Introduction
- Calculus is the study of rates of change.
- Differential calculus determines the rate of change of a quantity, while integral calculus finds the quantity where the rate of change is known.
- Calculus is used in a multitude of fields that you wouldn’t ordinarily think would make use of its concepts. Among them are physics, engineering, economics, statistics, and medicine.
- Russell, Deb. “What Is Calculus? Definition and Practical Applications.” ThoughtCo, Aug. 28, 2020.
- Textbook: Calculus: Single and Multivariable, by Hughes-Hallet et al., 7th edition, ISBN 9781119330387
Calculus Courses
Calculus I
Introduces the intuitive, numerical and theoretical concepts of limits, continuity, differentiation and integration. Includes the study of extrema, curve sketching, and applications involving algebraic, exponential, logarithmic and trigonometric functions. Designed for mathematics, science and engineering majors.
Calculus II
Continues course of study begun in Calculus I. Covers integration techniques, numerical integration, improper integrals, some differential equations, sequences, series and applications. Prerequisite: MATH 1510.
Calculus III
The purpose of this course, which is a continuation of MATH 132, is to study the methods of calculus in more detail. The course will cover the material in the textbook from Chapters 10-14.Vectors in the plane and 3-space, vector calculus in two-dimensions, partial differentiation, multiple integration, topics in vector calculus, and complex numbers and functions.
Applications of Calculus
An algebraic and graphical study of derivatives and integrals, with an emphasis on applications to business, social science, economics and the sciences. Prerequisite: MATH 1150 or MATH 1220.
Student Learning Objectives
Calculus I:
- 1. Limits
- a. Use limit notation.
- b. Compute limits or determine when a limit does not exist.
- c. Use limits to decide if a function is continuous.
- d. Use limits to decide if a function is differentiable.
- e. Use limits to determine asymptotes.
- 2. Derivatives
- a. Determine the derivative of a simple function, at a point as well as more generally, using the definition of the derivative.
- b. Determine the derivatives of algebraic and transcendental functions using the General Power, Product, Quotient, Chain Rules, implicit differentiation and the linearity of the differential operator.
- c. Describe the meaning of the derivative as a rate of change in a variety of contexts.
- d. Use derivatives to sketch graphs of functions with details showing critical points and their natures, inflection points, noting monotonicity, and concavity, connecting these to features found algebraically, such as intercepts and asymptotes.
- e. Compute local linear approximation.
- 3. Integrals
- a. Compute definite integrals using the limit definition and sigma notation.
- b. Approximate definite integrals using finite sums. 642 Revised 5/31/2019
- c. Compute indefinite integrals by identifying them with antiderivatives.
- d. Compute definite and indefinite integrals using substitution. e. Describe the meaning of the integral in a variety of contexts.
- 4. Applications of calculus
- a. Solve optimization problems, related rate problems and motion problems involving position, velocity, speed and acceleration using differentiation and integration.
- b. Compute area bounded by functions and vertical lines.
- c. Be able to apply theorems of calculus such as the Fundamental Theorem, the Intermediate Value Theorem, the Mean Value Theorem, the Mean Value Theorem of Integration, and the Extreme Value Theorem.
Calculus II:
- Integration
- a. Determine the indefinite integrals and compute definite integrals of algebraic and transcendental functions using various techniques of integration including integration by parts, trigonometric substitution, and partial fraction decomposition.
- b. Compute improper integrals using the appropriate limit definitions.
- c. Solve problems involving separable differential equations.
- 2. Sequences and Series
- a. Compute the limit of sequences.
- b. Compute the sum of a basic series using its nth partial sum.
- c. Compute the sum of geometric and telescoping series.
- d. Determine if a series converges using the appropriate test, such as the nth term, integral, p-series, comparison, limit comparison, ratio, root, and alternating series tests.
- e. Determine if a series converges absolutely, converges conditionally or diverges.
- 3. Properties of power series
- a. Compute the radius and interval of convergence of a power series.
- b. Compute the Taylor polynomials of functions.
- c. Compute basic Taylor series using the definition.
- d. Compute Taylor series using function arithmetic, composition, differentiation, and integration.
- e. Compute limits with Taylor series.
- f. Approximate definite integrals with Taylor series and estimate the error of approximation.
- g. Determine the sum of a convergent series using Taylor series.
- 4. Applications of integration
- a. Compute volumes and areas of surfaces of solids of revolution.
- b. Compute length of curves.
- c. Apply integration using alternative coordinate forms and using a parameter.
Calculus III:
- 1. Vectors in 3-dimensional space
- a. Use vector notation correctly.
- b. Perform vector operations, including dot product, cross product, differentiation and integration, and demonstrate their geometric interpretations.
- c. Perform operations on vector valued functions and functions of a parameter.
- 2. Functions of multiple variables
- a. Identify and graph the equations of cylinders and quadratic surfaces in 3-dimensional space.
- b. Determine the domain of continuity of a vector valued function and of a function of multiple variables.
- 3. Applications of differentiation
- a. Compute partial derivatives, generally and at a point, and sketch their graphical representation on a surface in space. 658 Revised 5/31/2019
- b. Recognize when the chain rule is needed when differentiating functions of multiple variables, parametric equations and vector valued functions, and be able to use the chain rule in these situations.
