CALCULUS

Students will:

•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. Know the definition of convergence and divergence of a function as the domain variable approaches either a number or infinity:

a) Prove and use theorems evaluating the limits of sums, products, quotients, and composition of functions

b) Use graphical calculators to verify and estimate limits

c) Prove and use special limits, such as the limits of (sin(x))/x and (1-cos(x))/x as x tends to 0

•Demonstrate knowledge of both the formal definition and the graphical interpretation of continuity of a function

•Demonstrate an understanding and the application of the intermediate value theorem and the extreme value theorem

•Demonstrate an understanding of the formal definition of the derivative of a function at a point and the notion of differentiability:

a) Demonstrate an understanding of the derivative of a function as the slope of the tangent line to the graph of the function

b) Demonstrate an understanding of the interpretation of the derivative as an instantaneous rate of change. Use derivatives to solve a variety of problems from physics, chemistry, economics, and so forth that involve the rate of change of a function

c) Understand the relation between differentiability and continuity

d) Derive derivative formulas and use them to find the derivatives of algebraic, trigonometric, inverse trigonometric, exponential, and logarithmic functions

•Know the chain rule and its proof and applications to the calculation of the derivative of a variety of composite functions

•Find the derivatives of parametrically defined functions and use implicit differentiation in a wide variety of problems in physics, chemistry, economics, and so forth

•Compute derivatives of higher orders

•Know and can apply Rolle's theorem, the mean value theorem, and L´Hôpital's rule

•Use differentiation to sketch, by hand, graphs of functions. Identify maxima, minima, inflection points, and intervals in which the function is increasing and decreasing

•Know Newton's method for approximating the zeros of a function

•Use differentiation to solve optimization (maximum-minimum problems) in a variety of pure and applied contexts

•Use differentiation to solve related rate problems in a variety of pure and applied contexts

•Know the definition of the definite integral by using Riemann sums. Use this definition to approximate integrals

•Apply the definition of the integral to model problems in physics, economics, and so forth, obtaining results in terms of integrals

•Demonstrate knowledge and proof of the fundamental theorem of calculus and use it to interpret integrals as antiderivatives

•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

•*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. Combine these techniques when appropriate

•Know the definitions and properties of inverse trigonometric functions and the expression of these functions as indefinite integrals

•Compute, by hand, the integrals of rational functions by combining the techniques in standard (*) above with the algebraic techniques of partial fractions and completing the square

•Compute the integrals of trigonometric functions by using the techniques noted above

•Understand the algorithms involved in Simpson's rule and Newton's method. Use calculators or computers or both to approximate integrals numerically

•Understand improper integrals as limits of definite integrals

•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

•Differentiate and integrate the terms of a power series in order to form new series from known ones

•Calculate Taylor polynomials and Taylor series of basic functions, including the remainder term

•Know the techniques of solution of selected elementary differential equations and their applications to a wide variety of situations, including growth-and-decay problems

•Understand and compute the radius (interval) of the convergence of power series