StudySmarter: Study help & AI tools

4.5 • +22k Ratings

More than 22 Million Downloads

Free

Taylor's Theorem

Immerse yourself in the world of engineering mathematics with a detailed look at Taylor's Theorem - a key concept in mathematical analysis. This theorem is a prime mover behind countless engineering calculations and models. The article breaks down the intricacies of the Taylor's Theorem Series, elucidating its meanings, components, and historical evolution. You'll gain insight into its practical applications, the potential for error, and the process behind its proof. As a bonus, you'll explore the important relationship it shares with the Mean Value Theorem, the role it plays in multivariate functions, and its real-world applications in engineering mathematics.

Explore our app and discover over 50 million learning materials for free.

- Design Engineering
- Engineering Fluid Mechanics
- Engineering Mathematics
- Acceptance Sampling
- Addition Rule of Probability
- Algebra Engineering
- Application of Calculus in Engineering
- Area under curve
- Basic Algebra
- Basic Derivatives
- Basic Matrix Operations
- Bayes' Theorem
- Binomial Series
- Bisection Method
- Boolean Algebra
- Boundary Value Problem
- CUSUM
- Cartesian Form
- Causal Function
- Centroids
- Cholesky Decomposition
- Circular Functions
- Complex Form of Fourier Series
- Complex Hyperbolic Functions
- Complex Logarithm
- Complex Trigonometric Functions
- Conservative Vector Field
- Continuous and Discrete Random Variables
- Control Chart
- Convergence Engineering
- Convergence of Fourier Series
- Convolution Theorem
- Correlation and Regression
- Covariance and Correlation
- Cramer's rule
- Cross Correlation Theorem
- Curl of a Vector Field
- Curve Sketching
- D'alembert Wave Equation
- Damping
- Derivative of Polynomial
- Derivative of Rational Function
- Derivative of a Vector
- Directional Derivative
- Discrete Fourier Transform
- Divergence Theorem
- Divergence Vector Calculus
- Double Integrals
- Eigenvalue
- Eigenvector
- Engineering Analysis
- Engineering Graphs
- Engineering Statistics
- Euler's Formula
- Exact Differential Equation
- Exponential and Logarithmic Functions
- Fourier Coefficients
- Fourier Integration
- Fourier Series
- Fourier Series Odd and Even
- Fourier Series Symmetry
- Fourier Transform Properties
- Fourier Transform Table
- Gamma Distribution
- Gaussian Elimination
- Half Range Fourier Series
- Higher Order Integration
- Hypergeometric Distribution
- Hypothesis Test for a Population Mean
- Implicit Function
- Improved Euler Method
- Interpolation
- Inverse Laplace Transform
- Inverse Matrix Method
- Inverse Z Transform
- Jacobian Matrix
- Laplace Shifting Theorem
- Laplace Transforms
- Large Sample Confidence Interval
- Least Squares Fitting
- Logic Gates
- Logical Equivalence
- Maths Identities
- Maxima and Minima of functions of two variables
- Maximum Likelihood Estimation
- Mean Value and Standard Deviation
- Method of Moments
- Modelling waves
- Multiple Regression
- Multiple Regression Analysis
- Newton Raphson Method
- Non Parametric Statistics
- Nonlinear Differential Equation
- Nonlinear Regression
- Numerical Differentiation
- Numerical Root Finding
- One Way ANOVA
- P Value
- Parseval's Theorem
- Partial Derivative
- Partial Derivative of Vector
- Partial Differential Equations
- Particular Solution for Differential Equation
- Phasor
- Piecewise Function
- Polar Form
- Polynomial Regression
- Probability Engineering
- Probability Tree
- Quality Control
- RMS Value
- Radians vs Degrees
- Rank Nullity Theorem
- Rank of a Matrix
- Reliability Engineering
- Runge Kutta Method
- Scalar & Vector Geometry
- Second Order Nonlinear Differential Equation
- Simple Linear Regression Model
- Single Sample T Test
- Standard Deviation of Random Variable
- Superposition
- System of Differential Equations
- System of Linear Equations Matrix
- Taylor's Theorem
- Three Way ANOVA
- Total Derivative
- Transform Variables in Regression
- Transmission Line Equation
- Triple Integrals
- Triple Product
- Two Sample Test
- Two Way ANOVA
- Unit Vector
- Vector Calculus
- Wilcoxon Rank Sum Test
- Z Test
- Z Transform
- Z Transform vs Laplace Transform
- Engineering Thermodynamics
- Materials Engineering
- Professional Engineering
- Solid Mechanics
- What is Engineering

Lerne mit deinen Freunden und bleibe auf dem richtigen Kurs mit deinen persönlichen Lernstatistiken

Jetzt kostenlos anmeldenNie wieder prokastinieren mit unseren Lernerinnerungen.