- c. Compute curvature of a parameterized vector representation of a curve in 2- and 3-dimensional space and be able to explain its meaning.
- d. Compute the unit tangent and unit normal vectors to a curve and be able to sketch them with the curve.
- e. Computationally move among position vector, velocity vector, speed, and acceleration vectors; recognize and demonstrate their use as applied to motion in space.
- f. Determine the equation of the tangent plane to a surface at a point.
- g. Use the tangent plane to a surface to approximate values on the surface and estimate error in approximation using differentials.
- h. Compute directional derivatives and represent them graphically relative to the inherent surface.
- i. Compute the gradient vector; represent it graphically relative to the inherent surface and use it to maximize or minimize rate of change of the function.
- j. Locate local and global maxima and minima of a function.
- k. Use Lagrange multipliers to maximize output with one or two constraints.
- 4. Application of Integration
- a. Compute arc length and be able to explain its derivation as a limit.
- b. Calculate double and triple integrals independently and with their geometric representations as surfaces, areas and volumes.
- c. Calculate iterated integrals in polar, cylindrical and spherical coordinate systems.
Applications of Calculus:
- 1. Find limits algebraically and graphically, and use limits to analyze continuity.
- 2. Find the derivative of a function by applying appropriate techniques (limit of the difference quotient, general derivative rules, product rule, quotient rule, chain rule, and higher order derivatives).
- 3. Perform implicit differentiation. Use implicit differentiation to solve related rate application problems.
- 4. Use the derivative to describe the rate of change and slope of a curve in general and at particular points. Compare and contrast average rates of change to instantaneous rates of change.
- 5. Find the maxima, minima, points of inflections, and determine concavity of a function by applying the first and second derivatives. Use these results to sketch graphs of functions and to solve optimization problems in context.
- 6. Find the antiderivative and indefinite integral functions to include integration by substitution. Apply the Fundamental Theorem of Calculus in computing definite integrals of functions.
- 7. Approximate the area under the curve using Riemann sums.
- 8. Use the integral to determine the area under a curve and to find the accumulated value of a function in context. 641 Revised 5/31/2019
- 9. Solve contextual problems by identifying the appropriate type of function given the context, creating a formula based on the information given, applying knowledge of algebra and calculus, and interpreting the results in context.
- 10. Communicate mathematical information using proper notation and verbal explanations.
Resources
Calculus I
Fall 2021:
MATH 1510-Test I (Tamang)
MATH 1510-Quiz 1 (Tamang)
MATH 1510-Test II (Tamang)
MATH 1510-Quiz 2 (Tamang)
MATH 1510-Test III (Tamang)
MATH 1510-Quiz 3 (Tamang)
MATH 1510-Final Exam (Tamang)
Fall 2023:
MATH 1510-Quiz 1 (Tamang)
MATH 1510-Test I (Tamang)
MATH 1510-Quiz 2 (Tamang)
MATH 1510-Test II (Tamang)
MATH 1510-Quiz 3 (Tamang)
MATH 1510-Test III (Tamang)
MATH 1510-Final Exam (Tamang)
Fall 2024:
MATH 1510-Quiz 1 (Tamang)
MATH 1510-Test I (Tamang)
MATH 1510-Quiz 2 (Tamang)
MATH 1510-Test II (Tamang)
MATH 1510-Quiz 3 (Tamang)
MATH 1510-Test III (Tamang)
MATH 1510-Final Exam (Tamang)
Calculus II
Spring 2022:
MATH 1520-Test I (Tamang)
MATH 1520-Quiz 1 (Tamang)
MATH 1520-Test II (Tamang)
MATH 1520-Quiz 2 (Tamang)
MATH 1520-Test III (Tamang)
MATH 1520-Quiz 3 (Tamang)
MATH 1520-Final Exam (Tamang)
Spring 2024:
MATH 1520-Test I (Tamang)
MATH 1520-Quiz 1 (Tamang)
MATH 1520-Test II (Tamang)
MATH 1520-Quiz 2 (Tamang)
MATH 1520-Test III (Tamang)
MATH 1520-Quiz 3 (Tamang)
MATH 1520-Final Exam (Tamang)
Calculus III
Fall 2022:
MATH 2530-Test I (Tamang)
MATH 2530-Quiz 1 (Tamang)
MATH 2530-Test II (Tamang)
MATH 2530-Quiz 2 (Tamang)
MATH 2530-Test III (Tamang)
MATH 2530-Quiz 3 (Tamang)
MATH 2530-Final Exam (Tamang)
Fall 2024:
MATH 2530-Quiz 1 (Tamang)
MATH 2530-Test I (Tamang)
MATH 2530-Quiz 2 (Tamang)
MATH 2530-Test II (Tamang)
MATH 2530-Quiz 3 (Tamang)
MATH 2530-Test III (Tamang)
MATH 2530-Final Exam (Tamang)
Other Calculus Resources
https://compscicentral.com/calculus-resources/
Contact
Dr. Sundar Tamang
Assistant Professor of Applied Mathematics / Statistics
Phone: (575) 538 6330
Email: Dr. Sundar Tamang
Office: Global Resource Center Rm. 213