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This document discusses solving for the area under the curve of a function algebraically without using a graphing calculator. It first finds the derivative of the function f(x) = ln(2x^2 + 3) using the chain rule. It then takes the integral of the derivative from 0 to 4 to find the area beneath the curve, which equals 2.4567.
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Solution4
The document discusses deriving the derivative of a quotient function using the quotient rule. It shows applying the quotient rule to the function f(x)=(x^2+3x+2)/(x^2+2) to get the derivative f'(x)=-3(x^2-2)(x^4+4x^2+4)/(x^4+4x^2+4). It then takes the limit as x approaches infinity of this function, simplifying it to lim f(x)=1 as x->∞, demonstrating that the limit exists.
The chain rule
The document discusses the chain rule and how to use it to differentiate and integrate composite functions. The chain rule states that if h(x) = g(f(x)), then h'(x) = g'(f(x))f'(x). It provides examples of applying the chain rule to differentiate functions like sin(x2 - 4) and integrate functions like ∫(3x2 + 4)3 dx. It also discusses how to integrate functions of the form f'(x)g(f(x)) by recognizing them as derivatives of composite functions.
Chain rule
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
Engineering Mathematics - Total derivatives, chain rule and derivative of imp...
This document discusses the total derivative and methods for finding derivatives of functions with multiple variables.
The total derivative expresses the total differential of a function u with respect to time t as the sum of the partial derivatives of u with respect to each variable x1, x2,...xn, multiplied by the rate of change of that variable with respect to time.
The chain rule is used to take derivatives of composite functions, where the output of one function is an input to another. The derivative is expressed as the product of the partial derivatives of each nested function.
Derivatives can also be taken for implicit functions, where not all variables can be solved for explicitly. The derivative of one variable with respect to another in an
4.1 the chain rule
The document discusses the chain rule for taking derivatives of composite functions. It explains that the chain rule allows you to break down composite functions into their constituent parts. The chain rule states that the derivative of the outside function is the derivative of the inside function multiplied by the derivative of the inside with respect to the outside. The document provides an example of using the chain rule to take the derivative of a polynomial function composed of simpler functions. It also notes that the general power rule can be used to take derivatives when the outside function is a power and the inside is a differentiable function.
Chain Rule
The document discusses the chain rule, which is used to find the derivative of composite functions. It provides examples of applying the chain rule to functions of the form f(g(x)) by taking the derivative of the outside function with respect to the inside function, and multiplying by the derivative of the inside function with respect to x. The chain rule can be used repeatedly when a function is composed of multiple nested functions. Derivative formulas themselves incorporate the chain rule. The chain rule is essential for finding derivatives and is the most common mistake made by students on tests.
Implicit differentiation
To summarize:
1) To find the derivative of an implicit function y=y(x) defined by an equation F(x,y)=0, take the derivative of both sides with respect to x.
2) This will give a new equation involving x, y, and dy/dx that can be solved for dy/dx.
3) The examples show applying this process to find derivatives and tangent lines for various implicit equations.
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Solution4
The document discusses deriving the derivative of a quotient function using the quotient rule. It shows applying the quotient rule to the function f(x)=(x^2+3x+2)/(x^2+2) to get the derivative f'(x)=-3(x^2-2)(x^4+4x^2+4)/(x^4+4x^2+4). It then takes the limit as x approaches infinity of this function, simplifying it to lim f(x)=1 as x->∞, demonstrating that the limit exists.
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The document discusses the chain rule and how to use it to differentiate and integrate composite functions. The chain rule states that if h(x) = g(f(x)), then h'(x) = g'(f(x))f'(x). It provides examples of applying the chain rule to differentiate functions like sin(x2 - 4) and integrate functions like ∫(3x2 + 4)3 dx. It also discusses how to integrate functions of the form f'(x)g(f(x)) by recognizing them as derivatives of composite functions.
Chain rule
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
Engineering Mathematics - Total derivatives, chain rule and derivative of imp...
This document discusses the total derivative and methods for finding derivatives of functions with multiple variables.
The total derivative expresses the total differential of a function u with respect to time t as the sum of the partial derivatives of u with respect to each variable x1, x2,...xn, multiplied by the rate of change of that variable with respect to time.
The chain rule is used to take derivatives of composite functions, where the output of one function is an input to another. The derivative is expressed as the product of the partial derivatives of each nested function.
Derivatives can also be taken for implicit functions, where not all variables can be solved for explicitly. The derivative of one variable with respect to another in an
4.1 the chain rule
The document discusses the chain rule for taking derivatives of composite functions. It explains that the chain rule allows you to break down composite functions into their constituent parts. The chain rule states that the derivative of the outside function is the derivative of the inside function multiplied by the derivative of the inside with respect to the outside. The document provides an example of using the chain rule to take the derivative of a polynomial function composed of simpler functions. It also notes that the general power rule can be used to take derivatives when the outside function is a power and the inside is a differentiable function.
