# Courses tagged with "Department of Economics" (22)

Sample questions from the A.P. Calculus AB and BC exams (both multiple choice and free answer). 2011 Calculus AB Free Response #1a. 2011 Calculus AB Free Response #1 parts b c d. 2011 Calculus AB Free Response #2 (a & b). 2011 Calculus AB Free Response #2 (c & d). 2011 Calculus AB Free Response #3 (a & b). 2011 Calculus AB Free Response #3 (c). 2011 Calculus AB Free Response #4a. 2011 Calculus AB Free Response #4b. 2011 Calculus AB Free Response #4c. 2011 Calculus AB Free Response #4d. 2011 Calculus AB Free Response #5a. 2011 Calculus AB Free Response #5b. 2011 Calculus AB Free Response #5c.. 2011 Calculus AB Free Response #6a. 2011 Calculus AB Free Response #6b. 2011 Calculus AB Free Response #6c. AP Calculus BC Exams: 2008 1 a. AP Calculus BC Exams: 2008 1 b&c. AP Calculus BC Exams: 2008 1 c&d. AP Calculus BC Exams: 2008 1 d. Calculus BC 2008 2 a. Calculus BC 2008 2 b &c. Calculus BC 2008 2d. 2011 Calculus BC Free Response #1a. 2011 Calculus BC Free Response #1 (b & c). 2011 Calculus BC Free Response #1d. 2011 Calculus BC Free Response #3a. 2011 Calculus BC Free Response #3 (b & c). 2011 Calculus BC Free Response #6a. 2011 Calculus BC Free Response #6b. 2011 Calculus BC Free Response #6c. 2011 Calculus BC Free Response #6d. 2011 Calculus AB Free Response #1a. 2011 Calculus AB Free Response #1 parts b c d. 2011 Calculus AB Free Response #2 (a & b). 2011 Calculus AB Free Response #2 (c & d). 2011 Calculus AB Free Response #3 (a & b). 2011 Calculus AB Free Response #3 (c). 2011 Calculus AB Free Response #4a. 2011 Calculus AB Free Response #4b. 2011 Calculus AB Free Response #4c. 2011 Calculus AB Free Response #4d. 2011 Calculus AB Free Response #5a. 2011 Calculus AB Free Response #5b. 2011 Calculus AB Free Response #5c.. 2011 Calculus AB Free Response #6a. 2011 Calculus AB Free Response #6b. 2011 Calculus AB Free Response #6c. AP Calculus BC Exams: 2008 1 a. AP Calculus BC Exams: 2008 1 b&c. AP Calculus BC Exams: 2008 1 c&d. AP Calculus BC Exams: 2008 1 d. Calculus BC 2008 2 a. Calculus BC 2008 2 b &c. Calculus BC 2008 2d. 2011 Calculus BC Free Response #1a. 2011 Calculus BC Free Response #1 (b & c). 2011 Calculus BC Free Response #1d. 2011 Calculus BC Free Response #3a. 2011 Calculus BC Free Response #3 (b & c). 2011 Calculus BC Free Response #6a. 2011 Calculus BC Free Response #6b. 2011 Calculus BC Free Response #6c. 2011 Calculus BC Free Response #6d.

Minima, maxima, and critical points. Rates of change. Optimization. Rates of change. L'Hopital's rule. Mean value theorem. Minima, maxima and critical points. Testing critical points for local extrema. Identifying minima and maxima for x^3 - 12x - 5. Concavity, concave upwards and concave downwards intervals. Recognizing concavity exercise. Recognizing concavity. Inflection points. Graphing using derivatives. Another example graphing with derivatives. Minimizing sum of squares. Optimizing box volume graphically. Optimizing box volume analytically. Optimizing profit at a shoe factory. Minimizing the cost of a storage container. Expression for combined area of triangle and square. Minimizing combined area. Rates of change between radius and area of circle. Rate of change of balloon height. Related rates of water pouring into cone. Falling ladder related rates. Rate of change of distance between approaching cars. Speed of shadow of diving bird. Mean Value Theorem. Introduction to L'H

