Demonstrate knowledge of both the
formal
definition and the graphical interpretation
of limit of values of functions (one-sided, infinite, and infinity, convergence
and divergence).
Demonstrate knowledge of both the formal definition
and the graphical interpretation of continuity of a function.
Demonstrate and apply the Intermediate Value Theorem
for Derivatives and the Extreme Value Theorem.
Demonstrate understanding of the formal definition
of the derivative of a function at a point and the notion of differentiability
(slope of tangent line, instantaneous rate of change, algebraic derivative
shortcuts).
Know and apply the chain rule to calculate the derivative
of a variety of composite functions.
Find the derivatives of
parametrically
defined functions and apply implicit differentiation
to a wide variety of problems.
Students compute higher order derivatives.
Student’s know and apply Rolle’s theorem, the Mean
Value Theorem, and L’Hôpital’s rule.
Use differentiation to sketch by hand the graphs
of functions identifying extrema, points of inflection, and intervals of
increase, decrease, and concavity.
Know Newton’s method for approximating the zeros
of a function.
Use differentiation to solve optimization problems.
Use differentiation to solve related rate problems.
Know the definition of the definite integral by using
Riemann sums and use Riemann sums to approximate definite integrals.
Apply the definition of the integral to model problems
obtaining results in integral form.
Demonstrate knowledge and proof of the fundamental
theorem of calculus and use it to interpret integrals as antiderivatives.
Use definite integrals to solve problems of area,
velocity, acceleration, volume, area of surface,
arc
length, and work.
Compute by hand integrals of functions using substitution,
integration
by parts, and trigonometric
substitution.
Know and use the properties of inverse trigonometric
functions for finding indefinite integrals.
Compute by hand integrals using the techniques of
partial
fractions and
completing
the square.
Compute the integrals of trigonometric functions.
Understand and apply the algorithms of Simpson’s
rule and Newton’s method.
Use improper integrals as limits of definite integrals.
Demonstrate understanding of the definitions of convergence
and divergence of sequences and series of real numbers and apply comparison
test, ration test, and alternative series test to determine convergence
of a series.
Compute the radius (interval) of the convergence
of a power series.
Differentiate and integrate the terms of a power
series in order to form new series from know series.
Calculate Taylor polynomials and Taylor series of
basic functions.
Know how to solve selected elementary differential
equations and apply them to a variety of problems.