# Expected Learning Outcomes

## Math1010, Intermediate Algebra

Upon successful completion of this course, a student should be able to:- Work with functions presented in tables, graphs, with algebraic expression, or in words. Determine if relations presented in any of these formats are functions. The functions covered in this course are linear, exponential, logarithmic, quadratic, square root, and power.
- Find the domain, x-intercepts, y-intercepts, output given the input, and input given the output for all functions presented in tables, graphs or with algebra. Find the range of functions presented graphically.
- Decide if a given function is linear, quadratic, exponential, or none of the above for functions presented algebraically, graphically, or in tables.
- Know the shapes of the graphs of all the above functions. Be able to recognize when a sufficient portion of the graph is sketched in order to indicate the graph shape.
- Graphing Techniques Vary by Function:
- Graph linear functions using either two points or one point and a slope.
- Graph quadratic functions using both these approaches--
- the x-intercept(s) and the vertex (or line of symmetry).
- the vertex and an efficient table (be able to use symmetry and possibly the y-intercept).

- Graph logarithmic functions with no transformations by switching the x- and y-values of the corresponding exponential graph.
- Graph exponential and square root functions, when given in transformation form, using efficient tables.
- Graph power functions with no transformations using tables.

- Solve linear, quadratic, exponential, logarithmic and square root equations. This includes solving quadratic equations using the zero-product property, completing the square, and the quadratic formula.
- Solve linear inequalities and give answers in inequality, interval and graphical (number line) format.
- Determine the slope of a line; find the equations of lines given information about them. Decide if lines are parallel, perpendicular, or neither.
- Solve 2x2 systems of linear equations and functions using graphical and substitution methods.
- Be able to factor quadratic expressions or decide if they cannot be factored; complete the square.
- Perform composition of functions presented in tables, graphs or with algebra.
- Decide if a function presented with a table or graph is invertible and give the inverse in the same format.
- Find the algebraic inverse of a linear function presented algebraically.
- Make sense of exponent rules, negative exponents, and rational exponents. Use exponent rules to simplify exponential expressions.
- Construct algebraic models to describe real-life situations. Be
able to decide what type of model fits the situation best:
- Use linear functions to model constant rates of growth.
- Use exponential functions to model constatn percent change.
- Use quadratic functions to model constant acceleration.

- Analyze and use linear, exponential, and quadratic models to answer questions about the situations they represent. In particular, relate graphical features (like the x- and y-intercepts and the vertex of a parabola) to specific aspects of the situation being modeled. For quadratics, be able to rewrite the function appropriately in order to find the information desired.

## Math1030, Introduction to Quantitative Reasoning

Upon successful completion of this course, a student should be able to:- Use Venn diagrams to examine relationships between sets and the validity of simple deductive arguments.
- Use an appropriate sentence to describe both the absolute and percent change in a given quantity and interpret such statements about the change.
- Use simple and compound units, making conversions when necessary, and develop accurate comparisons between units.
- Evaluate the impact of compound interest on simple financial decisions.
- Use the savings plan and loan formulas to calculate the payment amount into the savings plan when a certain financial goal needs to be achieved, to calculate the mortgage payment or interest paid over the life of the loan and discuss whether those results are realistic (or not), compare several loans with different interest rates in order to financial decisions.
- Compare and illustrate the features of linear and exponential growth using practical examples.
- Determine simple areas, volumes, and explain the differential effect of scaling on perimeter, area, volume as well as some of the practical implications of scaling.

## Math1040, Introduction to Statistics and Probability

Upon successful completion of this course, a student should be able to:- Recognize different ways of collecting data and decide what method would be the best for a given situation.
- Make a difference between various sampling techniques and decide what sampling technique would work the best in the given situation.
- Use different tables and graphs to organize and analyze data.
- Calculate the mean, median, mode, range, quartiles, interquartile range, outlier(s), find percentile that corresponds to a value and interpret the results in variety of ways.
- Find the z-score (the standard score) and compare z-scores from different data sets.
- Find the mean, variance and standard deviation; interpret standard deviation using Empirical Rule (68-95-99.7 rule) for the bell-shaped distribution; interpret standard deviation in connection with a distribution that is not bell-shaped or it is unknown using Chebychev’s Theorem; find the standard deviation for grouped data (using classes and midpoints).
- Determine the probability of an event using the Fundamental Counting Principle, conditional probability, the multiplication rule, and the addition rule.
- Create and use the probability distribution.
- Recognize a binomial experiment and calculate the binomial distribution using the Binomial Probability Formula.
- Recognize normal (bell-shaped) distribution and standard normal distribution; calculate the areas/probabilities using the standard normal distribution table.
- Explain the meaning of different values of the correlation coefficient and relate the concept to the strength/weakness of linear relationship between two variables when looking at different scatter plots.
- Find the equation of the regression line (line of best fit) and predict values using the equation of the regression line.