Jetzt kostenlos anmeldenImmerse yourself in the world of engineering mathematics with a detailed look at Taylor's Theorem - a key concept in mathematical analysis. This theorem is a prime mover behind countless engineering calculations and models. The article breaks down the intricacies of the Taylor's Theorem Series, elucidating its meanings, components, and historical evolution. You'll gain insight into its practical applications, the potential for error, and the process behind its proof. As a bonus, you'll explore the important relationship it shares with the Mean Value Theorem, the role it plays in multivariate functions, and its real-world applications in engineering mathematics.

Taylor's Theorem essentially provides a way to express a function as an infinite sum of terms. These are calculated from the function's derivatives at a certain point.

A Taylor series is a representation of a function as an infinite sum of terms derived from its derivatives at a single point.

- \( f(a) \): This is known as the zeroth derivative term. This term is simply the function evaluated at the point \( a \).
- \( f'(a)(x-a) \): This is the first derivative term.
- \( f''(a)(x-a)^2/2! \): This is the second-order derivative term, and so on.

Interestingly, the fascinating property of Taylor's Theorem is that it can approximate any function, no matter how complex, using simpler polynomial terms if you have sufficient terms in the series.

Here are steps you following to find the Taylor series: 1. Calculate the function and its derivatives at \( x = 0 \): \( f(0) = e^0 = 1 \) \( f'(0) = e^0 = 1 \) \( f''(0) = e^0 = 1 \) You'll notice that every derivative of \( e^x \) turns out to be \( e^x \) itself. Hence all terms will be equal to one. 2. Substituting back into the Taylor series equation: The Taylor series for \( f(x) = e^x \) around \( a = 0 \) thus becomes: \[ f(x) = 1 + x + \frac{{x^2}}{2!} + \frac{{x^3}}{3!} + \ldots \]

Let's follow the steps to utilise Taylor's theorem here: 1. Determine the first few derivatives of \( f(x) = \sqrt{x} \) and evaluate them at \( a = 9 \): \( f'(x) = \frac{1}{2\sqrt{x}} \) , \( f''(x) = - \frac{1}{4x^{3/2}} \) \( f'(9) = \frac{1}{6} \), \( f''(9) = -\frac{1}{108} \) 2. Considering only the first two terms of the Taylor series (assuming \( (x - a) \) to be small), we get: \( \sqrt{x} \approx f(9) + f'(9)(x - 9) \) Plugging \( x = 9.1 \) and \( a = 9 \), we get: \( \sqrt{9.1} \approx 3 + \frac{1}{6}(0.1) \approx 3.01667 \) This is close to the exact value of \( \sqrt{9.1} = 3.01662 \), showing how Taylor's theorem helps us estimate function values.

Following our method, you'll find that: 1. The derivatives of \( \sin(x) \) at \( x = 0 \) are: \( f(0) = \sin(0) = 0 \) \( f'(0) = \cos(0) = 1 \) \( f''(0) = -\sin(0) = 0 \) \( f'''(0) = -\cos(0) = -1 \) And you'll notice that the pattern repeats after this. 2. Substituting these values into the Taylor series equation, you get: \( \sin(x) = x - \frac{{x^3}}{3!} + \frac{{x^5}}{5!} - \frac{{x^7}}{7!} + \ldots \) By considering more terms, you get a better approximation. For situations when you don't have a calculator and need to compute \( \sin(x) \), this series approach proves to be exceedingly useful.

Factors Leading to Error | Explanation |

Truncation of the Series | Limiting the infinite series to a finite number of terms introduces discrepancies in the approximation. |

Choice of Point of Approximation \( a \) | The point used for approximation holds a key role. Optimal results occur when \( a \) is close to \( x \). |

Nature of the Function | The function's nature can impact the accuracy, particularly if it diverges rapidly from the approximating polynomial. |

Given that all derivatives of \( e^x \) return \( e^x \) themselves, you obtain the following: \( |R_3(x)| \leq \frac{{e^c|x^4|}}{4!} \) Now, select \( c \) between 0 and 0.5 (keeping your \( a \) and \( x \) in mind). With \( e^x \) being an increasing function, the largest value of \( e^c \) will be at \( c = 0.5 \). Hence, the maximum error becomes: \( |R_3(x)|_{max} \leq \frac{{e^{0.5}(0.5)^4}}{4!} \approx 0.0024801587 \) Comparing the approximate value that you obtain by using Taylor's third-order polynomial and the actual value of \( e^x \), you'll observe that the calculated error here is true to its prediction: \( e^{0.5} \approx 1.6487212707 \) \( P_3(0.5) = 1 + 0.5 + \frac{(0.5)^2}{2!} + \frac{(0.5)^3}{3!} = 1.6458333333 \) The actual difference turns out to be \( e^{0.5} - P_3(0.5) = 0.0028879374 \), which indeed lies within the estimated error bounds.

