Students demonstrate knowledge of both the formal definition and the graphical interpretation of limit of values of functions. This knowledge includes one-sided limits, infinite limits, and limits at infinity. Students know the definition of convergence and divergence of a function as the domain variable approaches either a number or infinity:
1.1
Prove and use limit theorems for combined functions
Students prove and use theorems evaluating the limits of sums, products, quotients, and composition of functions.
1.2
Use graphical calculators to verify and estimate limits
Students use graphical calculators to verify and estimate limits.
1.3
Prove and use special trigonometric limits
Students prove and use special limits, such as the limits of (sin(x))/x and (1−cos(x))/x as x tends to 0.
2.0
Understand formal/graphical definition of continuity
Students demonstrate knowledge of both the formal definition and the graphical interpretation of continuity of a function.
3.0
Apply the intermediate value and extreme value theorems
Students demonstrate an understanding and the application of the intermediate value theorem and the extreme value theorem.
4.0
Understand the formal definition of the derivative and differentiability
Students demonstrate an understanding of the formal definition of the derivative of a function at a point and the notion of differentiability:
4.1
Understand the derivative as the slope of the tangent line
Students demonstrate an understanding of the derivative of a function as the slope of the tangent line to the graph of the function.
4.2
Interpret the derivative as instantaneous rate of change
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
Understand the relation between differentiability and continuity
Students understand the relation between differentiability and continuity.
4.4
Derive and use derivative formulas for common function types
Students derive derivative formulas and use them to find the derivatives of algebraic, trigonometric, inverse trigonometric, exponential, and logarithmic functions.
5.0
Know and apply the chain rule
Students know the chain rule and its proof and applications to the calculation of the derivative of a variety of composite functions.
6.0
Find derivatives of parametric functions and use implicit differentiation
Students find the derivatives of parametrically defined functions and use implicit differentiation in a wide variety of problems in physics, chemistry, economics, and so forth.
7.0
Compute higher-order derivatives
Students compute derivatives of higher orders.
8.0
Apply Rolle's Theorem, the Mean Value Theorem, and L'Hopital's rule
Students know and can apply Rolle’s Theorem, the mean value theorem, and L’Hôpital’s rule.
9.0
Use differentiation to sketch graphs by hand
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.
10.0
Know Newton's method for approximating zeros
Students know Newton’s method for approximating the zeros of a function.
11.0
Use differentiation to solve optimization problems
Students use differentiation to solve optimization (maximum-minimum problems) in a variety of pure and applied contexts.
12.0
Use differentiation to solve related rate problems
Students use differentiation to solve related rate problems in a variety of pure and applied contexts.
13.0
Define and approximate the definite integral using Riemann sums
Students know the definition of the definite integral by using Riemann sums. They use this definition to approximate integrals.
14.0
Apply the definition of the integral to model applied problems
Students apply the definition of the integral to model problems in physics, economics, and so forth, obtaining results in terms of integrals.
15.0
Prove and use the Fundamental Theorem of Calculus
Students demonstrate knowledge and proof of the fundamental theorem of calculus and use it to interpret integrals as antiderivatives.
16.0
Use definite integrals for area, velocity, volume, and work
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
Compute integrals using substitution, parts, and trig substitution
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.
18.0
Know properties of inverse trig functions as indefinite integrals
Students know the definitions and properties of inverse trigonometric functions and the expression of these functions as indefinite integrals.
19.0
Compute integrals of rational functions using partial fractions
Students compute, by hand, the integrals of rational functions by combining the techniques in standard 17.0 with the algebraic techniques of partial fractions and completing the square.
20.0
Compute integrals of trigonometric functions
Students compute the integrals of trigonometric functions by using the techniques noted above.
21.0
Approximate integrals numerically using Simpson's rule and Newton's method
Students understand the algorithms involved in Simpson’s rule and Newton’s method. They use calculators or computers or both to approximate integrals numerically.
22.0
Understand improper integrals as limits of definite integrals
Students understand improper integrals as limits of definite integrals.
23.0
Determine convergence/divergence of sequences and series
Students demonstrate an understanding of the definitions of convergence and divergence of sequences and series of real numbers. By using such tests as the comparison test, ratio test, and alternate series test, they can determine whether a series converges.
24.0
Compute the radius of convergence of power series
Students understand and can compute the radius (interval) of the convergence of power series.
25.0
Differentiate and integrate power series term by term
Students differentiate and integrate the terms of a power series in order to form new series from known ones.
26.0
Calculate Taylor polynomials and Taylor series
Students calculate Taylor polynomials and Taylor series of basic functions, including the remainder term.
27.0
Solve elementary differential equations including growth-and-decay problems
Students know the techniques of solution of selected elementary differential equations and their applications to a wide variety of situations, including growth-and-decay problems.