# Intellectual Need-Provoking Tasks

We have created an “Intellectual Need-Provoking Task” corresponding to each topic in first-semester calculus.

These tasks are meant for students to work on prior to and/or after watching the corresponding instructional videos.

Each task is set up in a way so that the corresponding mathematical concept is required for solving or understanding the task. In this way, the tasks are designed to help students understand why the mathematical concept is important or useful.

For each task, there is a corresponding video of two students attempting to solve the problem; these videos are designed to highlight the various ways of thinking students might employ while working on the task, as well as to demonstrate the various ways students might struggle with the task.

TopicTask and Related FilesStudent Problem-Solving Video
Constant Rates of ChangeStudent Problem Solving: Pouring Water into a Cylinder
Graphing Constant Rate of ChangeStudent Problem Solving: Cannon Cow!
Varying Rates of ChangeStudent Problem Solving: Pouring Water into an Erlenmeyer Flask
Graphing Varying Rates of ChangeStudent Problem Solving: Filling a Spherical Flask
Average Rates of ChangeStudent Problem Solving: Two Race Cars, Constant Rates, and Average Rates
Approximating Instantaneous Rates of ChangeStudent Problem Solving: The Stationary Baseball
ContinuityStudent Problem Solving: Continuity
Differentiability and Local LinearityStudent Problem Solving: Growth of a Rabbit Population
Limit Definition of DerivativeStudent Problem Solving: Rate of Absorbing Ibuprofen
Using the Limit Definition of DerivativeStudent Problem Solving: Using Limits to Compute Derivatives
Interpreting DerivativesStudent Problem Solving: Interpreting Derivatives
Slopes of Secant and Tangent LinesStudent Problem Solving: The Imprecision of Tangents
Graphing DerivativesStudent Problem Solving: Graphing the Speed of a Baseball
Basic Derivative RulesStudent Problem Solving: Trying to Use the Limit Definition
The Product RuleStudent Problem Solving: Products of Polynomials
The Quotient RuleStudent Problem Solving: Derivatives of Quotients
The Chain RuleStudent Problem Solving: A Ripple in a Pond
l'Hopital's RuleStudent Problem Solving: Evaluating Indeterminate Limits
Mean Value TheoremStudent Problem Solving
Related RatesStudent Problem Solving
Implicit DifferentiationStudent Problem Solving: A Complicated Tangent Line
Introduction to OptimizationStudent Problem Solving: Maximizing Fuel Economy
Optimization: Algebraic ModelingStudent Problem Solving: Maximizing an Animal Pen
Introduction to Riemann SumsStudent Problem Solving: Dust Accumulation on the Mars Rover
Riemann Sum NotationStudent Problem Solving: Writing a Riemann Sum Two Ways
Definite IntegralsStudent Problem Solving: Mars Rover Using a Formula
AntiderivativesStudent Problem Solving: Antiderivatives
The Fundamental Theorem of Calculus, Part 1Student Problem Solving: Computing Total Accumulation
The Fundamental Theorem of Calculus, Part 2Student Problem Solving: Cumulative Probability from a Normal Distribution
U-SubstitutionStudent Problem Solving: Evaluating Indefinite Integrals