Math 121 Final Exam Objectives

The objectives most relevant to Part A of the exam are highlighted in green

For the final exam, 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 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
• 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 elemenatary 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
• understand how tangent lines are used in the Newton-Raphson method for approximating roots and be able to implement the Newton-Raphson method for a given function and a given initial value
• 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
• state and understand the hypotheses and conclusion of the Second Fundamental Theorem of Calculus
• use the Second Fundamental Theorem of Calculus to compute the derivative of an accumulation function