Math 121 Course Objectives
The objectives most relevant to Part A of the exam are highlighted in
green.
In successfully completing this course, a well-prepared student should be
able to
- understand and use algebra, trigonometry, and geometry with
mastery at the high school level
- understand and use basic facts about the real numbers as composed
of the rational and irrational numbers
- state a definition, equivalent to that used
in the text or class, for each relevant term (for example, function,
limit, continuity, derivative, antiderivative, definite
integral)
- determine the domain, range, and rule for a given function
- build a function that models a given real-world situation
- understand and articulate what a given limit statement tells us
about the relevant function or functions
- evaluate a simple given limit problem using
informal reasoning
- evaluate a given limit problem using formal
reasoning and tools such as algebraic manipulation, change of variable,
and L'Hopital's rule
- use the precise definition of limit to prove a given simple limit
statement
- analyze continutity for a given function
- give a geometric interpretation of a difference quotient
- analyze the limit of a difference quotient for simple
functions (both for a specific input and for a variable input)
- sketch a qualitatively correct plot of the graph of a
derivative given a plot of the graph of a function
- state each rule for computing derivatives (constant multiple,
sum, product, quotient, chain)
- state the derivative of each basic function (power, sin, cos, exp,
ln)
- use the rules and derivatives of basic functions to
differentiate any elementary function (i.e., combinations of the
basic functions)
- compute higher derivatives of a given
function
- determine the equation of a tangent line for
a given function at a given point
- state and use an appropriate interpretation
of derivative (as slope of tangent line or rate of change)
- compute a derivative given an implicit relation involving two
variables
- use implicit differentiation to find the derivative of an
inverse function knowing the derivative of the original function
- solve applied problems that involve relating two or more rates
of change
- understand the language and notation of the linear (or
tangent line) approximation and differentials
- use the linear approximation or differentials to relate the
error (or percentage error) in a measured quantity to the error (or
percentage error) in a computed quantity
- state and understand the hypotheses and conclusion of the Extreme Value
Theorem
- find and classify (as local minimizer, local
maximizer, or neither) all critical points for a given function on a
given domain
- find the global minimum and global maximum for a given function
on a given domain
- solve applied problems that involve optimizing a geometric
quantity (e.g., maximize a volume or minimize an area)
- find and analyze any vertical asymptotes of a given function
- analyze any horizontal asymptotes of a given function
- find all intervals of inputs for which a given function is
positive and all intervals of inputs for which the function is
negative
- find all intervals of inputs for which a given function is
increasing and all intervals of inputs for which the function is
decreasing
- find all intervals of inputs for which a given function is
concave up and all intervals of inputs for which the function is
concave down
- use calculus techniques to plot the graph of a given function
in a window that shows all essential features of the graph
- use the Second Derivative Theorem to classify a critical input
as a local minimum or local maximum
- state and understand the hypotheses and conclusion of the Mean Value
Theorem
- apply the Mean Value Theorem to prove a given mathematical
statement
- apply the Mean Value Theorem to analyze a given real-world
situation
- read and use summation notation
- approximate the value of a given definite
integral
- compute the exact value of a given simple definite integral
using the definition of definite integral
- give "easy-to-find" lower and upper bounds for the value of a
given definite integral
- state and use an appropriate interpretation
of definite integral (as area or accumulation)
- find antiderivatives of a given function
using knowledge of derivative rules and derivatives of specific
functions (power, trigonometric and inverse trigonometric, exponential
and logarithmic)
- state and understand the hypotheses and conclusion of the First
Fundamental Theorem of Calculus
- use the First Fundamental Theorem of Calculus to compute the
value of a given definite integral