Math 180 Course Objectives

The objectives most relevant to Part A of the final 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
• 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
• determine the domain, range, and rule for the inverse of a given function
• build a function that models a given real-world situation
• state a precise definition of limit equivalent to that given in the text
• use the precise definition of limit to prove a very simple limit statement
• understand and articulate what a given limit statement tells us about the relevant function or functions
• understand and articulate the similarities and differences among limit, one-sided limit, and limit at infinity
• conjecture a given limit using an input/output table or a graph
• evaluate a simple given limit using informal reasoning
• evaluate a given limit problem using formal reasoning and tools such as algebraic manipulation, change of variable, relating to known limits, and L'Hopital's rule
• use the Sandwich Theorem to prove a given limit statement
• analyze continuity for a given function
• give a geometric interpretation of a difference quotient
• compute a derivative starting from the definition
• sketch a qualitatively correct plot of the graph of a derivative given a plot of the graph of a function
• state the derivative of each basic function (constant, power, sin, cos, exp, ln)
• state each rule for computing derivatives (constant multiple, sum, difference, product, quotient, chain)
• use results for basic functions and rules for combinations of functions to differentiate a given function
• compute higher derivatives of a given function
• state and use an appropriate interpretation of derivative (as rate of change or slope of tangent line)
• determine the equation of a tangent line for a given function at a given point
• compute and use 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
• compute and use the linear approximation for a given function at a given point
• understand the language and notation of differentials
• use differentials to relate the error (or percentage error) in a measured quantity to the error (or percentage error) in a computed quantity
• state conditions under which a function on a stated domain is guaranteed to have a global minimum and a global maximum
• find the global minimum and global maximum of a given continuous function on a closed interval
• find and classify (as local minimum, local maximum, or neither) all critical points for a given function on a given domain
• state and understand the hypotheses and conclusion of the Mean Value Theorem
• apply the Mean Value Theorem to relate an average rate of change to an instantaneous rate of change
• apply the Mean Value Theorem to justify conclusions about the number of zeros for a given function
• use limits when appropriate to understand relevant aspects of the graph of a 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 to classify a critical input as a local minimum or local maximum
• set up and solve applied problems that involve minimizing or maximizing a specified quantity
• find antiderivatives of a given function using knowledge of derivative results and rules
• use differentiation to go from position to velocity to acceleration and use antidifferentiation to go from acceleration to velocity to position
• 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 (as limit of a Riemann sum)
• state and understand the hypotheses and conclusion of the Second Fundamental Theorem of Calculus
• use the Second Fundamental Theorem of Calculus to compute the value of a given definite integral
• set up and evaluate an appropriate definite integral to compute the area of a given region
• set up and evaluate an appropriate definite integral to compute the accumulated change for a given rate of change and interval