Divergence theorem intuition. Divergence theorem examples and proofs. Types of regions in 3D. 3-D Divergence Theorem Intuition. Divergence Theorem Example 1. Why we got zero flux in Divergence Theorem Example 1. Type I Regions in Three Dimensions. Type II Regions in Three Dimensions. Type III Regions in Three Dimensions. Divergence Theorem Proof (part 1). Divergence Theorem Proof (part 2). Divergence Theorem Proof (part 3). Divergence Theorem Proof (part 4). Divergence Theorem Proof (part 5). 3-D Divergence Theorem Intuition. Divergence Theorem Example 1. Why we got zero flux in Divergence Theorem Example 1. Type I Regions in Three Dimensions. Type II Regions in Three Dimensions. Type III Regions in Three Dimensions. Divergence Theorem Proof (part 1). Divergence Theorem Proof (part 2). Divergence Theorem Proof (part 3). Divergence Theorem Proof (part 4). Divergence Theorem Proof (part 5).

Volume under a surface with double integrals. Triple integrals as well. Double Integral 1. Double Integrals 2. Double Integrals 3. Double Integrals 4. Double Integrals 5. Double Integrals 6. Triple Integrals 1. Triple Integrals 2. Triple Integrals 3. Double Integral 1. Double Integrals 2. Double Integrals 3. Double Integrals 4. Double Integrals 5. Double Integrals 6. Triple Integrals 1. Triple Integrals 2. Triple Integrals 3.

Indefinite integral as anti-derivative. Definite integral as area under a curve. Integration by parts. U-substitution. Trig substitution. Antiderivatives and indefinite integrals. Indefinite integrals of x raised to a power. Antiderivative of hairier expression. Basic trig and exponential antiderivatives. Antiderivative of x^-1. Simple Riemann approximation using rectangles. Generalizing a left Riemann sum with equally spaced rectangles. Rectangular and trapezoidal Riemann approximations. Trapezoidal approximation of area under curve. Riemann sums and integrals. Deriving integration by parts formula. Antiderivative of xcosx using integration by parts. Integral of ln x. Integration by parts twice for antiderivative of (x^2)(e^x). Integration by parts of (e^x)(cos x). U-substitution. U-substitution example 2. U-substitution Example 3. U-substitution with ln(x). Doing u-substitution twice (second time with w). U-substitution and back substitution. U-substitution with definite integral. (2^ln x)/x Antiderivative Example. Another u-substitution example. Riemann sums and integrals. Intuition for Second Fundamental Theorem of Calculus. Evaluating simple definite integral. Definite integrals and negative area. Area between curves. Area between curves with multiple boundaries. Challenging definite integration. Introduction to definite integrals. Definite integrals (part II). Definite Integrals (area under a curve) (part III). Definite Integrals (part 4). Definite Integrals (part 5). Definite integral with substitution. Introduction to trig substitution. Another substitution with x=sin (theta). Integrals: Trig Substitution 1. Trig and U substitution together (part 1). Trig and U substitution together (part 2). Trig substitution with tangent. Integrals: Trig Substitution 2. Integrals: Trig Substitution 3 (long problem). Fundamental theorem of calculus. Applying the fundamental theorem of calculus. Swapping the bounds for definite integral. Both bounds being a function of x. Proof of Fundamental Theorem of Calculus. Connecting the first and second fundamental theorems of calculus. Introduction to improper integrals. Improper integral with two infinite bounds. Divergent improper integral. Antiderivatives and indefinite integrals. Indefinite integrals of x raised to a power. Antiderivative of hairier expression. Basic trig and exponential antiderivatives. Antiderivative of x^-1. Simple Riemann approximation using rectangles. Generalizing a left Riemann sum with equally spaced rectangles. Rectangular and trapezoidal Riemann approximations. Trapezoidal approximation of area under curve. Riemann sums and integrals. Deriving integration by parts formula. Antiderivative of xcosx using integration by parts. Integral of ln x. Integration by parts twice for antiderivative of (x^2)(e^x). Integration by parts of (e^x)(cos x). U-substitution. U-substitution example 2. U-substitution Example 3. U-substitution with ln(x). Doing u-substitution twice (second time with w). U-substitution and back substitution. U-substitution with definite integral. (2^ln x)/x Antiderivative Example. Another u-substitution example. Riemann sums and integrals. Intuition for Second Fundamental Theorem of Calculus. Evaluating simple definite integral. Definite integrals and negative area. Area between curves. Area between curves with multiple boundaries. Challenging definite integration. Introduction to definite integrals. Definite integrals (part II). Definite Integrals (area under a curve) (part III). Definite Integrals (part 4). Definite Integrals (part 5). Definite integral with substitution. Introduction to trig substitution. Another substitution with x=sin (theta). Integrals: Trig Substitution 1. Trig and U substitution together (part 1). Trig and U substitution together (part 2). Trig substitution with tangent. Integrals: Trig Substitution 2. Integrals: Trig Substitution 3 (long problem). Fundamental theorem of calculus. Applying the fundamental theorem of calculus. Swapping the bounds for definite integral. Both bounds being a function of x. Proof of Fundamental Theorem of Calculus. Connecting the first and second fundamental theorems of calculus. Introduction to improper integrals. Improper integral with two infinite bounds. Divergent improper integral.

