This document discusses implicit differentiation and finding the slope of tangent lines using implicit differentiation. It begins with an example problem of finding the slope of the tangent line to the curve x^2 + y^2 = 1 at the point (3/5, -4/5). It then explains how to set up and solve the implicit differentiation problem to find the slope. The document emphasizes that even when a relation is not explicitly a function, it can often be treated as locally functional to apply implicit differentiation and find tangent slopes. It provides another example problem and discusses horizontal and vertical tangent lines.
- The product of two differentiable functions is itself differentiable. The derivative of the product of two functions f and g is equal to the derivative of the first function times the second function, plus the derivative of the second function times the first function.
- Gottfried Leibniz originally wrote the formula for the product rule, motivated by the expression (x+dx) (y+dy) - xy. Subtracting dx dy from this expression resulted in the traditional differential form of the product rule.
Solving Systems of Equations using Substitution
Step 1) Solve one equation for one variable.
Step 2) Substitute the expression from Step 1 into the other equation.
Step 3) Solve the resulting equation to find the value of the variable. Step 4) Plug this value back into either original equation to find the value of the other variable. Step 5) Check that the solution satisfies both original equations.
The chain rule helps us find a derivative of a composition of functions. It turns out that it's the product of the derivatives of the composed functions.
The document discusses higher order derivatives. It defines the nth derivative of a function f(x) as f(n)(x). The first example finds the first five derivatives of f(x)=2x^4 - x^3 - 2. The second example finds the first three derivatives of f(x)=-x^2/3. The third example finds the first four derivatives of f(x)=ln(x) and discusses how derivatives of rational functions become more complicated with higher orders. It also provides examples of finding derivatives of other functions like sin(x).
1. The differentiation formulas for the trigonometric functions sin(u), cos(u), tan(u), cot(u), sec(u), and csc(u) are presented in terms of their derivatives du/dx.
2. The derivatives are obtained by applying the basic differentiation formulas for sin(u) and cos(u) along with the quotient, product, and chain rules.
3. Formulas are also provided for deriving the derivatives of inverse trigonometric functions like arcsin, arccos, arctan, etc.
The document defines a polynomial function as a function of the form f(x) = anxn + an-1xn-1 +...+ a0, where n is a nonnegative integer and an, an-1,...a0 are real numbers with an ≠ 0. The degree of a polynomial is the highest exponent of its terms. Examples are provided to illustrate how to determine the degree and number of terms of polynomial functions. The document also asks questions to check understanding of identifying polynomial functions and determining their degree.
3.2 implicit equations and implicit differentiationmath265
The document discusses implicit equations and implicit differentiation. It begins by explaining the difference between explicit and implicit forms of equations, using the example of y=1/x which can be written explicitly as y=1/x or implicitly as xy=1. It then introduces the concept of implicit differentiation, which involves taking the derivative of an implicit equation with respect to x and solving for the derivative of y with respect to x (y’). This allows one to find the slope of the curve at a point, even if the explicit form of the relation between x and y is difficult to determine from the implicit equation.
This document discusses simplifying rational expressions by dividing out common factors, factoring numerators and denominators, and dividing common factors. It provides examples of simplifying various rational expressions step-by-step and explains how to identify excluded values that would make denominators equal to zero.
- The product of two differentiable functions is itself differentiable. The derivative of the product of two functions f and g is equal to the derivative of the first function times the second function, plus the derivative of the second function times the first function.
- Gottfried Leibniz originally wrote the formula for the product rule, motivated by the expression (x+dx) (y+dy) - xy. Subtracting dx dy from this expression resulted in the traditional differential form of the product rule.
Solving Systems of Equations using Substitution
Step 1) Solve one equation for one variable.
Step 2) Substitute the expression from Step 1 into the other equation.
Step 3) Solve the resulting equation to find the value of the variable. Step 4) Plug this value back into either original equation to find the value of the other variable. Step 5) Check that the solution satisfies both original equations.
The chain rule helps us find a derivative of a composition of functions. It turns out that it's the product of the derivatives of the composed functions.
The document discusses higher order derivatives. It defines the nth derivative of a function f(x) as f(n)(x). The first example finds the first five derivatives of f(x)=2x^4 - x^3 - 2. The second example finds the first three derivatives of f(x)=-x^2/3. The third example finds the first four derivatives of f(x)=ln(x) and discusses how derivatives of rational functions become more complicated with higher orders. It also provides examples of finding derivatives of other functions like sin(x).
1. The differentiation formulas for the trigonometric functions sin(u), cos(u), tan(u), cot(u), sec(u), and csc(u) are presented in terms of their derivatives du/dx.
2. The derivatives are obtained by applying the basic differentiation formulas for sin(u) and cos(u) along with the quotient, product, and chain rules.
3. Formulas are also provided for deriving the derivatives of inverse trigonometric functions like arcsin, arccos, arctan, etc.
The document defines a polynomial function as a function of the form f(x) = anxn + an-1xn-1 +...+ a0, where n is a nonnegative integer and an, an-1,...a0 are real numbers with an ≠ 0. The degree of a polynomial is the highest exponent of its terms. Examples are provided to illustrate how to determine the degree and number of terms of polynomial functions. The document also asks questions to check understanding of identifying polynomial functions and determining their degree.