## Math1050, College Algebra

Upon successful completion of this course, a student should be able to:- Sketch the graph of basic polynomials (second and third order), rational, radical, exponential, logarithmic, and piecewise functions with or without transformations. Be able to identify important points such as x and y intercepts, maximum or minimum values; domain and range; and any symmetry.
- For rational functions, identify x and y intercepts, vertical, horizontal and oblique asymptotes (end behavior), and domain. Use information to sketch graphs of functions.
- For polynomial functions, identify all zeros (real and complex), factors, x and y intercepts, end behavior and where the function is positive or negative. Use information to sketch graphs.
- Understand the connections between graphic, algebraic, and verbal descriptions of functions.
- Given the graph of a function, be able to identify the domain, range, any asymptotes and/or symmetry, x and y intercepts, as well as find a rule for the function if it is obtained from a standard function through transformations.
- Define i as the square root of -1 and know the complex arithmetic necessary for solving quadratic equations with complex roots.
- Solve absolute value, linear, polynomial, rational, radical, exponential and logarithmic equations and inequalities.
- Find the inverse of a function algebraiclly and graphically.
- Perform composition of functions and operations on functions.
- Understand sequences and be able to differentiate between geometric, arithmetic, and others such as Fibonacci-type sequences giving direct formulas where available.
- Understand series notation and know how to compute sums of finite or infinite arithmetic or geometric series.
- Solve systems of equations (3x3 linear) and non-linear equations in two variables.
- Make sense of algebraic expressions and explain relationship among algebraic quantities including quadratic, exponential, logarithmic, rational, radical, and polynomial expressions, equations and functions.
- Represent and interpret "real world" situations using quadratic, exponential, logarithmic, rational, radical and polynomial expressions, equations, and functions.

## Math1060, Trigonometry

Upon successful completion of this course, a student should be able to:- Understand trigonometric function definitions in the context of the right triangles and on the unit circle.
- Graph basic trigonometric functions and those with basic transformations. Be able to write an equation given a graph. Identify amplitude, periods, phase shifts from graphic and algebraic representations of functions.
- Solve applications problems using principles in trigonometry.
- Represent and interpret “real world” contexts situations using radian trigonometric functions.
- Use trigonometric inverses correctly, understanding the domain/range restrictions.
- Verify trigonometric identities, using proper logic and use trigonometric identities to evaluate expressions.
- Solve trigonometric equations.
- Solve for all measurements in any triangle, using the Pythagorean Theorem, trigonometric functions, the Law of Sines, and Law of Cosines in a variety of contexts and applications.
- Be able to convert to and from rectangular and trigonometric-form coordinates (polar coordinates).
- Graph complex numbers in a plane, perform operations on such numbers and use DeMoivre’s theorem to find roots and powers of complex numbers.
- Understand geometry and arithmetic operations with vectors and use vectors in application problems.
- Use parametric equations in application problems and be able to convert between parametric and non-parametric representation of functions.
- Understand and explain arithmetic with complex numbers using trigonometry.
- Recognize the formulas for parabolas, hyperbolas and ellipses (including circles). Be able to manipulate these basic conics to find foci, any asymptotes, and important points and to graph these conics. Use conics in real world context situation.

## Math1070, Introduction to Statistical Inference

Upon successful completion of this course, a student should be able to:- Be able to summarize the data using charts, graphs, histograms, and to calculate basic descriptive statistics like the mean, standard deviation, median and quartiles.
- Work with the normal distribution and use table to find probabilities.
- Understand the difference between correlation and causation.
- Perform regression analysis and compute correlation.
- Understand the Central Limit Theorem and the normality assumption.
- Understand the basics of tests of significance and confidence intervals including z-tests, t-tests, proportion tests, Chi-square tests, ANOVA and non-parametric tests.
- Be able to perform simple statistical analysis of large data sets using spreadsheets (throughout the whole course).