Limit introduction, squeeze theorem, and epsilon-delta definition of limits. Introduction to limits. Limit at a point of discontinuity. Determining which limit statements are true. Limit properties. Limit example 1. Limits 1. One-sided limits from graphs. One-sided limits from graphs. Introduction to Limits. Limit Examples (part 1). Limit Examples (part 2). Limit Examples (part 3). Limit Examples w/ brain malfunction on first prob (part 4). More Limits. Limits 1. Limits and infinity. Limits at positive and negative infinity. More limits at infinity. Limits with two horizontal asymptotes. Limits 2. Squeeze Theorem. Proof: lim (sin x)/x. Limit intuition review. Building the idea of epsilon-delta definition. Epsilon-delta definition of limits. Proving a limit using epsilon-delta definition. Limits to define continuity. Continuity. Epsilon Delta Limit Definition 1. Epsilon Delta Limit Definition 2. Introduction to limits. Limit at a point of discontinuity. Determining which limit statements are true. Limit properties. Limit example 1. Limits 1. One-sided limits from graphs. One-sided limits from graphs. Introduction to Limits. Limit Examples (part 1). Limit Examples (part 2). Limit Examples (part 3). Limit Examples w/ brain malfunction on first prob (part 4). More Limits. Limits 1. Limits and infinity. Limits at positive and negative infinity. More limits at infinity. Limits with two horizontal asymptotes. Limits 2. Squeeze Theorem. Proof: lim (sin x)/x. Limit intuition review. Building the idea of epsilon-delta definition. Epsilon-delta definition of limits. Proving a limit using epsilon-delta definition. Limits to define continuity. Continuity. Epsilon Delta Limit Definition 1. Epsilon Delta Limit Definition 2.

Line integral of scalar and vector-valued functions. Green's theorem and 2-D divergence theorem. Introduction to the Line Integral. Line Integral Example 1. Line Integral Example 2 (part 1). Line Integral Example 2 (part 2). Position Vector Valued Functions. Derivative of a position vector valued function. Differential of a vector valued function. Vector valued function derivative example. Line Integrals and Vector Fields. Using a line integral to find the work done by a vector field example. Parametrization of a Reverse Path. Scalar Field Line Integral Independent of Path Direction. Vector Field Line Integrals Dependent on Path Direction. Path Independence for Line Integrals. Closed Curve Line Integrals of Conservative Vector Fields. Example of Closed Line Integral of Conservative Field. Second Example of Line Integral of Conservative Vector Field. Green's Theorem Proof Part 1. Green's Theorem Proof (part 2). Green's Theorem Example 1. Green's Theorem Example 2. Constructing a unit normal vector to a curve. 2 D Divergence Theorem. Conceptual clarification for 2-D Divergence Theorem. Introduction to the Line Integral. Line Integral Example 1. Line Integral Example 2 (part 1). Line Integral Example 2 (part 2). Position Vector Valued Functions. Derivative of a position vector valued function. Differential of a vector valued function. Vector valued function derivative example. Line Integrals and Vector Fields. Using a line integral to find the work done by a vector field example. Parametrization of a Reverse Path. Scalar Field Line Integral Independent of Path Direction. Vector Field Line Integrals Dependent on Path Direction. Path Independence for Line Integrals. Closed Curve Line Integrals of Conservative Vector Fields. Example of Closed Line Integral of Conservative Field. Second Example of Line Integral of Conservative Vector Field. Green's Theorem Proof Part 1. Green's Theorem Proof (part 2). Green's Theorem Example 1. Green's Theorem Example 2. Constructing a unit normal vector to a curve. 2 D Divergence Theorem. Conceptual clarification for 2-D Divergence Theorem.