3.2 implicit equations and implicit differentiationmath265
The document discusses implicit equations and implicit differentiation. It begins by explaining the difference between explicit and implicit forms of equations, using the example of y=1/x which can be written explicitly as y=1/x or implicitly as xy=1. It then introduces the concept of implicit differentiation, which involves taking the derivative of an implicit equation with respect to x and solving for the derivative of y with respect to x (y’). This allows one to find the slope of the curve at a point, even if the explicit form of the relation between x and y is difficult to determine from the implicit equation.
This document discusses simplifying rational expressions by dividing out common factors, factoring numerators and denominators, and dividing common factors. It provides examples of simplifying various rational expressions step-by-step and explains how to identify excluded values that would make denominators equal to zero.
Polynomial functions have graphs that are smooth and continuous curves without sharp corners or breaks. Odd-degree polynomials have opposite behavior at their ends, while even-degree polynomials have the same behavior. The x-intercepts of a polynomial function are its zeros, which are found by setting the polynomial equal to 0. A polynomial function's graph can have at most n-1 turning points if the function is of degree n.
The document summarizes different types of derivatives. It discusses simple derivatives where there is one input and output, and defines them. It then discusses implicit derivatives where a relationship between two variables is given and the derivative of one with respect to the other is sought using implicit differentiation. An example finds the derivative of u with respect to v and v with respect to u for the equation 2u^2 - v^3 = 2 - uv. Reciprocal relationships between the derivatives are noted.
2/27/12 Special Factoring - Sum & Difference of Two Cubesjennoga08
The document is about factoring polynomials, specifically factoring the sum and difference of cubes. It provides the formulas for factoring the sum and difference of cubes, along with examples of factoring expressions using those formulas. It also discusses factoring out the greatest common factor from polynomials.
Here are the remainders when dividing the given polynomials by the specified polynomials:
1. The remainder is 0. Therefore, x-1 is a factor of x3+3x2-4x+2.
2. The remainder is 5.
3. The remainder is 0. Therefore, x+2 is a factor of 2x3+5x2+3x+11.
4. The remainder is 4.
5. The remainder is 7.
6. The remainder is 2.
The document defines exponential functions as functions of the form f(x) = bx, where b is a positive constant other than 1. It discusses how the graph of an exponential function depends on whether b is greater than or less than 1. Specifically, if b > 1 the graph increases to the right, and if 0 < b < 1 the graph decreases to the right. The document also covers transformations of exponential functions, including vertical and horizontal shifting, reflecting, and stretching/shrinking. It introduces the special number e, defines it as the limit of (1 + 1/n)n as n approaches infinity, and discusses its role in compound interest formulas.
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.
Factor Theorem and Remainder Theorem. Mathematics10 Project under Mrs. Marissa De Ocampo. Prepared by Danielle Diva, Ronalie Mejos, Rafael Vallejos and Mark Lenon Dacir of 10- Einstein. CNSTHS.
1. There are several methods for factoring polynomials outlined in the document: factoring using the distributive property, factoring the difference of two squares/cubes, factoring a perfect square trinomial, and factoring general trinomials using trial and error or grouping.
2. Factoring trinomials involves determining the signs in the factors based on the signs of the terms, then finding two factors of the constant term that satisfy the middle term.
3. More complex polynomials can be factored by grouping like terms or using special factoring patterns like the difference of squares/cubes.
This document discusses multiplying polynomials. It begins with examples of multiplying monomials by using the properties of exponents. It then covers multiplying a polynomial by a monomial using the distributive property. Examples are provided for multiplying binomials by binomials using both the distributive property and FOIL method. The document concludes by explaining methods for multiplying polynomials with more than two terms, such as using the distributive property multiple times, a rectangle model, or a vertical method similar to multiplying whole numbers.
This document provides instructions and examples for simplifying radical expressions. It defines a radical as a square root expression. It then provides 5 problems with step-by-step explanations and solutions for simplifying radical expressions by finding perfect squares under the radical signs. The problems cover simplifying radicals of variables, combining like radicals, and simplifying fractional radicals.
The document provides an introduction to the precise definition of a limit in calculus. It begins with a heuristic definition of a limit using an error-tolerance game between two players. It then presents Cauchy's precise definition, where the limit is defined using epsilon-delta relationships such that for any epsilon tolerance around the proposed limit L, there exists a corresponding delta tolerance around the point a such that the function values are within epsilon of L when the input values are within delta of a. Examples are provided to illustrate the definition. Pathologies where limits may not exist are also discussed.
I have added to the original presentation in response to one of the comments.... the result of 'x' is correct on slide 7, take a look at the new version of this ppt to clear up any confusion about why...
The document discusses how to graph functions by applying transformations to basic graphs. These transformations include vertical and horizontal shifts which move the graph up/down or left/right, reflections which flip the graph across an axis, and stretching or shrinking which make the graph taller or narrower. By identifying the transformations in an equation, one can graph it by applying the corresponding operations to the original function graph.
This document discusses how to graph linear equations on a coordinate plane using different methods:
1) Using two points from the linear equation to plot and connect those points
2) Using the x-intercept and y-intercept, which are the points where the line crosses the x- and y-axes
3) Using the slope of the line and the y-intercept, where the slope is rise over run and the y-intercept is where the line crosses the y-axis
4) Using the slope of the line and one point from the linear equation
Real-world examples of linear equations include stock market trends and car payment plans.
The document discusses evaluating functions by replacing the variable with a value from the domain and computing the result. It provides examples of evaluating various functions at different values of x. These include evaluating f(x) = 2x + 1 at x = 1.5, q(x) = x^2 - 2x + 2 at x = 2, and other functions at different values. It also discusses cases where a function cannot be evaluated, such as when the value is not in the domain of the function.
