Richard Wong

Assistant Teaching Professor in Mathematics at Rice University.

You can find the course syllabus here.

How can we describe the physical world mathematically? How can we use mathematics to describe phenomena in physics, biology, chemistry, or other STEM fields?

Calculus is the mathematical language that allows us to describe and model the behavior of the physical world around us, such as the speed and acceleration at which we travel, as well as our distance and displacement; or how a population grows and changes over time; or the rate at which chemicals react or move towards equilibrium.

In this course, you will develop the reasoning and questioning skills needed to explore these concepts mathematically. Moreover, you will become fluent in communicating your ideas through the mathematical language of calculus.

Schedule

  Learning Outcome Textbook Section Lectures
CI1 Limits and continuity. Evaluate limits algebraically, graphically, and/or numerically. Geometrically interpret limits and continuity. Determine if a function is continuous at a point or over an interval. Describe and interpret real-world functions in terms of continuity. Apply the Intermediate Value Theorem. 2.2-2.3, 2.5 1-6
CI2 Defining the derivative. Use the limit definition to compute derivatives. Geometrically interpret the derivative. Describe and interpret real-world scenarios using derivatives. 2.7-2.8 7-10
CI3 Applications of the derivative. Describe and interpret higher derivatives. Compute various derivatives of common functions. Estimate functions using linear approximation. 3.1, 3.3, 3.10 11-14
CI4 Calculating derivatives. Identify and apply various derivative rules, including the chain rule. Compute derivatives using implicit differentiation. Combine multiple derivative rules. Apply the inverse function theorem. 3.2, 3.4-3.6 15-21
CI5 Understanding functions via calculus. Use the Extreme Value Theorem to identify and classify local extrema. Sketch the shape of a function using its first two derivatives. Describe asymptotic behavior of functions. Evaluate limits at infinity and resolve indeterminate forms. State and apply L’Hôpital’s rule when appropriate. 2.6, 4.2-4.4 22-26
CI6 Optimization and related rates. Describe and solve real-world problems using optimization. Describe and interpret real-world scenarios using related rates. 3.9, 4.1, 4.7 27-30
CI7 Defining the integral. Approximate the area under a curve using Riemann sums. Describe the relationship between limits and definite integrals. Compute integrals using anti-derivatives. Describe and interpret real-world scenarios using integrals. 5.1-5.2, 5.4 31-34
CI8 The fundamental theorem of calculus. Describe the relationship between derivatives and integrals. Apply the fundamental theorem of calculus to evaluate integrals. Compare and connect various interpretations of the integral. 4.9, 5.3 35-37
CI9 Applications of integrals. Construct and evaluate integrals that represent real-world concepts, including area, volume, work, cost, mass density, etc. 6.1-6.3 38-41

Learning Objectives:

The goals of the course are that you: 

  • learn how to use the tools of calculus (differentiation and integration) to describe and model the real world.
  • Develop the reasoning and questioning skills needed to explore these (mathematical) topics and apply them to real-life situations.
  • Develop the collaboration and communication skills needed to convey your (mathematical) ideas.

Below you will find the explicit learning objectives associated to each of these goals.

Calculus I Objectives (CI)

  1. Limits and continuity. Evaluate limits algebraically, graphically, and/or numerically. Geometrically interpret limits and continuity. Determine if a function is continuous at a point or over an interval. Describe and interpret real-world functions in terms of continuity. Apply the Intermediate Value Theorem.
  2. Defining the derivative. Use the limit definition to compute derivatives. Geometrically interpret the derivative. Describe and interpret real-world scenarios using derivatives.
  3. Applications of the derivative. Describe and interpret higher derivatives. Compute various derivatives of common functions. Estimate functions using linear approximation.
  4. Calculating derivatives. Identify and apply various derivative rules, including the chain rule. Compute derivatives using implicit differentiation. Combine multiple derivative rules. Apply the inverse function theorem.
  5. Understanding functions via calculus. Use the Extreme Value Theorem to identify and classify local extrema. Sketch the shape of a function using its first two derivatives. Describe asymptotic behavior of functions. Evaluate limits at infinity and resolve indeterminate forms. State and apply L'Hôpital's rule when appropriate.
  6. Optimization and related rates. Describe and solve real-world problems using optimization. Describe and interpret real-world scenarios using related rates.
  7. Defining the integral. Approximate the area under a curve using Riemann sums. Describe the relationship between limits and definite integrals. Compute integrals using anti-derivatives. Describe and interpret real-world scenarios using integrals.
  8. The fundamental theorem of calculus. Describe the relationship between derivatives and integrals. Apply the fundamental theorem of calculus to evaluate integrals. Compare and connect various interpretations of the integral.
  9. Applications of integrals. Construct and evaluate integrals that represent real-world concepts, including area, volume, work, cost, mass density, etc.

Mathematical Reasoning Objectives (MR)

  1. Reason abstractly and quantitatively.  Use mathematics to model real-world situations and to interpret and solve problems.  Make appropriate assumptions and approximations to simplify a complicated situation. Draw pictures or study examples to provide insight.  Attend to the meaning of quantities instead of just computing them. Consider the units involved.
  2. Make sense of problems and persevere in solving them.  Understand what a problem is asking for. Analyze the givens, constraints, and goals of a problem. Break large/complex problems into smaller/simpler problems. Check answers using alternate methods.
  3. Build intuition.  Investigate and create specific examples and counterexamples. Look for and make use of structure or repeated patterns. Look for both general methods and shortcuts.  Seek to understand unexpected results.
  4. Use appropriate tools strategically.  Consider the available tools when solving a mathematical problem. Understand and use stated assumptions, definitions, and previously established results.
  5. Construct viable arguments.  Make conjectures, and use assumptions, definitions, and/or previously established results to prove or disprove them. Use strategies such as direct proofs, contrapositive statements, proof by cases, or proof by contradiction.
  6. Be creative and explore mathematics. Do more than find a solution. Look for novel or elegant solutions. See what changes when you add, change, or weaken hypotheses. Draw connections between topics and ideas.

Mathematical Communication Objectives (MC)

  1. Ask questions.  Notice, identify, and clarify sources of confusion. Look for connections and relationships between ideas. Explore topics and ideas deeply.
  2. Use and develop mathematical fluency.  Use standard mathematical notations and terms (as discussed in class or demonstrated in course materials).  Clearly indicate and explain any use of non-standard shorthand, notation, or tools.
  3. Analyze and constructively critique the reasoning of others.  Actively listen and summarize key ideas to check comprehension. Test conjectures against examples and potential counterexamples. Assess and reconcile various approaches to problems. Work together to find errors and fix flaws.
  4. Explain and justify your reasoning.  Communicate and justify your conclusions to others. Indicate the general strategy or argument, and identify the key step or idea(s). Listen and reflect to the critiques of others.  Work together to find errors and fix flaws.
  5. Attend to precision.  Communicate precisely to others. Use clear definitions, and carefully state any assumptions or results used.  Be able to explain heuristics/arguments in depth.
  6. Be clear and concise.  Use the appropriate amount of generality or specificity in arguments.  Avoid use of any extraneous assumptions, hypotheses, or statements. Indicate any figures or examples that you have in mind.
  7. Review, reflect, and revise. Review previous work and assess the positives and negatives. Reflect on your strategies and look for ways to improve. Use feedback to grow and develop your mathematical reasoning or communication skills.