## Math1080, Precalculus

Upon successful completion of this course, a student should be able to:- Solve absolute value linear inequalities and polynomial/rational inequalities.
- Graph polynomial, rational, radical, exponential, logarithmic, trigonometric, and piecewise functions, using transformations as well as information about the domain, asymptotes, symmetry, and/or intercepts.
- Given the graph of a function, be able to identify the domain, range, asymptotes, symmetry and zeros, as well as find the rule for the function if it is obtained from a standard function through transformations.
- Find the inverse of a function algebraically and graphically.
- Understand and be able to find the domain of functions. Perform composition of functions and operations on functions.
- Find the difference quotient of a function and use this to find lines related to curves of functions.
- Understand the connections between graphic, algebraic, and verbal descriptions of functions, in particular polynomials.
- Find all zeros, including complex, of a polynomial function.
- Solve exponential, logarithmic, rational, radical, trigonometric, and polynomial equations.
- Use the Binomial Theorem and Pascal's Triangle to expand a binomial expression.
- Solve systems of linear equations with matrices, using Gauss-Jordan elimination and inverse matrices.
- Perform matrix arithmetic and compute inverse matrices.
- Recognize the formulas for and graph parabolas, hyperbolas and ellipses (including circles).
- Understand trigonometric function definitions in the context of the right triangle and on the unit circle.
- Be able to convert to and from rectangular and trigonometric-form coordinates (polar coordinates not explicitly covered).
- Use trigonometric inverses correctly, understanding the domain/range restrictions.
- Verify trigonometric identities, using proper logic and use trigonometric identities to evaluate expressions.
- Solve for all measurements in any triangle, using the Pythagorean Theorem, trigonometric functions of angles, the Law of Sines and Law of Cosines, along with applications.
- Graph complex numbers in a plane, perform operations on such numbers and use DeMoivre’s theorem to find roots and powers of complex numbers.
- Understand sequences and be able to differentiate between geometric, arithmetic and Fibonacci-type sequences, giving direct formulas where available.
- Understand series notation and know how to compute sum of finite arithmetic and geometric series.

## Math1090, College Algebra for Business and Social Sciences

Upon successful completion of this course, a student should be able to:- Graph and analyze quadratic, exponential and logarithmic functions; solve quadratic, exponential and logarithmic equations.
- Understand what a mathematical function is and know how to use linear, quadratic, logarithmic and exponential functions to model real world examples.
- Know how to solve a system of linear or quadratic equations that arise in business applications.
- Find solutions to linear programming problems, to maximize a function over a geometric region.
- Perform simple matrix algebra computations.
- Use matrices to solve systems of linear equations.
- Understand what an inverse function is and be able to find the inverse function, when it exists.
- Distinguish between simple and compound interest situations.
- Calculate future and present value of annuities, and know when to use which formula for the life application.
- Compute an amortization schedule and loan payments, such as automobile or mortgage payments.

## Math1100, Quantitative Analysis

Upon successful completion of this course, a student should be able to:- Have a basic conceptual understanding of limits.
- Know how to differentiate and integrate polynomial, rational, logarithmic, and exponential functions.
- Use derivatives to gather information about the shape of the curve and use that information to graph the curve y = f(x), for polynomial, logarithmic, exponential and simple rational functions.
- Understand how to use differentiation to optimize functions for business applications, such as maximizing profit examples.
- Use integration to find area under curves and for business examples such as average value.
- Take partial derivatives of basic functions of two variables.

## Math1170, Calculus for Biologists I

Upon successful completion of this course, a student should be able to:- Translate from the language of modern wet labs to mathematical problems and translate the answer back into biological context.
- Derive and analyze discrete-time dynamical systems, including finding equilibria, cobwebbing, and analyzing stability.
- Understand the mathematical and scientific meaning of limits and continuous functions.
- Compute the derivatives of functions built from polynomial, exponential and trigonometric components.
- Use derivatives to graph functions and understand their leading behavior at zero and infinity.
- Apply differentiation to optimization and related rates problems.
- Approximate functions with the Taylor series and apply this to Newton's method for finding roots.
- Understand the Fundamental Theorem of Calculus and its relationship to differential equations.
- Compute definite and indefinite integrals of polynomials and special functions, and apply them to finding areas and volumes.
- Use modern software to practice hands-on applications of acquired mathematical concepts in computational projects.