This document provides an overview of exponential and logarithmic functions. It defines one-to-one functions and inverse functions. It explains how to find the inverse of a one-to-one function and shows that the inverse of f(x) is f-1(x). Properties of exponential functions like f(x)=ax and logarithmic functions like f(x)=logax are described. The product, quotient, and power rules for logarithms are outlined along with examples. Finally, it discusses how to solve exponential and logarithmic equations using properties of these functions.
This document outlines key concepts and examples for factoring polynomials. It discusses factoring trinomials of the forms x^2 + bx + c, ax^2 + bx + c, and x^2 + bx + c by grouping. Examples are provided to demonstrate finding the greatest common factor of terms, factoring trinomials by finding two numbers whose product and sum meet the given criteria, and checking factoring results using FOIL. Sections cover the greatest common factor, factoring trinomials of different forms, and solving quadratic equations by factoring.
The document discusses the factor theorem and how to determine if a polynomial is a factor of another polynomial. It provides examples of using the factor theorem to show that (x + 1) is a factor of 2x^3 + 5x^2 - 3 and that (x - 2) is a factor of x^4 + x^3 - x^2 - x - 18. It also gives an example of finding a polynomial function given its zeros as -2, 1, -1. The document provides exercises for using the factor theorem to determine unknown values in polynomials.
The document discusses composition of functions and the chain rule. It provides examples of finding the composition of various functions f and g, written as f ∘ g(x) = f(g(x)). It also gives examples of using the chain rule to find the derivative of composite functions.
The document discusses how to find the domain and range of ordered pairs, graphs, and functions. It provides examples of finding the domain, which is the set of first coordinates for ordered pairs or the values of x included in the solution set for graphs, and the range, which is the set of second coordinates for ordered pairs or the values of y included in the solution set for graphs. The document gives the domain and range for several examples of ordered pairs, graphs and functions.
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.
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.
Polynomial functions have graphs that are smooth and continuous curves without sharp corners or breaks. Odd-degree polynomials have opposite behavior at their ends, while even-degree polynomials have the same behavior. The x-intercepts of a polynomial function are its zeros, which are found by setting the polynomial equal to 0. A polynomial function's graph can have at most n-1 turning points if the function is of degree n.
The document summarizes different types of derivatives. It discusses simple derivatives where there is one input and output, and defines them. It then discusses implicit derivatives where a relationship between two variables is given and the derivative of one with respect to the other is sought using implicit differentiation. An example finds the derivative of u with respect to v and v with respect to u for the equation 2u^2 - v^3 = 2 - uv. Reciprocal relationships between the derivatives are noted.
2/27/12 Special Factoring - Sum & Difference of Two Cubesjennoga08
The document is about factoring polynomials, specifically factoring the sum and difference of cubes. It provides the formulas for factoring the sum and difference of cubes, along with examples of factoring expressions using those formulas. It also discusses factoring out the greatest common factor from polynomials.
Here are the remainders when dividing the given polynomials by the specified polynomials:
1. The remainder is 0. Therefore, x-1 is a factor of x3+3x2-4x+2.
2. The remainder is 5.
3. The remainder is 0. Therefore, x+2 is a factor of 2x3+5x2+3x+11.
4. The remainder is 4.
5. The remainder is 7.
6. The remainder is 2.
The document defines exponential functions as functions of the form f(x) = bx, where b is a positive constant other than 1. It discusses how the graph of an exponential function depends on whether b is greater than or less than 1. Specifically, if b > 1 the graph increases to the right, and if 0 < b < 1 the graph decreases to the right. The document also covers transformations of exponential functions, including vertical and horizontal shifting, reflecting, and stretching/shrinking. It introduces the special number e, defines it as the limit of (1 + 1/n)n as n approaches infinity, and discusses its role in compound interest formulas.
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.
Factor Theorem and Remainder Theorem. Mathematics10 Project under Mrs. Marissa De Ocampo. Prepared by Danielle Diva, Ronalie Mejos, Rafael Vallejos and Mark Lenon Dacir of 10- Einstein. CNSTHS.
1. There are several methods for factoring polynomials outlined in the document: factoring using the distributive property, factoring the difference of two squares/cubes, factoring a perfect square trinomial, and factoring general trinomials using trial and error or grouping.
2. Factoring trinomials involves determining the signs in the factors based on the signs of the terms, then finding two factors of the constant term that satisfy the middle term.
3. More complex polynomials can be factored by grouping like terms or using special factoring patterns like the difference of squares/cubes.
This document discusses multiplying polynomials. It begins with examples of multiplying monomials by using the properties of exponents. It then covers multiplying a polynomial by a monomial using the distributive property. Examples are provided for multiplying binomials by binomials using both the distributive property and FOIL method. The document concludes by explaining methods for multiplying polynomials with more than two terms, such as using the distributive property multiple times, a rectangle model, or a vertical method similar to multiplying whole numbers.
This document provides instructions and examples for simplifying radical expressions. It defines a radical as a square root expression. It then provides 5 problems with step-by-step explanations and solutions for simplifying radical expressions by finding perfect squares under the radical signs. The problems cover simplifying radicals of variables, combining like radicals, and simplifying fractional radicals.
The document provides an introduction to the precise definition of a limit in calculus. It begins with a heuristic definition of a limit using an error-tolerance game between two players. It then presents Cauchy's precise definition, where the limit is defined using epsilon-delta relationships such that for any epsilon tolerance around the proposed limit L, there exists a corresponding delta tolerance around the point a such that the function values are within epsilon of L when the input values are within delta of a. Examples are provided to illustrate the definition. Pathologies where limits may not exist are also discussed.
