The document discusses integrating functions of time to calculate changes in displacement, distance, velocity, and speed based on position, velocity, and acceleration graphs over time. It provides examples of how integrating areas under curves relates to these physical quantities. Derivative graphs and their relationships are also summarized, along with how different function types integrate or differentiate into other graph types. An example problem calculating the times when a particle is at rest and its maximum velocity is also worked through.
In this section we look at problems where changing quantities are related. For instance, a growing oil slick is changing in diameter and volume at the same time. How are the rates of change of these quantities related? The chain rule for derivatives is the key.
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.
Dsp U Lec07 Realization Of Discrete Time Systemstaha25
This document provides an overview of discrete-time systems and digital signal processing. It discusses discrete-time system components like unit delays and adders. It also covers discrete system networks including FIR and IIR networks. Various realizations of discrete systems are presented, including direct form I and II, cascaded, and parallel realizations. Digital filters are defined and the advantages and disadvantages as well as types (FIR and IIR) are discussed. Design steps and specifications for digital filters are also outlined.
The document discusses modeling service compositions and orchestrations using dynamic simulation. It aims to predict the behavior of service provisioning over time by modeling workflows as continuous-time Petri nets. This allows simulating how varying input request rates map to outputs while considering the structure and characteristics of the composition. The approach converts Petri nets into systems of ordinary differential equations to capture the dynamics and derive models for activities, places, and asynchronous messaging between components.
Numerical smoothing and hierarchical approximations for efficient option pric...Chiheb Ben Hammouda
1. The document presents a numerical smoothing technique to improve the efficiency of option pricing and density estimation when analytic smoothing is not possible.
2. The technique involves numerically determining discontinuities in the integrand and computing the integral only over the smooth regions. It also uses hierarchical representations and Brownian bridges to reduce the effective dimension of the problem.
3. The numerical smoothing approach outperforms Monte Carlo methods for high dimensional cases and improves the complexity of multilevel Monte Carlo from O(TOL^-2.5) to O(TOL^-2 log(TOL)^2).
System 1 and System 2 were basic early systems for image matching that used color and texture matching. Descriptor-based approaches like SIFT provided more invariance but not perfect invariance. Patch descriptors like SIFT were improved by making them more invariant to lighting changes like color and illumination shifts. The best performance came from combining descriptors with color invariance. Representing images as histograms of visual word occurrences captured patterns in local image patches and allowed measuring similarity between images. Large vocabularies of visual words provided more discriminative power but were costly to compute and store.
This document discusses noncommutative quantum field theory, where the coordinates do not commute. It begins by motivating noncommutativity from theories of quantum gravity and string theory. It then introduces the Moyal product to write actions for noncommutative fields. While Lorentz symmetry is broken, the actions are still invariant under a twisted Poincaré algebra. Representations are classified by mass and spin as in ordinary theories. The document considers both space-like and time-like noncommutativity, but argues that time-like noncommutativity poses challenges for perturbative unitarity.
In this section we look at problems where changing quantities are related. For instance, a growing oil slick is changing in diameter and volume at the same time. How are the rates of change of these quantities related? The chain rule for derivatives is the key.
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.
Dsp U Lec07 Realization Of Discrete Time Systemstaha25
This document provides an overview of discrete-time systems and digital signal processing. It discusses discrete-time system components like unit delays and adders. It also covers discrete system networks including FIR and IIR networks. Various realizations of discrete systems are presented, including direct form I and II, cascaded, and parallel realizations. Digital filters are defined and the advantages and disadvantages as well as types (FIR and IIR) are discussed. Design steps and specifications for digital filters are also outlined.
The document discusses modeling service compositions and orchestrations using dynamic simulation. It aims to predict the behavior of service provisioning over time by modeling workflows as continuous-time Petri nets. This allows simulating how varying input request rates map to outputs while considering the structure and characteristics of the composition. The approach converts Petri nets into systems of ordinary differential equations to capture the dynamics and derive models for activities, places, and asynchronous messaging between components.
Numerical smoothing and hierarchical approximations for efficient option pric...Chiheb Ben Hammouda
1. The document presents a numerical smoothing technique to improve the efficiency of option pricing and density estimation when analytic smoothing is not possible.
2. The technique involves numerically determining discontinuities in the integrand and computing the integral only over the smooth regions. It also uses hierarchical representations and Brownian bridges to reduce the effective dimension of the problem.