Thinking about forms of derivatives in multi-dimensions and for vector-valued functions: partial derivatives, gradient, divergence and curl. Partial Derivatives. Partial Derivatives 2. Gradient 1. Gradient of a scalar field. Divergence 1. Divergence 2. Divergence 3. Curl 1. Curl 2. Curl 3. Partial Derivatives. Partial Derivatives 2. Gradient 1. Gradient of a scalar field. Divergence 1. Divergence 2. Divergence 3. Curl 1. Curl 2. Curl 3.

Sequences, series and approximating functions. Maclaurin and Taylor series. Sequences and Series (part 1). Sequences and series (part 2). Maclaurin and Taylor Series Intuition. Cosine Taylor Series at 0 (Maclaurin). Sine Taylor Series at 0 (Maclaurin). Taylor Series at 0 (Maclaurin) for e to the x. Euler's Formula and Euler's Identity. Visualizing Taylor Series Approximations. Generalized Taylor Series Approximation. Visualizing Taylor Series for e^x. Error or Remainder of a Taylor Polynomial Approximation. Proof: Bounding the Error or Remainder of a Taylor Polynomial Approximation. Polynomial approximation of functions (part 1). Polynomial approximation of functions (part 2). Approximating functions with polynomials (part 3). Polynomial approximation of functions (part 4). Polynomial approximations of functions (part 5). Polynomial approximation of functions (part 6). Polynomial approximation of functions (part 7). Taylor Polynomials. Sequences and Series (part 1). Sequences and series (part 2). Maclaurin and Taylor Series Intuition. Cosine Taylor Series at 0 (Maclaurin). Sine Taylor Series at 0 (Maclaurin). Taylor Series at 0 (Maclaurin) for e to the x. Euler's Formula and Euler's Identity. Visualizing Taylor Series Approximations. Generalized Taylor Series Approximation. Visualizing Taylor Series for e^x. Error or Remainder of a Taylor Polynomial Approximation. Proof: Bounding the Error or Remainder of a Taylor Polynomial Approximation. Polynomial approximation of functions (part 1). Polynomial approximation of functions (part 2). Approximating functions with polynomials (part 3). Polynomial approximation of functions (part 4). Polynomial approximations of functions (part 5). Polynomial approximation of functions (part 6). Polynomial approximation of functions (part 7). Taylor Polynomials.

Using definite integrals with the shell and disc methods to find volumes of solids of revolution. Disk method around x-axis. Generalizing disc method around x-axis. Disc method around y-axis. Disc method (washer method) for rotation around x-axis. Generalizing the washer method. Disc method rotation around horizontal line. Washer method rotating around non-axis. Part 2 of washer for non axis rotation. Disc method rotating around vertical line. Calculating integral disc method around vertical line. Washer or ring method for vertical line rotation. Evaluating integral for washer method around vertical line. Shell method for rotating around vertical line. Evaluating integral for shell method example. Shell method for rotating around horizontal line. Shell method with two functions of x. Calculating integral with shell method. Shell method with two functions of y. Part 2 of shell method with 2 functions of y. Disc method: function rotated about x-axis. Disc method (rotating f(x) about x axis). Volume of a sphere. Disc method with outer and inner function boundaries. Shell method to rotate around y-axis. Disk method: rotating x=f(y) around the y-axis. Shell method around a non-axis line. Shell method around a non-axis line 2. Disk method around x-axis. Generalizing disc method around x-axis. Disc method around y-axis. Disc method (washer method) for rotation around x-axis. Generalizing the washer method. Disc method rotation around horizontal line. Washer method rotating around non-axis. Part 2 of washer for non axis rotation. Disc method rotating around vertical line. Calculating integral disc method around vertical line. Washer or ring method for vertical line rotation. Evaluating integral for washer method around vertical line. Shell method for rotating around vertical line. Evaluating integral for shell method example. Shell method for rotating around horizontal line. Shell method with two functions of x. Calculating integral with shell method. Shell method with two functions of y. Part 2 of shell method with 2 functions of y. Disc method: function rotated about x-axis. Disc method (rotating f(x) about x axis). Volume of a sphere. Disc method with outer and inner function boundaries. Shell method to rotate around y-axis. Disk method: rotating x=f(y) around the y-axis. Shell method around a non-axis line. Shell method around a non-axis line 2.