The Mean Value Theorem states that for a function \( f \) which is continuous over an interval \([a, b]\) and differentiable on \((a, b)\), there exists a point \( c \) in \((a, b)\) where the instantaneous rate of change (the derivative) equals the average rate of change over the interval \([a, b]\), formalised as: \( f'(c) = \frac{{f(b) - f(a)}}{{b - a}} \).

Consider the function \( f(x) = e^x \), and we wish to approximate \( f(1) \) using a Taylor polynomial. When you develop Taylor's series for \( e^x \) around \( a = 0 \), to the 3rd degree, it appears as: \( P_3(x) = 1 + x + \frac{{x^2}}{2!} + \frac{{x^3}}{3!} \) \( |R_3(x)| \leq \frac{{e^\xi|x^4|}}{4!} \) For \( x = 1 \), the maximum error in the approximation becomes: \( |R_3(1)| \leq \frac{{e^\xi}}{4!} = 0.0183156389 \) Upon comparing this anticipated error with the actual difference between \( e \) and \( P_3(1) = 1 + 1 + 0.5 + 0.1666666 = 2.6666666 \), one can see that it lies within the predicted bounds, i.e. \( |e - P_3(1)| \leq |R_3(1)| \) Therefore, the Mean Value Theorem guarantees the existence of \( \xi \) and renders a substantial component to Taylor's Theorem via the estimation of errors in its approximations.

The Taylor polynomial \( P_a \) of function \( f \) at point \( a \) can be represented as: \[ P_a(x) = f(a) + (Df(a))(x-a) + \frac{1}{2}(x-a)^T(D^2f(a))(x-a) \] The function \( f \) can then be described as: \[ f(x) = P_a(x) + R_a(x) \] where \( R_a(x) \) denotes the error term. \( Df(a) \) and \( D^2f(a) \) represent the first and second derivatives of the function at point \( a \), respectively. \( (x-a)^T(D^2f(a))(x-a) \) signifies the application of the second derivative to the vector \( (x-a) \).

\[ \frac{{\partial f}}{{\partial x}} = 2x + y \] Thus, \(\frac{{\partial f}}{{\partial x}}(1,1) = 3\)The partial derivative of \( f \) with respect to \( y \):

\[ \frac{{\partial f}}{{\partial y}} = x + 2y \] So, \(\frac{{\partial f}}{{\partial y}}(1,1) = 3\)

With the derivatives ready, we move on to determine the Taylor polynomial \( P_{(1,1)} \): \[ P_{(1,1)}(x, y) = f(1,1) + \frac{{\partial f}}{{\partial x}}(1,1) * (x-1) + \frac{{\partial f}}{{\partial y}}(1,1) * (y-1) \] Solve to get the polynomial: \[ P_{(1,1)}(x, y) = 3 + 3(x - 1) + 3(y - 1) \]This example illustrates the usage of Taylor's theorem for approximation of a multivariate function. Despite the high dimensions and complexity, the example underlines the theorem's potency in simplifying and providing insights into the function's characteristics. The beauty of Taylor's Theorem in multivariate settings unfolds in grasping such applications, and exploring them further can provide profound perspectives on a wide array of mathematical problems.

Optimisation forms the crux of engineering design. An engineering system can be conceived as a function that is influenced by different variables. To optimise this system entails finding the values of these variables that either maximise or minimise the output of the function - a process where Taylor's Theorem proves invaluable.

**Example 1:** In Electrical Engineering, one of the challenges is to model and analyse non-linear systems like Diodes and Transistors. These systems are primarily non-linear, making their behaviour challenging to predict. Here, Taylor's theorem comes to the rescue. It is used to derive the Small Signal Models for these devices. By expanding the non-linear I-V characteristics around the Bias point, one can arrive at linear approximation models which simplify analysis and design.

**Example 2:** In Civil and Mechanical Engineering, Taylor's Theorem forms a backbone to Finite Element Methods (FEM). These methodologies are widely used for solving complex geometrical problems in structures, heat transfer, fluid dynamics, and more. At the core lies the need to approximate a continuous function with a discrete or piecewise continuous function, essentially an application of Taylor's Theorem.

**Example 3:** In Economics, Taylor's Theorem often takes centre stage. The Taylor series is extensively used for it offers easy-to-use approximations for complex functions. For example, in macroeconomics, the Taylor Rule is promulgated which guides central banks in setting the nominal interest rate. This rule uses a first-order Taylor series approximation around an equilibrium level.