3. The numerical smoothing approach outperforms Monte Carlo methods for high dimensional cases and improves the complexity of multilevel Monte Carlo from O(TOL^-2.5) to O(TOL^-2 log(TOL)^2).
System 1 and System 2 were basic early systems for image matching that used color and texture matching. Descriptor-based approaches like SIFT provided more invariance but not perfect invariance. Patch descriptors like SIFT were improved by making them more invariant to lighting changes like color and illumination shifts. The best performance came from combining descriptors with color invariance. Representing images as histograms of visual word occurrences captured patterns in local image patches and allowed measuring similarity between images. Large vocabularies of visual words provided more discriminative power but were costly to compute and store.
This document discusses noncommutative quantum field theory, where the coordinates do not commute. It begins by motivating noncommutativity from theories of quantum gravity and string theory. It then introduces the Moyal product to write actions for noncommutative fields. While Lorentz symmetry is broken, the actions are still invariant under a twisted Poincaré algebra. Representations are classified by mass and spin as in ordinary theories. The document considers both space-like and time-like noncommutativity, but argues that time-like noncommutativity poses challenges for perturbative unitarity.
This document discusses curvilinear motion and kinematics. It introduces position vectors, path coordinates, velocity vectors, and acceleration vectors for particles moving in three-dimensional space. Key concepts covered include defining the position vector r(t) from a reference point to the particle, the instantaneous velocity vector v as the time derivative of r(t), and the acceleration vector a as the time derivative of v. When working in Cartesian coordinates, the derivatives of vector components are simply the derivatives of the individual x, y, z components.
The document discusses rigidity, gap theorems, and maximum principles for Ricci solitons. It defines a Ricci soliton as rigid if it has the form of N ×Γ Rk, where N is an Einstein manifold and Γ acts freely on N and orthogonally on Rk. It presents theorems showing that compact Ricci solitons that are locally conformally flat or have harmonic Weyl tensor are Einstein. It also discusses results for complete noncompact shrinking Ricci solitons, showing they are rigid if the Weyl tensor is harmonic.
The document provides an overview of key physics equations and concepts related to forces and motion, including equations for relative deviation, prefixes, units of area and volume, average speed, velocity, acceleration, momentum, Newton's laws of motion, and impulse. Key variables and their units are defined for each equation. Examples of displacement-time and velocity-time graphs are also included to illustrate the relationships between displacement, velocity, time, and acceleration.
On estimating the integrated co volatility usingkkislas
This document proposes a method to estimate the integrated co-volatility of two asset prices using high-frequency data that contains both microstructure noise and jumps.
It considers two cases - when the jump processes of the two assets are independent, and when they are dependent. For the independent case, it proposes an estimator that is robust to jumps. For the dependent case, it proposes a threshold estimator that combines pre-averaging to remove noise with a threshold method to reduce the effect of jumps. It proves the estimators are consistent and establishes their central limit theorems. Simulation results are also presented to illustrate the performance of the proposed methods.
The document explains the angle addition postulate, which states that for any angle with a point in its interior, the measure of the two smaller angles equals the measure of the larger total angle. It provides examples of drawing angles and finding the measure of interior angles. It also shows how to use the postulate to find the measure of an unknown interior angle given the measures of other angles.
Avoidance of Microstructural Heterogeneities by Hot Rolling Design in Thin Sl...Pello Uranga
The document describes a new microstructural model for predicting microstructural heterogeneity in thin slab direct rolled niobium microalloyed steels. The model uses grain size distributions measured from real thin slabs as inputs. It then outputs recrystallized and unrecrystallized grain size histograms and retained strain. Rolling simulations using the model showed that entry temperatures of 1060°C can result in heterogeneous final austenite structure, while 1100°C promotes a homogeneous structure. Optimized rolling schedules and minimum entry temperatures were determined to be 1090-1070°C to avoid heterogeneities. Increasing the initial slab thickness was also found to provide higher retained strain without affecting homogeneity.
A current perspectives of corrected operator splitting (os) for systemsAlexander Decker
This document discusses operator splitting methods for solving systems of convection-diffusion equations. It begins by introducing operator splitting, where the time evolution is split into separate steps for convection and diffusion. While efficient, operator splitting can produce significant errors near shocks.
The document then examines the nonlinear error mechanism that causes issues for operator splitting near shocks. When a shock develops in the convection step, it introduces a local linearization that neglects self-sharpening effects. This leads to splitting errors.