## Math1180, Calculus for Biologists II

Upon successful completion of this course, a student should be able to:- Derive and graphically analyze one-dimensional autonomous differential equations, and find their equilibria and stability.
- Use separation of variables to solve one-dimensional differential equations.
- Use the phase plane to analyze two-dimensional systems of autonomous differential equations.
- Understand and apply the axioms of probability, including the definition and interpretation of conditional probability and independence.
- Graphically display probability distributions and probability density functions of random variables, and compute the mean, median, variance and other descriptive statistics.
- Use joint distributions to describe relationships between random variables and summarize them with the covariance and correlation.
- Understand the assumptions behind the random variables that follow the binomial, geometric, Poisson, exponential, and normal distributions, and apply them to compute probabilities of events.
- Know how to use maximum likelihood to estimate parameters and compare models.
- Understand hypothesis testing, p-values and confidence limits, and correctly interpret results in scientific papers.
- Be familiar with the fundamental statistical methods of the t-test, chi-square test, and linear regression.
- Use modern software to practice hands-on applications of acquired mathematical concepts in computational projects.
- Use modern statistical software to analyze and display scientific data.

## Math1210, Calculus I

Upon successful completion of this course, a student should be able to:- Take limits of algebraic and trigonometric expressions of the form 0/0 (that simplify), non-zero number over 0, including limits that go to (positive or negative) infinity, limits that don't exist and limits that are finite.
- Use the limit definitions of derivative and definite integral for polynomial, rational and some trigonometric functions; understand definition of continuity.
- Differentiate all polynomial, rational, radical, and trigonometric functions and compositions of those functions; perform implicit differentiation and compute higher order derivatives.
- Use differentiation to find stationary, singular and inflection points, as well as domain and limit information to determine vertical and horizontal asymptotes, and then use all of that information to sketch the graph of a curve, y = f(x).
- Apply differentiation to optimization and related rates problems.
- Compute indefinite and definite integrals, using the power rule and basic u-substitution and the Fundamental Theorems of Calculus.
- Apply the definite integral to compute area between two curves, volumes of solids of revolutions, arc length, surface area for surfaces of revolution and center of mass.

## Math1220, Calculus II

Upon successful completion of this course, a student should be able to:- Compute derivatives and integrals for exponential, logarithmic, hyperbolic functions, and inverse trigonometric functions.
- Integrate integrable functions using integration by parts, u-substitution, trigonometric substitutions, rationalizing substitutions, partial fraction decomposition, and trigonometric identities. This includes knowing which techniques to apply to a given integral.
- Use L'Hopital's Rule to calculate indeterminate-type limits and also know what limits are the non-indeterminate forms and how to compute those limits.
- Compute improper integrals.
- Understand the difference between an infinite sequence and infinite series and determine if a sequence converges or diverges.
- Determine whether or not an infinite series of numbers converges or diverges using a variety of tests.
- Understand what it means for a Power Series to converge or diverge and be able to find the Taylor Series for a given function.
- Differentiate and integrate functions in polar coordinates.

## Math2210, Calculus III

Upon successful completion of this course, a student should be able to:- Compute dot and cross products of two vectors, projection of one vector onto another vector.
- Convert between cylindrical, rectangular and spherical coordinates. Understand when it's prudent to switch to one coordinate system over another in computing an integral.
- Determine the equation of a plane in 3-d, including a tangent plane to a surface in 3-d.
- Find the parametric equations of a line in 3-d.
- Perform calculus operations on functions of several variables, including limits, partial derivatives, directional derivatives, and gradients; understand what the gradient means geometrically.
- Find maxima and minima of a function of two variables; use Lagrange Multipliers for constrained optimization problems.
- Understand divergence and curl of a vector field.
- Compute double and triple integrals in rectangular, spherical and cylindrical coordinates; proper use of double or triple integrals for finding surface area or volume of a 3-d region.
- Compute line and surface integrals.
- Determine if a vector field is conservative and if so, find the corresponding potential function.
- Use and understand when to apply Green's Theorem, Gauss' Divergence Theorem and Stokes Theorem.

