and second derivative tests to determine relative and absolute maximums and minimums; apply maxima and minima theory to graphing functions
5. Maxima and Minima Problems Calculus use maxima and minima theory to real world problems
6. Related Rates of Change Calculus use related rates to solve real world problems involving rates
7. The Mean Value Theorem Calculus understand and apply Rolle's theorem and the mean value theorem
8. l'Hopital's Rule Calculus understand and apply l'Hopital's Rule to evaluate specific limits

Integration

1. Indefinite Integrals Calculus determine antiderivatives (indefinite integrals) of functions; find velocity and solve differential equations
2. Substitution Method Calculus find integrals by the substitution method.
3. Integrals of Trigonometric Functions Calculus determine integrals of trigonometric functions.
4. Definite Integrals: The Area under a curve Calculus approximate the area between two curves using rectangles and the Riemann Integral and understand the properties of the definite integral.
5. Calculating Definite Integrals by summation Calculus calculate definite integrals by the summation method.
6. The Fundamental Theorem of Integral Calculus use the First Fundamental Theorem of Calculus, antiderivative, and the Second Fundamental Theorem of Calculus to calculate definite integrals; distinguish between integral area and total area.
7. Rules for Approximating Definite Integrals Calculus approximate a definite integral by the Trapezoidal Rule or Simpson's Rule and estimate the error for each.

Applications of Definite Integrals

1. The Net Change in Position and distance traveled by a moving body Calculus use integration to determine net change in position and distance traveled by a moving body.
2. Areas between Curves Calculus calculate the area between two curves using integration.
3. Calculating Volume: Volumes of Revolution, Slicing (Disk Method) Calculus calculate a volume of a surface of revolution using cross sections and the slice (disk) method.
4. Calculating Volume: Washer and Shell Methods Calculus calculate a volume of a surface of revolution using the washer and shell methods.
5. Lengths of Plane Curves Calculus calculate the length of a curve using integration; understand parametric equations and their derivatives.
6. The Area of a Surface of Revolution Calculus calculate the area of a surface of revolution using integrals.
7. The Average Value of a Function Calculus determine the average value of function and apply it to electricity, average daily inventory problems.
8. Moments and Centers of Mass Calculus understand the concepts of moments and center of mass; calculate the center of mass (gravity) of wires, rods, and areas (plates).
9. Work Calculus calculate the work to compress a spring, or pump water from a storage tank or similar problems.

Transcendental Functions

1. Inverse Functions Calculus understand inverse functions and calculate the derivative of the inverse of a function.
2. Inverse Trigonometric Functions Calculus understand inverse trigonometric functions; find the domain and range.
3. Derivatives of Inverse Trigonometric Functions Calculus find the derivatives and integrals of the inverse trigonometric functions.
4. The Natural Logarithm Calculus understand the natural logarithm and its properties; find its derivative and integral; apply logarithmic differentiation.
5. The Exponential Function ex Calculus understand the exponential function and its properties; find its derivative and integral.
6. Other Exponential Functions ax Calculus find derivative and integrals of functions like ax, xa, and xx.
7. Logarithmic Functions Calculus find derivatives and integrals of logarithmic functions and determine rates of growth.
8. Applications of Exponential and Logarithmic Functions Calculus apply logarithmic and exponential functions to problems such as continuous compound interest, decay, and heat transfer.

Methods of Integration

1. Basic Integration Formulas Calculus use basic integration formulas.
2. Integration by Parts Calculus find integrals of certain products by the method of integration by parts.
3. Products and Powers of Trigonometric Functions Calculus find integrals of products and powers of trigonometric functions.
4. Even Powers of Trigonometric Functions Calculus use methods of integration with even powers of trigonometric functions.
5. Trigonometric Substitution Calculus use trigonometric substitutions to find integrals involving binomials like a2 - u2, a2 + u2, and u2 - a2.
6. Integrals of Quadratic Expressions Calculus use methods of completing the square to find integrals of functions that involve quadratic expressions.
7. Integrals of Rational Function: Partial Fractions Calculus integrate certain rational functions by separating them into partial fractions.
8. Improper Integrals Calculus determine convergence or divergence of an improper integral.
9. Integral Tables Calculus use integral tables to find integrals.