- Students demonstrate knowledge of both the formal definition and the graphical interpretation of limit of values of functions. This 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 either approaches a number or infinity.
- Students prove and use theorems evaluating the limits of sums, products, quotients, and composition of functions.
- Students verify and estimate limits using graphical calculators.
- Students prove and use special limits such as the limits of (sin(x))/x and (1 - cos(x))/x as x tends to 0.
- Students demonstrate knowledge of both the formal definition and graphical interpretation of continuity of a function.
- Students demonstrate understanding and application of the Intermediate Value Theorem and the Extreme Value Theorem.
- Students demonstrate understanding of the formal definition of the derivative of a function at a point, and the notion of differentiability.
- Students demonstrate understanding of the derivative of a function as the slope of the tangent line to the graph of the function.
- Students demonstrate understanding of the interpretation of the derivative as instantaneous rate of change. Students can use derivatives to solve a variety of problems coming from physics, chemistry, economics, etc, that involve the rate of change of a function.
- Students understand the relation between differentiability and continuity.
- Students derive derivative formulas and use them to find the derivatives of algebraic, trigonometric, inverse trigonometric, exponential, and logarithmic functions.
- Students know the Chain Rule and its proof and applications to the calculation of the derivative of a variety of composite functions.
- Students find the derivatives of parametrically defined functions and use implicit differentiation in a wide variety of problems coming from physics, chemistry, economics, etc.
- Students compute derivatives of higher orders.
- Students know and can apply Rolle's theorem, the Mean Value Theorem, and L'Hopital's rule.
- Students use differentiation to sketch, by hand, graphs of functions. They can identify maxima, minima, inflection points, and intervals where the function is increasing and decreasing.
- Students know Newton's method for approximating the zeros of a function.
- Students use differentiation to solve optimization (maximum - minimum problems) in a variety of pure and applied contexts.
- Students use differentiation to solve related rate problems in a variety of pure and applied contexts.
- Students know the definition of the definite integral using Riemann sums. They use this definition to approximate integrals.
- Students apply the definition of the integral to model problems in physics, economics, etc, obtaining results in terms of integrals.
- Students demonstrate knowledge of and proof of the Fundamental Theorem of Calculus, and use it to interpret integrals as anti-derivatives.
- 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.
- Students compute, by hand, the integrals of a wide variety of functions using techniques of integration such as:
- Substitution
- Integration by parts
- Trigonometric substitution
They can also combine these techniques when appropriate.
- Students know the definitions and properties of inverse trigonometric functions, and their appearance as indefinite integrals.
- Students compute, by hand, the integrals of rational functions by combining the above techniques with the algebraic techniques of partial fractions and completing the square.
- Students compute the integrals of trigonometric functions using the above techniques.
- Students understand the algorithms involved in Simpson's rule and Newton's method. They use calculators and/or computers to approximate integrals numerically.
- Students understand improper integrals as limits of definite integrals.
- Students demonstrate understanding of the definitions of convergence and divergence of sequences and series of real numbers. They can determine whether a series converges using such tests as the comparison test, ratio test, and alternate series test.
- Students understand and can compute the radius (interval) of convergence of power series.
- Students differentiate and integrate the terms of a power series in order to form new series from known ones.
- Students calculate Taylor polynomials and Taylor series of basic functions, including the remainder term.
- 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.
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