Parameterizing a surface. Surface integrals. Stokes' theorem. Introduction to Parametrizing a Surface with Two Parameters. Determining a Position Vector-Valued Function for a Parametrization of Two Parameters. Partial Derivatives of Vector-Valued Functions. Introduction to the Surface Integral. Example of calculating a surface integral part 1. Example of calculating a surface integral part 2. Example of calculating a surface integral part 3. Surface Integral Example Part 1 - Parameterizing the Unit Sphere. Surface Integral Example Part 2 - Calculating the Surface Differential. Surface Integral Example Part 3 - The Home Stretch. Surface Integral Ex2 part 1 - Parameterizing the Surface. Surface Integral Ex2 part 2 - Evaluating Integral. Surface Integral Ex3 part 1 - Parameterizing the Outside Surface. Surface Integral Ex3 part 2 - Evaluating the Outside Surface. Surface Integral Ex3 part 3 - Top surface. Surface Integral Ex3 part 4 - Home Stretch. Conceptual Understanding of Flux in Three Dimensions. Constructing a unit normal vector to a surface. Vector representation of a Surface Integral. Stokes' Theorem Intuition. Green's and Stokes' Theorem Relationship. Orienting Boundary with Surface. Orientation and Stokes. Conditions for Stokes Theorem. Stokes Example Part 1. Part 2 Parameterizing the Surface. Stokes Example Part 3 - Surface to Double Integral. Stokes Example Part 4 - Curl and Final Answer. Evaluating Line Integral Directly - Part 1. Evaluating Line Integral Directly - Part 2. Stokes' Theorem Proof Part 1. Stokes' Theorem Proof Part 2. Stokes' Theorem Proof Part 3. Stokes' Theorem Proof Part 4. Stokes' Theorem Proof Part 5. Stokes' Theorem Proof Part 6. Stokes' Theorem Proof Part 7. Introduction to Parametrizing a Surface with Two Parameters. Determining a Position Vector-Valued Function for a Parametrization of Two Parameters. Partial Derivatives of Vector-Valued Functions. Introduction to the Surface Integral. Example of calculating a surface integral part 1. Example of calculating a surface integral part 2. Example of calculating a surface integral part 3. Surface Integral Example Part 1 - Parameterizing the Unit Sphere. Surface Integral Example Part 2 - Calculating the Surface Differential. Surface Integral Example Part 3 - The Home Stretch. Surface Integral Ex2 part 1 - Parameterizing the Surface. Surface Integral Ex2 part 2 - Evaluating Integral. Surface Integral Ex3 part 1 - Parameterizing the Outside Surface. Surface Integral Ex3 part 2 - Evaluating the Outside Surface. Surface Integral Ex3 part 3 - Top surface. Surface Integral Ex3 part 4 - Home Stretch. Conceptual Understanding of Flux in Three Dimensions. Constructing a unit normal vector to a surface. Vector representation of a Surface Integral. Stokes' Theorem Intuition. Green's and Stokes' Theorem Relationship. Orienting Boundary with Surface. Orientation and Stokes. Conditions for Stokes Theorem. Stokes Example Part 1. Part 2 Parameterizing the Surface. Stokes Example Part 3 - Surface to Double Integral. Stokes Example Part 4 - Curl and Final Answer. Evaluating Line Integral Directly - Part 1. Evaluating Line Integral Directly - Part 2. Stokes' Theorem Proof Part 1. Stokes' Theorem Proof Part 2. Stokes' Theorem Proof Part 3. Stokes' Theorem Proof Part 4. Stokes' Theorem Proof Part 5. Stokes' Theorem Proof Part 6. Stokes' Theorem Proof Part 7.