I have added to the original presentation in response to one of the comments.... the result of 'x' is correct on slide 7, take a look at the new version of this ppt to clear up any confusion about why...
The document discusses how to graph functions by applying transformations to basic graphs. These transformations include vertical and horizontal shifts which move the graph up/down or left/right, reflections which flip the graph across an axis, and stretching or shrinking which make the graph taller or narrower. By identifying the transformations in an equation, one can graph it by applying the corresponding operations to the original function graph.
This document discusses how to graph linear equations on a coordinate plane using different methods:
1) Using two points from the linear equation to plot and connect those points
2) Using the x-intercept and y-intercept, which are the points where the line crosses the x- and y-axes
3) Using the slope of the line and the y-intercept, where the slope is rise over run and the y-intercept is where the line crosses the y-axis
4) Using the slope of the line and one point from the linear equation
Real-world examples of linear equations include stock market trends and car payment plans.
The document discusses evaluating functions by replacing the variable with a value from the domain and computing the result. It provides examples of evaluating various functions at different values of x. These include evaluating f(x) = 2x + 1 at x = 1.5, q(x) = x^2 - 2x + 2 at x = 2, and other functions at different values. It also discusses cases where a function cannot be evaluated, such as when the value is not in the domain of the function.
This document provides an overview of exponential and logarithmic functions. It defines one-to-one functions and inverse functions. It explains how to find the inverse of a one-to-one function and shows that the inverse of f(x) is f-1(x). Properties of exponential functions like f(x)=ax and logarithmic functions like f(x)=logax are described. The product, quotient, and power rules for logarithms are outlined along with examples. Finally, it discusses how to solve exponential and logarithmic equations using properties of these functions.
This document outlines key concepts and examples for factoring polynomials. It discusses factoring trinomials of the forms x^2 + bx + c, ax^2 + bx + c, and x^2 + bx + c by grouping. Examples are provided to demonstrate finding the greatest common factor of terms, factoring trinomials by finding two numbers whose product and sum meet the given criteria, and checking factoring results using FOIL. Sections cover the greatest common factor, factoring trinomials of different forms, and solving quadratic equations by factoring.
The document discusses the factor theorem and how to determine if a polynomial is a factor of another polynomial. It provides examples of using the factor theorem to show that (x + 1) is a factor of 2x^3 + 5x^2 - 3 and that (x - 2) is a factor of x^4 + x^3 - x^2 - x - 18. It also gives an example of finding a polynomial function given its zeros as -2, 1, -1. The document provides exercises for using the factor theorem to determine unknown values in polynomials.
The document discusses composition of functions and the chain rule. It provides examples of finding the composition of various functions f and g, written as f ∘ g(x) = f(g(x)). It also gives examples of using the chain rule to find the derivative of composite functions.
The document discusses how to find the domain and range of ordered pairs, graphs, and functions. It provides examples of finding the domain, which is the set of first coordinates for ordered pairs or the values of x included in the solution set for graphs, and the range, which is the set of second coordinates for ordered pairs or the values of y included in the solution set for graphs. The document gives the domain and range for several examples of ordered pairs, graphs and functions.
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.
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.
This document provides an explanation of differentiation and examples of calculating limits and derivatives using the first principle definition of the derivative. It begins by defining the limit of a function and providing examples of evaluating limits. It then introduces the concept of the derivative as the slope of the tangent line to a curve and explains how to calculate derivatives using small changes in x and y. The document provides examples of finding derivatives using this first principle definition. It also discusses rules for deriving composite functions and products of polynomials. Exercises are provided throughout for students to practice differentiation.
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.
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.
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 define what it means for a function to have a maximum or minimum value, and explain the Extreme Value Theorem, which indicates these maxima and minima must be there under certain conditions.
Fermat's Theorem says that at differentiable extreme points, the derivative should be zero, and thus we arrive at a technique for finding extrema: look among the endpoints of the domain of definition and the critical points of the function.
There's also a little digression on Fermat's Last theorem, which is not related to calculus but is a big deal in recent mathematical history.
This document discusses limits involving infinity in calculus. It begins with definitions of infinite limits, both positive and negative infinity. It provides examples of common infinite limits, such as the limit of 1/x as x approaches 0. It also discusses vertical asymptotes and rules for infinite limits, such as the limit of a sum being infinity if both terms have infinite limits. The document contains examples evaluating limits at points where functions are not continuous.
The document discusses the chain rule for calculating the derivative of composite functions. It states that if a function F is composed of two differentiable functions g and φ, where F(x) = φ(g(x)), then F is differentiable and its derivative can be found using the chain rule. It asks the reader to identify the inner and outer functions for examples of composite functions.
This document discusses finding the maximum and minimum values of functions. It introduces the Extreme Value Theorem, which states that if a function is continuous on a closed interval, then it attains both a maximum and minimum value on that interval. It also discusses Fermat's Theorem, which relates local extrema of a differentiable function to its derivative. Examples are provided to illustrate these concepts.
Higher order derivatives for N -body simulationsKeigo Nitadori
This document discusses higher order derivatives that are useful for N-body simulations. It presents formulas for calculating higher order derivatives of power functions like y=xn, and applies this to derivatives of gravitational force f=mr-3. Specifically:
1) It derives recursive formulas for calculating higher order derivatives of power functions y=xn in terms of previous derivatives.
