Lesson 19: The Mean Value Theorem (slides)

1. The chain rule describes how to take the derivative of a function composed of other functions. It states that if z = f(x, y) and x and y are functions of t, then the derivative of z with respect to t is the sum of the partial derivatives multiplied by the derivatives of x and y with respect to t. 2. The chain rule is applied to find the derivative of two example functions, w = xy and w = xy + z, with respect to t along given paths for x, y, and z in terms of t. 3. The chain rule is generalized to the case of a function u of n variables, where each variable is a function of m other variables

The document introduces approximations of the area under a curve using Riemann sums with rectangles. It explains left and right Riemann sums, showing how to calculate them by dividing the area into equal subintervals and determining the height of each rectangle. While Riemann sums provide approximations, taking the widths of the subintervals to zero provides the exact area under a curve, as shown in a video clip about the concept. Riemann sums have applications in economics for determining consumer surplus and in science for modeling phenomena like blood flow.

The Rational Zero Theorem provides a method to determine all possible rational zeros of a polynomial function. It states that if p/q is a rational zero, then p is a factor of the constant term and q is a factor of the leading coefficient. Descartes' Rule of Signs can be used to determine the maximum number of positive and negative real zeros by counting the variations in sign of the polynomial function and its substitution of -x. It provides bounds on the number of positive and negative real zeros that are either the number of variations in sign or less by an even integer. The example demonstrates applying these methods to determine all 16 possible rational zeros and the bounds of 0 positive and either 3 or 1 negative real zeros for the given polynomial.

critical points/ stationary points , turning points,Increasing, decreasing functions, absolute maxima & Minima, Local Maxima & Minima , convex upward & convex downward - first & second derivative tests.

This document discusses finding the maximum and minimum values of functions, as well as points of inflection. It provides instructions on how to find the coordinates of turning points by setting the derivative of the function equal to zero and solving. It also discusses how to determine if a turning point is a maximum or minimum by taking the second derivative and checking if it is positive or negative. The document concludes by giving examples of how to apply these concepts to optimization word problems involving areas, volumes, or other quantities that need to be maximized or minimized under certain constraints.

The document provides information about Lagrangian interpolation, including: 1. It introduces Lagrangian interpolation as a method to find the value of a function at a discrete point using a polynomial that passes through known data points. 2. It gives the formula for the Lagrangian interpolating polynomial and provides an example of using it to find the velocity of a rocket at a certain time. 3. It discusses using higher order polynomials for interpolation, providing another example that calculates velocity using quadratic and cubic polynomials.

1) Ordinary differential equations relate a dependent variable to one or more independent variables by means of differential coefficients. They can be classified based on order, degree, whether they are linear or non-linear, and type (exact, separable variables, homogeneous). 2) First order differential equations can sometimes be solved by separation of variables, or by finding an integrating factor. Homogeneous equations can be transformed by substitution. 3) Second order linear differential equations can be reduced to a system of two first order equations. The complementary function and particular solutions combine to form the general solution. Unequal or equal roots of the characteristic equation determine the form of the complementary function.

An ellipse is defined algebraically as the set of all points where the sum of the distances to two fixed points (the foci) is a constant. Geometrically, an ellipse can be constructed by stretching a circle: using a piece of string fixed at both ends (the foci) and tracing the path of a pencil as it is moved around so that the total length of string remains constant. The standard equation of an ellipse is (x-h)2/a2 + (y-k)2/b2 = 1, where (h,k) is the center and a and b are the lengths of the semi-major and semi-minor axes. To graph an ellipse, one plots

1. The rational function f(x) = 1/x^2 has a vertical asymptote at x = 2 because the denominator is 0 at that point. 2. As x values approach 2 from either side, y values approach positive or negative infinity, respectively. 3. The graph gets closer and closer to the vertical line x = 2 but never touches it. This line is called a vertical asymptote. 4. The graph also has a horizontal asymptote at y = 0, as y values approach 0 as x values increase or decrease indefinitely.