Calculating derivatives. Power rule. Product and quotient rules. Chain Rule. Implicit differentiation. Derivatives of common functions. Newton Leibniz and Usain Bolt. Slope of a line secant to a curve. Slope of a secant line example 1. Slope of a secant line example 2. Slope of a secant line example 3. Approximating instantaneous rate of change word problem. Approximating equation of tangent line word problem. Slope of secant lines. Derivative as slope of a tangent line. Tangent slope as limiting value of secant slope example 1. Tangent slope as limiting value of secant slope example 2. Tangent slope as limiting value of secant slope example 3. Tangent slope is limiting value of secant slope. Calculating slope of tangent line using derivative definition. Derivatives 1. The derivative of f(x)=x^2 for any x. Formal and alternate form of the derivative. Formal and alternate form of the derivative for ln x. Formal and alternate form of the derivative example 1. The formal and alternate form of the derivative. Interpreting slope of a curve exercise. Recognizing slope of curves. Calculus: Derivatives 1. Calculus: Derivatives 2. Derivative Intuition Module. Derivative intuition. Graphs of functions and their derivatives example 1. Where a function is not differentiable. Identifying a function's derivative example. Figuring out which function is the the derivative. Graphs of functions and their derivatives. Intuitively drawing the derivative of a function. Intuitively drawing the antiderivative of a function. Visualizing derivatives exercise. Visualizing derivatives. Power Rule. Is the power rule reasonable. Derivative properties and polynomial derivatives. Power rule. Proof: d/dx(x^n). Proof: d/dx(sqrt(x)). Power rule introduction. Derivatives of sin x, cos x, tan x, e^x and ln x. Special derivatives. Chain rule introduction. Chain rule definition and example. Chain rule with triple composition. Chain rule for derivative of 2^x. Derivative of log with arbitrary base. Chain rule 1. Extreme Derivative Word Problem (advanced). The Chain Rule. Chain Rule Examples. Even More Chain Rule. More examples using multiple rules. Derivatives of sin x, cos x, tan x, e^x and ln x. Special derivatives. Applying the product rule for derivatives. Product rule for more than two functions. Product rule. Quotient rule from product rule. Quotient rule for derivative of tan x. Quotient rule. Using the product rule and the chain rule. Product Rule. Quotient rule and common derivatives. Equation of a tangent line. Implicit differentiation. Showing explicit and implicit differentiation give same result. Implicit derivative of (x-y)^2 = x + y + 1. Implicit derivative of y = cos(5x - 3y). Implicit derivative of (x^2+y^2)^3 = 5x^2y^2. Finding slope of tangent line with implicit differentiation. Implicit derivative of e^(xy^2) = x - y. Derivative of x^(x^x). Implicit differentiation. Proof: d/dx(ln x) = 1/x. Proof: d/dx(e^x) = e^x. Proofs of derivatives of ln(x) and e^x. Newton Leibniz and Usain Bolt. Slope of a line secant to a curve. Slope of a secant line example 1. Slope of a secant line example 2. Slope of a secant line example 3. Approximating instantaneous rate of change word problem. Approximating equation of tangent line word problem. Slope of secant lines. Derivative as slope of a tangent line. Tangent slope as limiting value of secant slope example 1. Tangent slope as limiting value of secant slope example 2. Tangent slope as limiting value of secant slope example 3. Tangent slope is limiting value of secant slope. Calculating slope of tangent line using derivative definition. Derivatives 1. The derivative of f(x)=x^2 for any x. Formal and alternate form of the derivative. Formal and alternate form of the derivative for ln x. Formal and alternate form of the derivative example 1. The formal and alternate form of the derivative. Interpreting slope of a curve exercise. Recognizing slope of curves. Calculus: Derivatives 1. Calculus: Derivatives 2. Derivative Intuition Module. Derivative intuition. Graphs of functions and their derivatives example 1. Where a function is not differentiable. Identifying a function's derivative example. Figuring out which function is the the derivative. Graphs of functions and their derivatives. Intuitively drawing the derivative of a function. Intuitively drawing the antiderivative of a function. Visualizing derivatives exercise. Visualizing derivatives. Power Rule. Is the power rule reasonable. Derivative properties and polynomial derivatives. Power rule. Proof: d/dx(x^n). Proof: d/dx(sqrt(x)). Power rule introduction. Derivatives of sin x, cos x, tan x, e^x and ln x. Special derivatives. Chain rule introduction. Chain rule definition and example. Chain rule with triple composition. Chain rule for derivative of 2^x. Derivative of log with arbitrary base. Chain rule 1. Extreme Derivative Word Problem (advanced). The Chain Rule. Chain Rule Examples. Even More Chain Rule. More examples using multiple rules. Derivatives of sin x, cos x, tan x, e^x and ln x. Special derivatives. Applying the product rule for derivatives. Product rule for more than two functions. Product rule. Quotient rule from product rule. Quotient rule for derivative of tan x. Quotient rule. Using the product rule and the chain rule. Product Rule. Quotient rule and common derivatives. Equation of a tangent line. Implicit differentiation. Showing explicit and implicit differentiation give same result. Implicit derivative of (x-y)^2 = x + y + 1. Implicit derivative of y = cos(5x - 3y). Implicit derivative of (x^2+y^2)^3 = 5x^2y^2. Finding slope of tangent line with implicit differentiation. Implicit derivative of e^(xy^2) = x - y. Derivative of x^(x^x). Implicit differentiation. Proof: d/dx(ln x) = 1/x. Proof: d/dx(e^x) = e^x. Proofs of derivatives of ln(x) and e^x.