To address this, the document discusses corrected operator splitting, which uses the wave structure from the convection step to identify where nonlinear splitting errors occur. Terms are added to the diffusion step to compensate for
The document discusses the Discrete Fourier Transform (DFT). It begins by explaining the limitations of the Discrete Time Fourier Transform (DTFT) and Discrete Fourier Series (DFS) from a numerical computation perspective. It then introduces the DFT as a numerically computable transform obtained by sampling the DTFT in the frequency domain. The DFT represents a periodic discrete-time signal using a sum of complex exponentials. It defines the DFT and inverse DFT equations. The document also discusses properties of the DFT such as linearity and time/frequency shifting. Finally, it notes that the Fast Fourier Transform (FFT) implements the DFT more efficiently by constraining the number of points to powers of two.
This document discusses various feature detectors used in computer vision. It begins by describing classic detectors such as the Harris detector and Hessian detector that search scale space to find distinguished locations. It then discusses detecting features at multiple scales using the Laplacian of Gaussian and determinant of Hessian. The document also covers affine covariant detectors such as maximally stable extremal regions and affine shape adaptation. It discusses approaches for speeding up detection using approximations like those in SURF and learning to emulate detectors. Finally, it outlines new developments in feature detection.
The document provides an overview of key physics equations and concepts for Form 4 students, including equations for relative deviation, prefixes, units for area and volume, equations for average speed, velocity, acceleration, momentum, Newton's laws of motion, and impulse. Key terms are defined for important concepts like displacement, time, mass, force, and velocity. Formulas are presented for calculations involving these fundamental physics quantities and relationships.
WAVELET-PACKET-BASED ADAPTIVE ALGORITHM FOR SPARSE IMPULSE RESPONSE IDENTIFI...bermudez_jcm
Presented at IEEE ICASSP-2007:
This paper proposes a wavelet-packet-based (WPB) algorithm for efficient identification of sparse impulse responses with arbitrary frequency spectra. The discrete wavelet packet transform (DWPT) is adaptively tailored to the energy distribution of the unknown system\'s response spectrum. The new algorithm leads to a reduced number of active coefficients and to a reduced computational complexity, when compared to competing wavelet-based algorithms. Simulation results illustrate the applicability of the proposed algorithm.
The document summarizes a simulation of news and insider influences on stock market price dynamics using a non-linear model. It describes two types of trader strategies (F and N), how aggregate excess demand is calculated, dynamic price adjustment, and how the share of each trader type changes over time based on past relative returns. It also discusses using the model to simulate dollar/ruble exchange rates and analyze patterns in the time series data.
Summary of "A Universally-Truthful Approximation Scheme for Multi-unit Auction"Thatchaphol Saranurak
1) The document summarizes a paper that presents a universally truthful approximation scheme for multi-unit auctions.
2) The paper introduces two key concepts - ∆-perturbed maximizer and consensus function with drop-outs - that are used to develop a randomized polynomial time approximation scheme (PTAS) for multi-unit auctions.
3) The ∆-perturbed maximizer adds random perturbations to bid valuations, which allows the multi-unit allocation problem to be modeled as a multiple-choice knapsack problem that can be solved in polynomial time. The consensus function combines allocation results from different perturbed instances in a way that maintains feasibility and truthfulness properties.
Second Order Perturbations During Inflation Beyond Slow-rollIan Huston
This document outlines research on second-order perturbations during inflation beyond the slow-roll approximation. It discusses perturbation theory at first and second order, and presents results on the source term and second-order perturbations for inflation models with features. The document also describes Pyflation, an open-source Python code for numerically calculating inflationary perturbations up to second order, and outlines future goals for the code including calculating the three-point function and incorporating multi-field models.
call for papers, research paper publishing, where to publish research paper, journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJEI, call for papers 2012,journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, research and review articles, engineering journal, International Journal of Engineering Inventions, hard copy of journal, hard copy of certificates, journal of engineering, online Submission, where to publish research paper, journal publishing, international journal, publishing a paper, hard copy journal, engineering journal
This document discusses Hidden Markov Models (HMMs) and Markov chains. It begins with an introduction to Markov processes and how HMMs are used in various domains like natural language processing. It then describes the properties of a Markov chain, which has a set of states that the system transitions between randomly at discrete time steps based on transition probabilities. The Markov property is explained as the conditional independence of future states from past states given the present state. HMMs extend Markov chains by making the state sequence hidden and only allowing observation of the output states.