## Math1250, Calculus for AP Students I

Upon successful completion of this course, a student should be able to:- Compute derivatives of polynomial, rational and power functions by definition.
- Differentiate all elementary functions; perform implicit differentiation and compute higher order derivatives. (By elementary functions we mean rational, power, exponential, logarithmic, trigonometric and inverse trigonometric functions, as well as functions obtained by them by addition, multiplication, and composition -- in other words: all functions which can de defined by formulas.)
- Use differentiation to find stationary, singular and inflection points, as well as domain and limit information to determine vertical and horizontal asymptotes, and then use all of that information to sketch the graph of a curve, y = f(x).
- Apply differentiation to optimization and related rates problems.
- Understand the concept of a parametric curve, in the context of related rates problems.
- Compute, whenever possible, indefinite and definite integrals of elementary functions, using formulas, integration by substitution and by parts and partial fraction techniques.
- Understand Fundamental Theorems of Calculus.
- Understand, and be able to prove trigonometric identities.
- Apply the definite integral to compute area between two curves, volumes of solids of revolutions, arc length, surface area for surfaces of revolution and center of mass.
- Use L'Hopital's Rule to calculate indeterminate-type limits and also know what limits are the non-indeterminate forms and how to compute those limits.
- Compute improper integrals.
- Compute using complex numbers; understand the connection between trigonometric and exponential functions.
- Understand the difference between an infinite sequence and infinite series and determine if a sequence converges or diverges.
- Determine whether or not an infinite series of numbers converges or diverges using a variety of tests.
- Understand what it means for a power series to converge or diverge and be able to find the Taylor series for a given function.
- Use Taylor's theorem to approximate functions by polynomials.
- Differentiate and integrate functions in polar coordinates.
- Be able to understand proofs of theorems demonstrated in class.
- Solve complex problems involving ideas from all of the above.

## Math1260, Calculus for AP Students II

Upon successful completion of this course, a student should be able to:- Have a thorough understanding of the following topics in 2 and 3 dimensional linear algebra:
- Vectors in plane and space
- Dot product, and projection of vectors
- Matrices and Determinants of size 2 and 3;
- Cross Products
- Equations of lines in planes in 3 variables; Distance formulas

- Differentiate and integrate functions of 2 and 3 variables.
- Find equation of tangent planes to surfaces.
- Understand the geometric meaning of the gradient.
- Find maxima and minima of a function of two variables; use Lagrange Multipliers for constrained optimization problems.
- Thoroughly understand parametric curves, velocity, acceleration and curvature.
- Have a good understanding of the following topics in Vector Analysis:
- Line and surface integrals
- Curl and Divergence
- Green, Stokes and Gauss theorems.

- Apply those theorems to concrete problems in Physics.
- Be able to understand proofs of theorems demonstrated in class.
- Solve complex problems involving ideas from all of the above.

## Math1310, Engineering Calculus I

Upon successful completion of this course, a student should be able to:- Understand how to transform functions into other functions through x- and y-translations and rescaling, re-parameterizations, and function composition.
- Know the properties of special classes of functions including logarithms, exponential functions, polynomials, and rational functions and know how to obtain function inverses when they exist.
- Master the concept of a limiting value of a function f(x) = y when x approaches a value c, know when limits exist, utilize limit laws, know how the property of continuity of a function at c relates to its limiting value, know how asymptotic behavior can be described by limits, and how limiting values can be specified even when the function is not defined at c.
- Understand how to use limits to compute the derivative of a function that describes rate of change of a function.
- Utilize derivatives to model how two related quantities change with respect to each other, including motion of objects in terms of velocity and acceleration.
- Know the methods of differentiation for different classes of functions including exponential, logarithmic, trigonometric, inverse trigonometric, power functions, and compositions, sums, products, quotients of functions, as well as knowing the how to differentiate functions that are only implicitly defined by an equation.
- Utilize the derivative in applied contexts, including function approximation, and how the average slope of a function relates to the derivative through the mean value theorem.
- Obtain the derivative of one quantity by knowing the derivative of the other, if two quantities are related by an equation.
- Use linear approximations to perform numerical/algorithmic equation solving via Newton's method.
- Utilize the derivative to find maximum, minimum, or otherwise "optimal" input valuess for equations important in science, business and engineering.
- Understand the definition of the integral of a function as the limiting value of an increasingly large average of function values.
- Relate the integral to the area under the function's curve, know how to approximate the integral by a finite sum, and how to integrate over infinite-length domains.
- Master specific integration techniques, including substitution, integration by parts, and partial fractions.
- Understand the key concept underlying definite integration, that it computes the net accumulation of a quantity through summation of the change in the quantity amount per unit of time or space, over a specified interval of time or space.
- Read and understand problem descriptions, then be able to formulate equations modeling the problem usually by applying geometric or physical principles.
- Select the appropriate calculus operations to apply to a given problem, execute them accurately, and interpret the results using numerical and graphical computational aids.
- Gain experience with problem solving in groups, be able to communicate effectively about problem objectives and the use of solving methods with peers, and solve problems in a team fashion. Students will also learn how to articulate questions effectively with both the instructor and TA, and be able to effectively communicate problem solutions.