2) It applies these formulas to calculate derivatives of the gravitational force f=mr-3 in terms of derivatives of r and q=r-3/2.
3) It also describes an alternative approach by Le Guyader (1993) for calculating derivatives of r and q in terms of dot products of r with itself.
Using the Mean Value Theorem, we can show the a function is increasing on an interval when its derivative is positive on the interval. Changes in the sign of the derivative detect local extrema. We also can use the second derivative to detect concavity and inflection points. This means that the first and second derivative can be used to classify critical points as local maxima or minima
1. The document discusses the chain rule for finding the derivative of a composition of two functions f and g.
2. The chain rule states that the derivative of the composition f(g(x)) is the product of the derivative of the outer function f evaluated at g(x) and the derivative of the inner function g.
3. An example using linear functions shows that the derivative of a composition of two linear functions results in another linear function whose slope is the product of the original slopes.
There's an obvious guess for the rule about the derivative of a product—which happens to be wrong. We explain the correct product rule and why it has to be true, along with the quotient rule. We show how to compute derivatives of additional trigonometric functions and new proofs of the Power Rule.
Implicit differentiation allows us to find slopes of lines tangent to curves that are not graphs of functions. Almost all of the time (yes, that is a mathematical term!) we can assume the curve comprises the graph of a function and differentiate using the chain rule.
Implicit differentiation allows us to find slopes of lines tangent to curves that are not graphs of functions. Almost all of the time (yes, that is a mathematical term!) we can assume the curve comprises the graph of a function and differentiate using the chain rule.
Implicit differentiation allows us to find slopes of lines tangent to curves that are not graphs of functions. Almost all of the time (yes, that is a mathematical term!) we can assume the curve comprises the graph of a function and differentiate using the chain rule.
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.
Lesson 11: Implicit Differentiation (Section 41 slides)Mel Anthony Pepito
This document provides an overview of implicit differentiation. It begins with a motivating example of finding the slope of the tangent line to the curve x^2 + y^2 = 1 at the point (3/5, -4/5). It is shown that while y is not explicitly defined as a function of x for this curve, it can be treated as such locally using implicit differentiation. The key steps are to take the derivative of the equation with respect to x, which introduces a term for dy/dx, and then solve for dy/dx. This reveals that implicit differentiation allows the derivative of implicitly defined functions to be found.
The document is about implicit differentiation and contains the following:
- It introduces implicit differentiation using the example of finding the slope of the curve x^2 + y^2 = 1 at the point (3/5, -4/5).
- It shows solving this problem explicitly by isolating y and taking the derivative, as well as implicitly by treating y as a function f(x) and differentiating the equation x^2 + f(x)^2 = 1.
- The objectives, outline, and motivation for implicit differentiation are provided to set up the key concepts covered in the section.
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.
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.
1. Graph and analyze the critical points, extrema, inflection points, intervals of increasing/decreasing, and intervals of concave up/down for 10 functions.
2. Review homework on finding derivatives using the definition of the difference quotient and evaluating limits. Find the derivatives of 6 functions.
3. Use implicit differentiation to find the derivative of one function defined implicitly and to find points with tangent lines of slope 1 for another implicit function.
4. Find the second derivatives of two functions.
This document contains multiple definitions and examples related to limits at infinity:
1) It defines limits at infinity and horizontal asymptotes, stating that a limit equals a value L if the function values can be made arbitrarily close to L by taking x sufficiently large or small.
2) Examples show computing limits by factoring out highest degree terms and applying limits laws, such as a limit equaling 1/2.
3) Additional examples provide strategies for determining limits at infinity, such as comparing exponential to geometric growth rates or rationalizing nondeterminate forms.
Lesson 27: Integration by Substitution (worksheet solutions)Matthew Leingang
1) The document provides solutions to 12 integrals using substitution techniques. Substitutions included letting u = functions of x and using trigonometric identities.
2) Integrals were evaluated as both definite integrals, giving numbers, and indefinite integrals, giving general antiderivatives.
3) Multiple methods were sometimes shown, such as using long division or direct substitution to solve one integral.
This document provides an outline and learning objectives for a midterm exam covering vectors and three-dimensional coordinate systems in a Math 21a course. The midterm will cover material up to and including section 11.4 in the textbook. It outlines key topics like three-dimensional coordinate systems, vectors, the dot and cross product, equations of lines and planes, and vector functions. Examples are provided for distance between points in space and rewriting an equation in standard form to identify what surface it represents. Learning objectives are stated for topics like three-dimensional coordinate systems, vectors, and vector addition.
Lesson 21: Curve Sketching II (Section 4 version)Matthew Leingang
The document provides guidance on graphing functions by outlining a checklist process involving 4 steps: 1) finding signs of the function, 2) taking the derivative to determine monotonicity and local extrema, 3) taking the second derivative to determine concavity, and 4) combining the information into a graph. An example function is then graphed in detail to demonstrate the full process.
Lesson 27: Integration by Substitution (worksheet)Matthew Leingang
The document provides a worksheet with 12 integration problems. Students are asked to find both definite and indefinite integrals using substitution techniques. They must determine their own substitutions for some problems and use trigonometric identities in others. The problems cover a range of integral types including rational, radical, and trigonometric functions.
1. The student's name is Azizaty Desiana. She is in class XI IPS 1. Her average score is 41.5. The test had 20 questions.
2. The document provides examples of math problems and questions about functions. It asks for the limit of various expressions as variables approach certain values.
3. The questions are assessing knowledge of functions, inverses, limits, and calculations involving algebraic expressions.
