Lesson 16: Implicit Differentiation

Nov 2, 20073 likes5,750 views

The document is a lesson on implicit differentiation and related concepts: 1) Implicit differentiation allows one to take the derivative of an implicitly defined relation between x and y, even if y is not explicitly defined as a function of x. 2) Examples are provided to demonstrate implicit differentiation, such as finding the slope of a tangent line to a curve. 3) The van der Waals equation is introduced to describe non-ideal gas properties, and implicit differentiation is used to find the isothermal compressibility of a van der Waals gas.

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This document contains notes from a calculus class lecture on evaluating definite integrals. It discusses using the evaluation theorem to evaluate definite integrals, writing derivatives as indefinite integrals, and interpreting definite integrals as the net change of a function over an interval. The document also contains examples of evaluating definite integrals, properties of integrals, and an outline of the key topics covered.

Lesson 25: Evaluating Definite Integrals (handout)Matthew Leingang

This document contains lecture notes from a Calculus I class covering Section 5.3 on evaluating definite integrals. The notes discuss using the Evaluation Theorem to calculate definite integrals, writing derivatives as indefinite integrals, and interpreting definite integrals as the net change of a function over an interval. Examples are provided to demonstrate evaluating definite integrals using the midpoint rule approximation. Properties of integrals such as additivity and the relationship between definite and indefinite integrals are also outlined.

Lesson 24: Areas and Distances, The Definite Integral (handout)Matthew Leingang

Lesson 24: Areas and Distances, The Definite Integral (slides)Matthew Leingang

Lesson 23: Antiderivatives (slides)Matthew Leingang

Lesson 22: Optimization Problems (handout)Matthew Leingang

Lesson 21: Curve Sketching (slides)Matthew Leingang

The document discusses curve sketching of functions by analyzing their derivatives. It provides: 1) A checklist for graphing a function which involves finding where the function is positive/negative/zero, its monotonicity from the first derivative, and concavity from the second derivative. 2) An example of graphing the cubic function f(x) = 2x^3 - 3x^2 - 12x through analyzing its derivatives. 3) Explanations of the increasing/decreasing test and concavity test to determine monotonicity and concavity from a function's derivatives.

Lesson 21: Curve Sketching (handout)Matthew Leingang

The document contains lecture notes on curve sketching from a Calculus I class. It discusses using the first and second derivative tests to determine properties of a function like monotonicity, concavity, maxima, minima, and points of inflection in order to sketch the graph of the function. It then provides an example of using these tests to sketch the graph of the cubic function f(x) = 2x^3 - 3x^2 - 12x.

Lesson 20: Derivatives and the Shapes of Curves (slides)Matthew Leingang

This document contains lecture notes on derivatives and the shapes of curves from a Calculus I class taught by Professor Matthew Leingang at New York University. The notes cover using derivatives to determine the intervals where a function is increasing or decreasing, classifying critical points as maxima or minima, using the second derivative to determine concavity, and applying the first and second derivative tests. Examples are provided to illustrate finding intervals of monotonicity for various functions.

Lesson 20: Derivatives and the Shapes of Curves (handout)Matthew Leingang

This document contains lecture notes on calculus from a Calculus I course. It covers determining the monotonicity of functions using the first derivative test. Key points include using the sign of the derivative to determine if a function is increasing or decreasing over an interval, and using the first derivative test to classify critical points as local maxima, minima, or neither. Examples are provided to demonstrate finding intervals of monotonicity for various functions and applying the first derivative test.

Lesson 19: The Mean Value Theorem (slides)Matthew Leingang