Calculus can be thought of as the mathematics of CHANGE. Because everything in the world is changing, calculus helps us track those changes. Algebra, by contrast, can be thought of as dealing with a large set of numbers that are inherently CONSTANT. Solving an algebra problem, like y = 2x + 5, merely produces a pairing of two predetermined numbers, although an infinite set of pairs. Algebra is even useful in rate problems, such as calculating how the money in your savings account increases because of the interest rate R, such as Y = X0+Rt, where t is elapsed time and X0 is the initial deposit. With compound interest, things get complicated for algebra, as the rate R is itself a function of time with Y = X0 + R(t)t. Now we have a rate of change which itself is changing. Calculus came to the rescue, as Isaac Newton introduced the world to mathematics specifically designed to handle those things that change. Calculus is among the most important and useful developments of human thought. Even though it is over…

This course is the second installment of Single-Variable Calculus. In Part I (MA101) [1], we studied limits, derivatives, and basic integrals as a means to understand the behavior of functions. In this course (Part II), we will extend our differentiation and integration abilities and apply the techniques we have learned. Additional integration techniques, in particular, are a major part of the course. In Part I, we learned how to integrate by various formulas and by reversing the chain rule through the technique of substitution. In Part II, we will learn some clever uses of substitution, how to reverse the product rule for differentiation through a technique called integration by parts, and how to rewrite trigonometric and rational integrands that look impossible into simpler forms. Series, while a major topic in their own right, also serve to extend our integration reach: they culminate in an application that lets you integrate almost any function you’d like. Integration allows us to calculat…

Multivariable Calculus is an expansion of Single-Variable Calculus in that it extends single variable calculus to higher dimensions. You may find that these courses share many of the same basic concepts, and that Multivariable Calculus will simply extend your knowledge of functions to functions of several variables. The transition from single variable relationships to many variable relationships is not as simple as it may seem; you will find that multi-variable functions, in some cases, will yield counter-intuitive results. The structure of this course very much resembles the structure of Single-Variable Calculus I and II. We will begin by taking a fresh look at limits and continuity. With functions of many variables, you can approach a limit from many different directions. We will then move on to derivatives and the process by which we generalize them to higher dimensions. Finally, we will look at multiple integrals, or integration over regions of space as opposed to intervals. The goal of Mu…

Prepare for the College Mathematics CLEP Exam through Education Portal's brief video lessons on mathematics. This course covers topics ranging from real number systems to probability and statistics. You'll learn to use the midpoint and distance formulas, graph inequalities and multiply binomials. You'll also explore the properties of various shapes and learn to determine their area and perimeter. Our lessons are taught by professional educators with experience in mathematics. In addition to designing the videos in this course, these educators have developed written transcripts and self-assessment quizzes to round out your learning experience.

This course is the second installment of Single-Variable Calculus. In Part I (MA101 [1]), we studied limits, derivatives, and basic integrals as a means to understand the behavior of functions. While this end goal remains the same, we will now focus on adapting what we have learned to applications. By the end of this course, you should have a solid understanding of functions and how they behave. You should also be able to apply the concepts we have learned in both Parts I and II of Single-Variable Calculus to a variety of situations. We will begin by revisiting and building upon what we know about the integral. We will then explore the mathematical applications of integration before delving into the second major topic of this course: series. The course will conclude with an introduction to differential equations. [1] http:///courses/ma101/…

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