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This document contains notes for a lesson on the chain rule from a Calculus 1 class. It defines the chain rule formula and provides an example of applying the chain rule to find the derivative of a function. It also includes another example problem and its step-by-step solution using the chain rule. The document concludes with a metaphor to help understand applying the chain rule.
Similar to Lesson 12: Implicit Differentiation (20)
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This document discusses electronic grading of paper assessments using PDF forms. Key points include:
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- Methods for stamping completed forms onto scanned documents including using pdftk or overlaying in TikZ.
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NOTE: the final location is section specific. Section 1 (morning) is in SILV 703, Section 11 (afternoon) is in CANT 200
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.
x
What can you say about g? 2 4 6 8 10f
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Lesson 12: Implicit Differentiation
1. Section 2.6
Implicit Differentiation
V63.0121, Calculus I
February 24/25, 2009
Announcements
Midterm in class March 4/5
ALEKS due Friday, 11:59pm
.
.
Image credit: Telstar Logistics
. . . . . .
2. Outline
The big idea, by example
Examples
Vertical and Horizontal Tangents
Chemistry
The power rule for rational powers
. . . . . .
3. y
.
Motivating Example
Problem
Find the slope of the line which is
tangent to the curve
. x
.
2 2
x +y =1
at the point (3/5, −4/5).
. . . . . .
4. y
.
Motivating Example
Problem
Find the slope of the line which is
tangent to the curve
. x
.
2 2
x +y =1
at the point (3/5, −4/5).
. . . . . .
5. y
.
Motivating Example
Problem
Find the slope of the line which is
tangent to the curve
. x
.
2 2
x +y =1
at the point (3/5, −4/5). .
. . . . . .
6. y
.
Motivating Example
Problem
Find the slope of the line which is
tangent to the curve
. x
.
2 2
x +y =1
at the point (3/5, −4/5). .
Solution (Explicit)
√
Isolate: y2 = 1 − x2 =⇒ y = − 1 − x2 . (Why the −?)
. . . . . .
7. y
.
Motivating Example
Problem
Find the slope of the line which is
tangent to the curve
. x
.
2 2
x +y =1
at the point (3/5, −4/5). .
Solution (Explicit)
√
Isolate: y2 = 1 − x2 =⇒ y = − 1 − x2 . (Why the −?)
−2x
dy x
=− √ =√
Differentiate:
2 1 − x2 1 − x2
dx
. . . . . .
8. y
.
Motivating Example
Problem
Find the slope of the line which is
tangent to the curve
. x
.
2 2
x +y =1
at the point (3/5, −4/5). .
Solution (Explicit)
√
Isolate: y2 = 1 − x2 =⇒ y = − 1 − x2 . (Why the −?)
−2x
dy x
=− √ =√
Differentiate:
2 1 − x2 1 − x2
dx
dy 3/5 3/5 3
=√ = =.
Evaluate:
1 − (3/5)
dx x=3/5 4/5 4
2
. . . . . .
9. y
.
Motivating Example
Problem
Find the slope of the line which is
tangent to the curve
. x
.
2 2
x +y =1
at the point (3/5, −4/5). .
Solution (Explicit)
√
Isolate: y2 = 1 − x2 =⇒ y = − 1 − x2 . (Why the −?)
−2x
dy x
=− √ =√
Differentiate:
2 1 − x2 1 − x2
dx
dy 3/5 3/5 3
=√ = =.
Evaluate:
1 − (3/5)
dx x=3/5 4/5 4
2
. . . . . .
10. We know that x2 + y2 = 1 does not define y as a function of x, but
suppose it did.
Suppose we had y = f(x), so that
x2 + (f(x))2 = 1
We could differentiate this equation to get
2x + 2f(x) · f′ (x) = 0
We could then solve to get
x
f′ (x) = −
f(x)
. . . . . .
11. The beautiful fact (i.e., deep theorem) is that this works!
y
.
“Near” most points on
the curve x2 + y2 = 1, the
curve resembles the
graph of a function.
So f(x) is defined “locally”
. x
.
and is differentiable
The chain rule then
applies for this local
.
choice.
. . . . . .
12. The beautiful fact (i.e., deep theorem) is that this works!
y
.
“Near” most points on
the curve x2 + y2 = 1, the
curve resembles the
graph of a function.
So f(x) is defined “locally”
. x
.
and is differentiable
The chain rule then
applies for this local
.
choice.
. . . . . .
13. The beautiful fact (i.e., deep theorem) is that this works!
y
.
“Near” most points on
the curve x2 + y2 = 1, the
curve resembles the
graph of a function.
So f(x) is defined “locally”
. x
.
and is differentiable
The chain rule then
applies for this local
.
choice.
l
.ooks like a function
. . . . . .
14. The beautiful fact (i.e., deep theorem) is that this works!
y
.
“Near” most points on
the curve x2 + y2 = 1, the .
curve resembles the
graph of a function.
So f(x) is defined “locally”
. x
.
and is differentiable
The chain rule then
applies for this local
choice.
. . . . . .
15. The beautiful fact (i.e., deep theorem) is that this works!
y
.
“Near” most points on
the curve x2 + y2 = 1, the .
curve resembles the
graph of a function.
So f(x) is defined “locally”
. x
.
and is differentiable
The chain rule then
applies for this local
choice.
. . . . . .
16. The beautiful fact (i.e., deep theorem) is that this works!
y
.