Chain Rule
The document discusses the chain rule, which is used to find the derivative of composite functions. It provides examples of applying the chain rule to functions of the form f(g(x)) by taking the derivative of the outside function with respect to the inside function, and multiplying by the derivative of the inside function with respect to x. The chain rule can be used repeatedly when a function is composed of multiple nested functions. Derivative formulas themselves incorporate the chain rule. The chain rule is essential for finding derivatives and is the most common mistake made by students on tests.
Implicit differentiation
To summarize:
1) To find the derivative of an implicit function y=y(x) defined by an equation F(x,y)=0, take the derivative of both sides with respect to x.
2) This will give a new equation involving x, y, and dy/dx that can be solved for dy/dx.
3) The examples show applying this process to find derivatives and tangent lines for various implicit equations.
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The document discusses the chain rule and how to use it to find derivatives of more complex equations. It provides examples of using the chain rule to take derivatives of functions involving exponents, trigonometric functions, radicals, and combinations of these. Key steps include identifying the inner and outer functions, taking the derivative of the inner function, and plugging into the chain rule formula. The document also contrasts using the chain rule method versus the inside-outside method for some problems.
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The document discusses applying the chain rule to find derivatives of composite functions. It provides examples of identifying composite functions and using the chain rule formula to take derivatives. It lists practice problems for students to find derivatives of various composite functions using the chain rule, including functions with exponents like (x3 + 1)8, as well as trigonometric functions like y = sin(4x).
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- The quotient rule is provided but not explained for finding the derivative of a function that is a quotient of two other functions.
Multiple ppt
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Solution2
The document discusses differentiating the area formula for a circle to find the rate of change of the radius with respect to time. It uses the given information that the area is 5 cm^2 and the radius is 45 cm. It determines that the rate of change of the radius is 1/90π cm/sec. It then derives the equation of a circle with radius 45 cm centered at the origin and takes the derivative to find the derivative formula.
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2) Solving exponential and natural logarithm equations by rewriting one in terms of the other with base e.
3) The derivatives of exponential functions with bases including b, e, and e to the power of u, and examples of differentiating specific exponential functions.
دالة الاكسبونيشل الرياضية والفريدة من نوعها و كيفية استخدامهال
The document discusses derivatives of exponential and logarithmic functions. It begins by introducing exponential functions and their role in modeling growth and decay. It then derives the derivative of the exponential function ex as ex and the derivative of the natural logarithm function ln x as 1/x. Finally, it generalizes these results to derivatives of exponential and logarithmic functions with arbitrary bases b > 0 and b ≠ 1.
Derivatives
This document provides information about derivatives and their applications:
1. It defines the derivative as the limit of the difference quotient, and explains how to calculate derivatives using first principles. It also covers rules for finding derivatives of sums, products, quotients, exponentials, and logarithmic functions.
2. Higher order derivatives are introduced, with examples of how to take second and third derivatives.
3. Applications of derivatives like finding velocity and acceleration from a position-time function are demonstrated. Maximum/minimum values and how to find local and absolute extrema are also discussed with an example.
.
The document discusses functions and their characteristics including domain, range, and inverse functions. It provides examples of evaluating, adding, multiplying, and dividing functions. It also covers compound functions, using graphs to determine domain and range, and recognizing functions using the vertical line test. Logarithms are also briefly introduced.
119 Powerpoint 2.2
This document discusses functions and how to work with them. It defines functions and explains that a function relates each input to a single unique output. It provides examples of relations that are and are not functions. It also discusses how to evaluate functions, find domains, use function notation like f(x), and calculate difference quotients.
Wati lg
The document discusses indefinite integrals and their relationship to derivatives. It states that the indefinite integral is the inverse of the derivative, so finding the integral of a function means finding its antiderivative. The document provides examples of using formulas to find the antiderivatives of functions like 5x^4 by recognizing it as the derivative of x^5. It also demonstrates how integrals follow properties like scalar multiplication and addition/subtraction.
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This document discusses various methods for finding the zeros or roots of polynomial functions, including factoring, factor theorem, synthetic division, and using the principle that every polynomial of degree n has n zeros. It provides examples of finding the zeros of polynomials by factorization, using a given zero to find other zeros through synthetic division, and identifying which numbers are zeros of various polynomials. Exercises are included for students to practice finding remaining zeros given one zero and identifying polynomial factors.
Calculus - Functions Review
This document provides an overview of functions and function notation that will be used in Calculus. It defines a function as an equation where each input yields a single output. Examples demonstrate determining if equations are functions and evaluating functions using function notation. The key concepts of domain and range of a function are explained. The document concludes by finding the domains of various functions involving fractions, radicals, and inequalities.