## Math1320, Engineering Calculus II

Upon successful completion of this course, a student should be able to:- Utilize methods of integration to compute volumes of objects with circular-shaped aspects, and compute lengths of curves. These applications introduce a higher-level concept of integration, involving the summation of small volume segments dV or small length segments ds, which are computed by performing an appropriate parameterization to a real-number-line integral in terms of dx.
- Use integration to compute problems important in physics and engineering.
- Know how to compute an average value of a function using the mean value theorem for integrals, the center of mass for objects, and the computation of energy as a force integrated over a distance.
- Utilize physical laws to formulate differential equations that solve for the motion of masses by forces of gravitation, friction, electrostatics, to name a few.
- Become familiar with the phenomenon of exponential growth and decay in science and engineering contexts.
- Become skilled in computations and applications of infinite sequences and sums.
- Be familiar with the properties of infinite sums to either converge to a finite value or diverge to an infinite value, and learn about methods to determine convergence.
- Represent functions as a Taylor series and use Taylor's Theorem to approximate functions and estimate error from using finitely many terms of the Taylor series.
- Become familiar with 2- and 3-dimensional coordinate systems, vectors and vector operations including the dot and cross products, equations of 3-d lines, planes and other surfaces.
- Represent motion of objects in 3-d using vector functions; represent velocity and acceleration using vector projections into tangential and centripetal coordinates of acceleration, and characterize curves in space by computing arc length and curvature.
- Characterize aspects of surfaces and volumes for functions in 3-d, using partial derivatives and the gradient vector.
- Describe approximating tangent planes to points on surfaces using partial derivatives.
- Compute derivatives of multi-dimensional function compositions using multi-dimensional versions of the chain rule.
- Use tools of integration of multivariate functions over areas and volumes and use iterated multiple integration.
- Use multi-dimensional change-of-variables technique to transform coordinates over which integration proceeds by utilizing the Jacobian. Specifically, sutdents will learn how to transform between an integral over an area or volume in Cartesian coordinates to cylindrical or spherical coordinates.
- Read and understand problem descriptions, then be able to formulate equations modeling the problem usually by applying geometric or physical principles.
- Select the appropriate calculus operations to apply to a given problem, execute them accurately, and interpret the results using numerical and graphical computational aids.
- Gain experience with problem solving in groups, be able to communicate effectively about problem objectives and the use of solving methods with peers, and solve problems in a team fashion. Students will also learn how to articulate questions effectively with both the instructor and TA, and be able to effectively communicate problem solutions.