“Near” most points on
the curve x2 + y2 = 1, the .
curve resembles the
graph of a function. l
.ooks like a function
So f(x) is defined “locally”
. x
.
and is differentiable
The chain rule then
applies for this local
choice.
. . . . . .
17. The beautiful fact (i.e., deep theorem) is that this works!
y
.
“Near” most points on
the curve x2 + y2 = 1, the
curve resembles the
graph of a function.
So f(x) is defined “locally”
. . x
.
and is differentiable
The chain rule then
applies for this local
choice.
. . . . . .
18. The beautiful fact (i.e., deep theorem) is that this works!
y
.
“Near” most points on
the curve x2 + y2 = 1, the
curve resembles the
graph of a function.
So f(x) is defined “locally”
. . x
.
and is differentiable
The chain rule then
applies for this local
choice.
. . . . . .
19. The beautiful fact (i.e., deep theorem) is that this works!
y
.
“Near” most points on
the curve x2 + y2 = 1, the
curve resembles the
graph of a function.
So f(x) is defined “locally”
. . x
.
and is differentiable
.
The chain rule then does not look like a
applies for this local function, but that’s
choice. OK—there are only
two points like this
. . . . . .
20. Problem
Find the slope of the line which is tangent to the curve x2 + y2 = 1 at the
point (3/5, −4/5).
Solution (Implicit, with Leibniz notation)
Differentiate. Remember y is assumed to be a function of x:
dy
2x + 2y = 0,
dx
dy
Isolate :
dx
dy x
=− .
dx y
Evaluate:
dy 3/5 3
= =.
dx ( 3 ,− 4 ) 4/5 4
5 5
. . . . . .
21. Summary
y
.
If a relation is given between x and
y,
“Most of the time” “at most
places” y can be assumed to .
be a function of x
.
we may differentiate the
relation as is
dy
Solving for does give the
dx
slope of the tangent line to
the curve at a point on the
curve.
. . . . . .
23. Outline
The big idea, by example
Examples
Vertical and Horizontal Tangents
Chemistry
The power rule for rational powers
. . . . . .
24. Example
Find the equation of the line
tangent to the curve
.
y2 = x2 (x + 1) = x3 + x2
at the point (3, −6).
.
. . . . . .
25. Example
Find the equation of the line
tangent to the curve
.
y2 = x2 (x + 1) = x3 + x2
at the point (3, −6).
.
Solution
Differentiating the expression implicitly with respect to x gives
3x2 + 2x
dy dy
2y = 3x2 + 2x, so = , and
dx dx 2y
3 · 32 + 2 · 3
dy 11
=− .
=
dx 2(−6) 4
(3,−6)
. . . . . .
26. Example
Find the equation of the line
tangent to the curve
.
y2 = x2 (x + 1) = x3 + x2
at the point (3, −6).
.
Solution
Differentiating the expression implicitly with respect to x gives
3x2 + 2x
dy dy
2y = 3x2 + 2x, so = , and
dx dx 2y
3 · 32 + 2 · 3
dy 11
=− .
=
dx 2(−6) 4
(3,−6)
11
Thus the equation of the tangent line is y + 6 = − (x − 3).
4
. . . . . .
29. Example
Find the horizontal tangent lines to the same curve: y2 = x3 + x2
Solution
We solve for dy/dx = 0:
3x2 + 2x
= 0 =⇒ 3x2 + 2x = 0 =⇒ x(3x + 2) = 0
2y
. . . . . .
30. Example
Find the horizontal tangent lines to the same curve: y2 = x3 + x2
Solution
We solve for dy/dx = 0:
3x2 + 2x
= 0 =⇒ 3x2 + 2x = 0 =⇒ x(3x + 2) = 0
2y
The possible solution x = 0 leads to y = 0, which is not a smooth
point of the function (the denominator in dy/dx becomes 0).
. . . . . .
31. Example
Find the horizontal tangent lines to the same curve: y2 = x3 + x2
Solution
We solve for dy/dx = 0:
3x2 + 2x
= 0 =⇒ 3x2 + 2x = 0 =⇒ x(3x + 2) = 0
2y
The possible solution x = 0 leads to y = 0, which is not a smooth
point of the function (the denominator in dy/dx becomes 0).
The possible solution x = − 2 yields y = ± 3√3 .
2
3
. . . . . .
33. Example
Find the vertical tangent lines to the same curve: y2 = x3 + x2
Solution
dx
= 0.
Tangent lines are vertical when
dy
Differentiating x implicitly as a function of y gives
dx dx
2y = 3x2 + 2x , so
dy dy
dx 2y
=2
3x + 2x
dy
. . . . . .
34. Example
Find the vertical tangent lines to the same curve: y2 = x3 + x2
Solution
dx
= 0.
Tangent lines are vertical when
dy
Differentiating x implicitly as a function of y gives
dx dx
2y = 3x2 + 2x , so
dy dy
dx 2y
=2
3x + 2x
dy
This is 0 only when y = 0.
. . . . . .
35. Example
Find the vertical tangent lines to the same curve: y2 = x3 + x2
Solution
dx
= 0.
Tangent lines are vertical when
dy
Differentiating x implicitly as a function of y gives
dx dx
2y = 3x2 + 2x , so
dy dy
dx 2y
=2
3x + 2x
dy
This is 0 only when y = 0.
We get the false solution x = 0 and the real solution x = −1.
. . . . . .
36. Ideal gases
The ideal gas law relates
temperature, pressure, and
volume of a gas:
PV = nRT
(R is a constant, n is the
amount of gas in moles)
.
.