Functions
A function transforms inputs (domain) to outputs (range). The inverse of a function reverses the inputs and outputs, so the domain of the inverse is the original function's range, and the range of the inverse is the original function's domain. To have an inverse, a function must be one-to-one, meaning each input maps to a single unique output. The graph of an inverse is a reflection of the original function's graph across the line y=x.
Lesson 2_Eval Functions.pptx
This document provides an overview of evaluating functions. It begins with defining a function and function notation. It then discusses the difference between solving a function and evaluating a function. The main steps for evaluating a function are outlined as: 1) write the original function, 2) substitute the given value for the variable, and 3) simplify using order of operations. Several examples are provided of evaluating different types of functions at given values by following these steps. The document concludes by restating the key steps for evaluating functions.
chapter3.ppt
The document discusses exponential and logarithmic functions. It defines exponential functions as functions of the form f(x) = bx, where b is the base. It provides examples of graphs of exponential functions with different bases. It then introduces logarithmic functions as the inverse of exponential functions, where logarithms are defined as logbx = y if x = by. It provides properties and examples involving logarithmic functions.
The Table Method for Derivatives
This document contains a couple of problems from the textbook for Calc 1, Boise State, Fall 2014. It also explains the table method for evaluating complicated derivatives
Exponential_Functions.ppt
This document discusses exponential functions. It defines exponential functions as having the form f(x)=Nx where N is greater than 0 and the variable is in the exponent. It covers properties of exponents used to simplify expressions and solve equations. Graphs of common exponential functions like f(x)=2x and f(x)=ex are shown and their key characteristics described. Exponential functions model growth and decay in real world situations.
6.3 integration by substitution
This document discusses integration by substitution. It provides an example of recognizing a composite function and rewriting the integral in terms of the inside and outside functions. Specifically, it shows rewriting the integral of (x2 +1)2x dx as the integral of the outside function (x2 + 1) with the inside function (x) plugged in, plus a constant. It then provides additional practice problems applying the technique of substitution to rewrite integrals in terms of u-substitutions.
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This document provides an introduction to symbolic math in MATLAB. It discusses differentiation and integration of functions using symbolic operators. Differentiation is defined as finding the rate of change of a function with respect to a variable. Integration finds the original function given its derivative. The document provides examples of differentiating and integrating simple functions in MATLAB's symbolic toolbox and exercises for the reader to practice.
Chapter 1 (math 1)
This document provides a summary of precalculus concepts including:
1. Functions and their graphs including function definitions, transformations, combinations, and compositions of functions.
2. Trigonometry including trigonometric functions, graphs of trigonometric functions, and trigonometric identities.
3. Graphs of second-degree equations including circles, parabolas, ellipses, and hyperbolas.
The document contains examples and explanations of key precalculus topics to serve as a review for a Math 131 course. It covers essential functions like polynomials, rational functions, and transcendental functions. It also discusses trigonometric functions and their graphs along with transformations of functions.
Lesson 2 - Functions and their Graphs - NOTES.ppt
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Module II Partition and Generating Function (2).ppt
This document provides an overview of module 2 of the course "Discrete Mathematics for IT". The module covers topics related to relations and functions, including properties of binary relations, closure of relations, Warshall's algorithm, equivalence relations and partitions, partial ordering relations, functions, composition of functions, invertible functions, and the pigeonhole principle. It also discusses discrete numeric functions, generating functions, and linear recurrence relations with constant coefficients.
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دالة الاكسبونيشل الرياضية والفريدة من نوعها و كيفية استخدامهال
دالة الاكسبونيشل الرياضية والفريدة من نوعها و كيفية استخدامهال
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Solution3
- 1. Area Under the Curve Solution
- 2. The first thing to do in order to solve this problem, is to solve for the derivative of the function… ln (2x 2 + 3)
- 3. This function is a composite function. It means, it has an outer function, ln(x) and an inner function, 2x 2 + 3 To solve this, we need to use the Chain Rule… dy f(g(x)) = f ’(g(x)) g ’(x) dx
- 4. f ‘ (x) = ln (2x 2 + 3) = 1 . 4x 2x 2 + 3 = 4x 2x 2 + 3 -Parent Function -According to the natural log rule, the derivative of ln (x) is 1/x. The derivative of the inner function is 4x according to the power rule of differentiation. -Simplifying the derivative formula
- 5. Now that we have the derivative function, we can find the area under the curve. Since we’re not allowed to use a graphing calculator, we need to find this algebraically…
- 6. Since we already have the antiderivative of the formula, we can just simply use the find the integral from 0 to 4… A = 4x 2x 2 + 3 = ln (2x 2 + 3) = ln (2(4) 2 + 3) - ln (2(0) 2 + 3 = 3.5553 - 1.0986 = 2.4567 0 4 Integral Notation Antidifferentiation Substitution Simplify Area Under the Curve