## Math1311, Accelerated Engineering Calculus I

Upon successful completion of this course, a student should be able to:- Understand how to transform functions into other functions through x- and y-translations and rescaling, re-parameterizations, and function composition.
- Know the properties of special classes of functions including logarithms, exponential functions, polynomials, and rational functions and know how to obtain function inverses when they exist.
- Master the concept of a limiting value of a function f(x) = y when x approaches a value c, know when limits exist, utilize limit laws, know how the property of continuity of a function at c relates to its limiting value, know how asymptotic behavior can be described by limits, and how limiting values can be specified even when the function is not defined at c.
- Understand how to use limits to compute the derivative of a function that describes rate of change of a function.
- Utilize derivatives to model how two related quantities change with respect to each other, including motion of objects in terms of velocity and acceleration.
- Know the methods of differentiation for different classes of functions including exponential, logarithmic, trigonometric, inverse trigonometric, power functions, and compositions, sums, products, quotients of functions, as well as knowing the how to differentiate functions that are only implicitly defined by an equation.
- Utilize the derivative in applied contexts, including function approximation, and how the average slope of a function relates to the derivative through the mean value theorem.
- Obtain the derivative of one quantity by knowing the derivative of the other, if two quantities are related by an equation.
- Use linear approximations to perform numerical/algorithmic equation solving via Newton's method.
- Utilize the derivative to find maximum, minimum, or otherwise "optimal" input valuess for equations important in science, business and engineering.
- Understand the definition of the integral of a function as the limiting value of an increasingly large average of function values.
- Relate the integral to the area under the function's curve, know how to approximate the integral by a finite sum, and how to integrate over infinite-length domains.
- Master specific integration techniques, including substitution, integration by parts, and partial fractions.
- Understand the key concept underlying definite integration, that it computes the net accumulation of a quantity through summation of the change in the quantity amount per unit of time or space, over a specified interval of time or space.
- Read and understand problem descriptions, then be able to formulate equations modeling the problem usually by applying geometric or physical principles.
- Select the appropriate calculus operations to apply to a given problem, execute them accurately, and interpret the results using numerical and graphical computational aids.
- Gain experience with problem solving in groups, be able to communicate effectively about problem objectives and the use of solving methods with peers, and solve problems in a team fashion. Students will also learn how to articulate questions effectively with both the instructor and TA, and be able to effectively communicate problem solutions.
- Utilize methods of integration to compute volumes of objects with circular-shaped aspects, and compute lengths of curves. These applications introduce a higher-level concept of integration, involving the summation of small volume segments dV or small length segments ds, which are computed by performing an appropriate parameterization to a real-number-line integral in terms of dx.

## Math1321, Accelerated Engineering Calculus II

Upon successful completion of this course, a student should be able to:- Use integration to compute problems important in physics and engineering.
- Know how to compute an average value of a function using the mean value theorem for integrals, the center of mass for objects, and the computation of energy as a force integrated over a distance.
- Utilize physical laws to formulate differential equations that solve for the motion of masses by forces of gravitation, friction, electrostatics, to name a few.
- Become familiar with the phenomenon of exponential growth and decay in science and engineering contexts.
- Become skilled in computations and applications of infinite sequences and sums.
- Be familiar with the properties of infinite sums to either converge to a finite value or diverge to an infinite value, and learn about methods to determine convergence.
- Represent functions as a Taylor series and use Taylor's Theorem to approximate functions and estimate error from using finitely many terms of the Taylor series.
- Become familiar with 2- and 3-dimensional coordinate systems, vectors and vector operations including the dot and cross products, equations of 3-d lines, planes and other surfaces.
- Represent motion of objects in 3-d using vector functions; represent velocity and acceleration using vector projections into tangential and centripetal coordinates of acceleration, and characterize curves in space by computing arc length and curvature.
- Characterize aspects of surfaces and volumes for functions in 3-d, using partial derivatives and the gradient vector.
- Describe approximating tangent planes to points on surfaces using partial derivatives.
- Compute derivatives of multi-dimensional function compositions using multi-dimensional versions of the chain rule.
- Use tools of integration of multivariate functions over areas and volumes and use iterated multiple integration.
- Use multi-dimensional change-of-variables technique to transform coordinates over which integration proceeds by utilizing the Jacobian. Specifically, students will learn how to transform between an integral over an area or volume in Cartesian coordinates to cylindrical or spherical coordinates.
- Become familiar with vector functions that define vector fields in the plane and 3D space, particularly conservative vector fields, represented by the gradient of a scalar function, which are important for gravitation and electrostatics. When masses or charged particles are pushed through fields such as these, along curved paths, the work done can be computed as a line integral.
- Learn how the fundamental theorem for line integrals for conservative vector fields reduces the integral to valuation of the potential at the endpoints of the path.
- Leanr the fundamental vector calculus integral theorems of Green, Stokes, and Divergence. The notion that one-dimensional integrals of functions can be computed from evaluation of a related function (e.g. an antiderivative or a potential function) on the endpoints of the interval of integration generalizes to integration over areas, surfaces and 3D domains. Integration over these domains can be computed by evaluation on the boundary of an area, surface or volume of the appropriate function.
- Compute and understand the meaning of the curl and divergence of a vector field and utilize them to compute area and volume integrals using Green's and Stokes', and Divergence Theorems respectively. Students will also learn how these theorems represent conservation principles for physical vector fields important in gravitation and electric fields.