Image credit: Scott Beale / Laughing Squid
. . . . . .
37. .
Definition
The isothermic compressibility of a fluid is defined by
dV 1
β=−
dP V
with temperature held constant.
.
Image credit: Neil Better
. . . . . .
38. .
Definition
The isothermic compressibility of a fluid is defined by
dV 1
β=−
dP V
with temperature held constant.
The smaller the β, the “harder” the fluid.
.
Image credit: Neil Better
. . . . . .
40. Example
Find the isothermic compressibility of an ideal gas.
Solution
If PV = k (n is constant for our purposes, T is constant because of the
word isothermic, and R really is constant), then
dP dV dV V
· V + P = 0 =⇒ =−
dP dP dP P
So
1 dV 1
β=− · =
V dP P
Compressibility and pressure are inversely related.
. . . . . .
41. Nonideal gasses
Not that there’s anything wrong with that
Example
The van der Waals equation H
..
makes fewer simplifications:
O.
( ) . xygen . .
H
n2
P + a 2 (V − nb) = nRT, .
V H
..
O. H
. ydrogen bonds
. xygen
where P is the pressure, V the
H
..
volume, T the temperature, n .
the number of moles of the
O.
. xygen . .
H
gas, R a constant, a is a
measure of attraction between
H
..
particles of the gas, and b a
measure of particle size.
. . . . . .
42. Nonideal gasses
Not that there’s anything wrong with that
Example
The van der Waals equation
makes fewer simplifications:
( )
n2
P + a 2 (V − nb) = nRT,
V
where P is the pressure, V the
volume, T the temperature, n
the number of moles of the
gas, R a constant, a is a
measure of attraction between
particles of the gas, and b a
measure of particle size. .
.
Image credit: Wikimedia Commons
. . . . . .
43. Let’s find the compressibility of a van der Waals gas. Differentiating
the van der Waals equation by treating V as a function of P gives
( ) ( )
an2 dV 2an2 dV
+ (V − bn) 1 − 3
P+ 2 = 0,
dP V dP
V
. . . . . .
44. Let’s find the compressibility of a van der Waals gas. Differentiating
the van der Waals equation by treating V as a function of P gives
( ) ( )
an2 dV 2an2 dV
+ (V − bn) 1 − 3
P+ 2 = 0,
dP V dP
V
so
V2 (V − nb)
1 dV
β=− =
2abn3 − an2 V + PV3
V dP
. . . . . .
45. Let’s find the compressibility of a van der Waals gas. Differentiating
the van der Waals equation by treating V as a function of P gives
( ) ( )
an2 dV 2an2 dV
+ (V − bn) 1 − 3
P+ 2 = 0,
dP V dP
V
so
V2 (V − nb)
1 dV
β=− =
2abn3 − an2 V + PV3
V dP
What if a = b = 0?
. . . . . .
46. Let’s find the compressibility of a van der Waals gas. Differentiating
the van der Waals equation by treating V as a function of P gives
( ) ( )
an2 dV 2an2 dV
+ (V − bn) 1 − 3
P+ 2 = 0,
dP V dP
V
so
V2 (V − nb)
1 dV
β=− =
2abn3 − an2 V + PV3
V dP
What if a = b = 0?
dβ
Without taking the derivative, what is the sign of ?
db
. . . . . .
47. Let’s find the compressibility of a van der Waals gas. Differentiating
the van der Waals equation by treating V as a function of P gives
( ) ( )
an2 dV 2an2 dV
+ (V − bn) 1 − 3
P+ 2 = 0,
dP V dP
V
so
V2 (V − nb)
1 dV
β=− =
2abn3 − an2 V + PV3
V dP
What if a = b = 0?
dβ
Without taking the derivative, what is the sign of ?
db
dβ
Without taking the derivative, what is the sign of ?
da
. . . . . .
48. Nasty derivatives
(2abn3 − an2 V + PV3 )(nV2 ) − (nbV2 − V3 )(2an3 )
dβ
=−
(2abn3 − an2 V + PV3 )2
db
(2 )
nV3 an + PV2
= −( )2 < 0
PV3 + an2 (2bn − V)
n2 (bn − V)(2bn − V)V2
dβ
=( )2 > 0
da PV3 + an2 (2bn − V)
(as long as V > 2nb, and it’s probably true that V ≫ 2nb).
. . . . . .
49. Outline
The big idea, by example
Examples
Vertical and Horizontal Tangents
Chemistry
The power rule for rational powers
. . . . . .
51. Using implicit differentiation to find derivatives
Example
√
dy
if y = x.
Find
dx
Solution
√
If y = x, then
y2 = x,
so
dy dy 1 1
= √.
= 1 =⇒ =
2y
dx dx 2y 2x
. . . . . .
52. The power rule for rational numbers
Example
dy
if y = xp/q , where p and q are integers.
Find
dx
. . . . . .
53. The power rule for rational numbers
Example
dy
if y = xp/q , where p and q are integers.
Find
dx
Solution
We have
p xp−1
dy dy
= · q−1
yq = xp =⇒ qyq−1 = pxp−1 =⇒
dx dx qy
. . . . . .
54. The power rule for rational numbers
Example
dy
if y = xp/q , where p and q are integers.
Find
dx
Solution
We have
p xp−1
dy dy
= · q−1
yq = xp =⇒ qyq−1 = pxp−1 =⇒
dx dx qy
Now yq−1 = xp(q−1)/q = xp−p/q so
xp−1
= xp−1−(p−p/q) = xp/q−1
yq−1
. . . . . .