- Taylor's Theorem helps in approximating complex functions but the estimates generated by it may harbor errors due to several factors:
- Truncation of the series: This happens when the infinite series in a Taylor's Theorem series is limited to a finite number of terms.
- Choice of point of approximation: This is another factor that can significantly affect the approximation's quality.
- Nature of the function: If the function diverges rapidly from the approximating polynomial, the higher order terms might hold more significance, causing larger errors.
- Calculation of error in the Taylor's Theorem approximation can be done using the remainder term in the Thorem and Lagrange's form of the remainder.
- The proof of Taylor's Theorem is a critical part of understanding its operational efficacy. The proof can be broken down into verification of existence of Taylor polynomial, derivation of an expression for the remainder term, and making key observations about the value of the function, its derivatives and the polynomial at the point of approximation.
- Common misconceptions about the Taylor's Theorem involve presumptions about the infinite Taylor series providing an exact representation and the approximation improving unquestionably with increasing number of terms, and the assumption that the function and its Taylor series share identical exact derivatives at the point of approximation.
- In the realm of multivariate functions, Taylor's Theorem forms the Taylor Polynomial of a multivariate function at a selected point and then describes the function using this polynomial and a remainder term that denote the error.

Taylor's Theorem is a fundamental principle in calculus that approximates a function near a point via its derivatives at that point. It permits functions to be expressed as a series, known as the Taylor series, enabling complex mathematical analyses and predictions.

Taylor's Theorem states that any function satisfying certain conditions can be expressed as a sum of its derivatives at a certain point. This sum, also known as the Taylor series, equals the function's value at that point, plus the first derivative times the displacement from that point, and so on.

An example of Taylor's theorem is the approximation of the exponential function e^x. The Taylor series expansion of e^x at any point 'a' is the infinite sum of (x-a)^n/n! for n=0 to infinity. This allows calculations of e^x to high precision without using the exponential function directly.

The complexity of Taylor's theorem can depend on an individual's understanding of calculus and mathematical analysis. However, with a strong foundation in these areas and some practice, it is possible to understand and apply Taylor's theorem effectively.

Taylor's theorem is a fundamental concept in calculus that provides an approximation of a function near a point using information about its derivatives. Maclaurin's theorem is a special case of Taylor's theorem where the function is approximated near the point zero.

What is Taylor's Theorem in Engineering Mathematics?

Taylor's Theorem provides a way to express a function as an infinite sum of terms calculated from the function's derivatives at a certain point. It's used in approximations and problem-solving methods in engineering mathematics.

What components make up a Taylor Series?

A Taylor Series is formed by evaluating a function and its derivatives at a specific point. It includes the function evaluated at a point (zeroth derivative term), the first derivative term, and increasing order derivative terms divided by the factorial of the term's order.

What are the steps to use Taylor's theorem to find the Taylor series of the function \( f(x) = e^x \) around the point \( a = 0 \)?

Calculate the function and its derivatives at \( x = 0 \), you'll notice that every derivative of \( e^x \) turns out to be \( e^x \) itself. Hence all terms will be equal to one. Substituting back into the Taylor series equation, the Taylor series for \( f(x) = e^x \) around \( a = 0 \) thus becomes: \[ f(x) = 1 + x + \frac{{x^2}}{2!} + \frac{{x^3}}{3!} + \ldots \]

Why is Taylor's theorem used in estimating the value of \( \sqrt{9.1} \)?

Taylor's theorem simplifies complicated calculations. In the case of \( \sqrt{9.1} \), we consider it as an approximation of a function \( f(x) = \sqrt{x} \) around the point \( a = 9 \). This is because \( \sqrt{9} = 3 \) can be calculated exactly, making the calculation simpler. Using Taylor series, the estimated value is very close to the exact value.

What are the main factors that lead to error in Taylor's Theorem?

The main factors that contribute to errors in Taylor's Theorem are the truncation of the series - since it's practically impossible to use all terms of an infinite series, the choice of the point of approximation (ideally this should be close to x), and the nature of the function being approximated.

How is the error, or remainder, represented in Taylor's Theorem, and how can it be calculated?

The error in Taylor's Theorem is represented by the remainder term, denoted as Rn(x). Lagrange's form of the remainder can be used to calculate the error, where the error signifies the deviation of the actual function from its Taylor polynomial approximation at a point in the interval between a and x.

Already have an account? Log in

Open in App
More about Taylor's Theorem

The first learning app that truly has everything you need to ace your exams in one place

- Flashcards & Quizzes
- AI Study Assistant
- Study Planner
- Mock-Exams
- Smart Note-Taking

Sign up to highlight and take notes. It’s 100% free.

Save explanations to your personalised space and access them anytime, anywhere!

Sign up with Email Sign up with AppleBy signing up, you agree to the Terms and Conditions and the Privacy Policy of StudySmarter.

Already have an account? Log in