The document discusses closures of relations, including reflexive closure and symmetric closure. It provides definitions and theorems related to closures. It also uses an example to illustrate finding the reflexive closure and symmetric closure of a relation. Additionally, it covers topics like paths in directed graphs, shortest paths, and transitive closure. It includes an example of calculating the transitive closure of a relation by finding its zero-one matrix.

The document discusses the Fundamental Theorem of Calculus, which has two parts. The first part states that if a function f is continuous on an interval, then the derivative of the integral of f is equal to f. This is proven using Riemann sums. The second part relates the integral of a function f to the integral of its derivative F'. Examples are provided to illustrate how the area under a curve relates to these concepts.

This document discusses several numerical analysis methods for finding roots of equations or solving systems of equations. It describes the bisection method for finding roots of continuous functions, the method of false positions for approximating roots between two values with opposite signs of a function, Gauss elimination for transforming a system of equations into triangular form, Gauss-Jordan method which further eliminates variables in equations below, and iterative methods which find solutions through successive approximations rather than direct computation.

This document defines metric spaces and discusses their basic properties. It begins by defining what a metric is and what constitutes a metric space. It provides some basic examples of metrics, such as the discrete metric and p-norm metrics. It then discusses metric topologies, defining open and closed balls and showing that the collection of open sets forms a topology. It also introduces the concept of topologically equivalent metrics.

1) The document defines partial derivatives of functions of multiple variables and discusses notation for partial derivatives. 2) It describes calculating a partial derivative by treating all other variables as constants and taking the ordinary derivative with respect to the variable of interest. 3) Higher order partial derivatives are discussed, which are partial derivatives of previously obtained partial derivatives. Schwartz' theorem states that for functions of two variables, mixed partial derivatives are equal if one of them exists and is continuous.

This document defines key concepts related to straight lines, including their various forms of equations, slope, intercepts, and how to calculate them. It discusses the general, standard, point-slope, and intercept forms of linear equations. It also explains how to find the slope between two points using the slope formula, and how to determine the x-intercept and y-intercept from a line's equation. Examples are provided to illustrate these concepts.

The document discusses inverse functions, including: - An inverse function undoes the output of the original function by relating the input and output variables. - For a function to have an inverse, it must be one-to-one so that each output is paired with a unique input. - To find the inverse of a function, swap the input and output variables and isolate the new output variable.

Integration is used in physics to determine rates of change and distances given velocities. Numerical integration is required when the antiderivative is unknown. It involves approximating the definite integral of a function as the area under its curve between bounds. The Trapezoidal Rule approximates this area using straight lines between points, while Simpson's Rule uses quadratic or cubic functions, achieving greater accuracy with fewer points. Both methods involve dividing the area into strips and summing their widths multiplied by the function values at strip points.

This geometry lesson covers properties of isosceles and equilateral triangles. It defines key terms like base angles theorem and equilateral triangle corollary. Examples are provided to find missing angle and side measures in various triangles. Students are guided through practice problems to apply the concepts before independent practice questions to find values of variables and perimeters of triangles.

The derivative of a composition of functions is the product of the derivatives of those functions. This rule is important because compositions are so powerful.

The document discusses limits and the limit laws. It introduces the concept of a limit using an "error-tolerance" game. It then proves some basic limits, such as the limit of x as x approaches a equals a, and the limit of a constant c equals c. It also proves the limit laws, such as the fact that limits can be combined using arithmetic operations and the rules for limits of quotients and roots.

This document provides an outline for a calculus lecture on basic differentiation rules. It includes objectives to understand key rules like the constant multiple rule, sum rule, and derivatives of sine and cosine. Examples are worked through to find the derivatives of functions like squaring, cubing, square root, and cube root using the definition of the derivative. Graphs and properties of derived functions are also discussed.

Many functions in nature are described as the rate of change of another function. The concept is called the derivative. Algebraically, the process of finding the derivative involves a limit of difference quotients.

Many problems in nature are expressible in terms of a certain differential equation that has a solution in terms of exponential functions. We look at the equation in general and some fun applications, including radioactivity, cooling, and interest.

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.

There are various reasons why we would want to find the extreme (maximum and minimum values) of a function. Fermat's Theorem tells us we can find local extreme points by looking at critical points. This process is known as the Closed Interval Method.

Infinity is a complicated concept, but there are rules for dealing with both limits at infinity and infinite limits.

The Taylor series provides a means to approximate a function value at one point based on the function value and its derivatives at another known point. It states that any smooth function can be approximated as a polynomial. The Taylor series expansion allows estimating the value of a function like x^100 at a point like x=20 by using the known value and derivatives of the function at another point, like x=1. Increasing the order of the Taylor series approximation or decreasing the step size between points improves the accuracy of the approximation.

The line tangent to a curve is also the line which best "fits" the curve near that point. So derivatives can be used for approximating complicated functions with simple linear ones. Differentials are another set of notation for the same problem.

The document discusses an introductory calculus class and provides announcements about homework due dates and a student survey. It also outlines guidelines for written homework assignments, the grading rubric, and examples of what to include and avoid in written work. The document aims to provide students information about course policies and expectations for written assignments.

The product rule is generally better because: - It is more systematic and avoids mistakes from expanding products - It works for any differentiable functions u and v, not just polynomials - It provides insight into the structure of the derivative that direct computation does not So in this example, using the product rule is preferable to direct multiplication.

1) The document provides an overview of continuity, including defining continuity as a function having a limit equal to its value at a point. 2) It discusses several theorems related to continuity, such as the sum of continuous functions being continuous and various trigonometric, exponential, and logarithmic functions being continuous on their domains. 3) The document also covers inverse trigonometric functions and their domains of continuity.

Using implicit differentiation we can treat relations which are not quite functions like they were functions. In particular, we can find the slopes of lines tangent to curves which are not graphs of functions.

We cover the inverses to the trigonometric functions sine, cosine, tangent, cotangent, secant, cosecant, and their derivatives. The remarkable fact is that although these functions and their inverses are transcendental (complicated) functions, the derivatives are algebraic functions. Also, we meet my all-time favorite function: arctan.

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.

This document provides an overview of different types of functions including: linear, polynomial, rational, power, trigonometric, and exponential functions. It discusses representing functions verbally, numerically, visually, and symbolically. Key topics covered include transformations of functions through shifting graphs vertically and horizontally, as well as composing multiple functions.

The function is the fundamental unit in calculus. There are many ways to describe functions: with words, pictures, symbols, or numbers.

This document discusses the mean value theorem and continuity in calculus. It defines Rolle's theorem, which states that if a function is continuous on a closed interval and differentiable on the open interval, and if the function is equal at the endpoints, then its derivative must be equal to zero for at least one value between the endpoints. It then uses Rolle's theorem to prove the mean value theorem, which states that the rate of change of a function over an interval is equal to the derivative of the function at some value between the endpoints. Finally, it introduces the Cauchy mean value theorem, which relates the rates of change of two functions over an interval to their derivatives at some interior point.

The document is a lecture on derivatives and the shapes of curves. It covers the mean value theorem, testing for monotonicity using the first derivative test, and finding intervals of monotonicity. It also discusses concavity and the second derivative test. Examples are provided to demonstrate how to find the intervals where a function is increasing or decreasing using the first derivative test.

We discuss the ideas of monotonicity (increasing or decreasing) and concavity (up or down) of a function. Because of the Mean Value Theorem, we can determine these characteristics using derivatives.

The Mean Value Theorem is the most important theorem in calculus! It allows us to infer information about a function from information about its derivative. Such as: a function whose derivative is zero must be a constant function.

1. Rolle's theorem states that if a function f(x) is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), and f(a) = f(b), then there exists at least one number c in (a,b) where the derivative f'(c) = 0. 2. The mean value theorem states that if a function f(x) is continuous on a closed interval [a,b] and differentiable on the open interval (a,b), there exists a number c in (a,b) such that f'(c) = (f(b) - f(a))/(b - a).