This document summarizes a paper that applies multivariate integer-valued autoregressive (MINAR) models to earthquake count data. Specifically, it generalizes existing bivariate integer-valued autoregressive (BINAR) models and applies the generalized BINAR model to study the joint dynamics of earthquake counts across different tectonic plates over time. The paper derives several theoretical properties of MINAR models, including unconditional mean, variance, autocorrelations, and forecasting moments. It then analyzes earthquake count data from various tectonic plate pairs to investigate space-time clustering of earthquakes and the impact of medium and large earthquakes on counts over time.
1) The document discusses 1D kinematics graphs of position, velocity, and acceleration over time.
2) The slope of a position vs. time graph gives velocity, while the slope of a velocity vs. time graph gives acceleration.
3) Taking the derivative of a position, velocity, or acceleration function provides the next variable (velocity, acceleration, or change in acceleration), just as taking the integral of acceleration or velocity functions provides position or change in velocity.
This presentation explains how the differentiation is applied to identify increasing and decreasing functions,identifying the nature of stationary points and also finding maximum or minimum values.
[shaderx5] 4.2 Multisampling Extension for Gradient Shadow Maps종빈 오
This document discusses techniques for improving gradient shadow maps, including merging gradient shadow maps with percentage-closer filtering (PCF) by calculating gradients over PCF areas and optimizing texture lookups. It introduces using slope-scale depth bias, fuzzy depth comparison, and linearly filtered depth values to reduce acne artifacts. It then describes combining gradient shadow maps and PCF by calculating average gradients over PCF areas to reduce sampling counts.
This document discusses various two-dimensional geometric transformations including translations, rotations, scaling, reflections, shears, and composite transformations. Translations move objects without deformation using a translation vector. Rotations rotate objects around a fixed point or pivot point. Scaling transformations enlarge or shrink objects using scaling factors. Reflections produce a mirror image of an object across an axis. Shearing slants an object along an axis. Composite transformations combine multiple basic transformations using matrix multiplication.
This document discusses curvilinear motion and kinematics. It introduces position vectors, path coordinates, velocity vectors, and acceleration vectors for particles moving in three-dimensional space. Key concepts covered include defining the position vector r(t) from a reference point to the particle, the instantaneous velocity vector v as the time derivative of r(t), and the acceleration vector a as the time derivative of v. When working in Cartesian coordinates, the derivatives of vector components are simply the derivatives of the individual x, y, z components.
The document discusses rigidity, gap theorems, and maximum principles for Ricci solitons. It defines a Ricci soliton as rigid if it has the form of N ×Γ Rk, where N is an Einstein manifold and Γ acts freely on N and orthogonally on Rk. It presents theorems showing that compact Ricci solitons that are locally conformally flat or have harmonic Weyl tensor are Einstein. It also discusses results for complete noncompact shrinking Ricci solitons, showing they are rigid if the Weyl tensor is harmonic.
The document provides an overview of key physics equations and concepts related to forces and motion, including equations for relative deviation, prefixes, units of area and volume, average speed, velocity, acceleration, momentum, Newton's laws of motion, and impulse. Key variables and their units are defined for each equation. Examples of displacement-time and velocity-time graphs are also included to illustrate the relationships between displacement, velocity, time, and acceleration.
On estimating the integrated co volatility usingkkislas
This document proposes a method to estimate the integrated co-volatility of two asset prices using high-frequency data that contains both microstructure noise and jumps.
It considers two cases - when the jump processes of the two assets are independent, and when they are dependent. For the independent case, it proposes an estimator that is robust to jumps. For the dependent case, it proposes a threshold estimator that combines pre-averaging to remove noise with a threshold method to reduce the effect of jumps. It proves the estimators are consistent and establishes their central limit theorems. Simulation results are also presented to illustrate the performance of the proposed methods.
The document explains the angle addition postulate, which states that for any angle with a point in its interior, the measure of the two smaller angles equals the measure of the larger total angle. It provides examples of drawing angles and finding the measure of interior angles. It also shows how to use the postulate to find the measure of an unknown interior angle given the measures of other angles.
Avoidance of Microstructural Heterogeneities by Hot Rolling Design in Thin Sl...Pello Uranga
The document describes a new microstructural model for predicting microstructural heterogeneity in thin slab direct rolled niobium microalloyed steels. The model uses grain size distributions measured from real thin slabs as inputs. It then outputs recrystallized and unrecrystallized grain size histograms and retained strain. Rolling simulations using the model showed that entry temperatures of 1060°C can result in heterogeneous final austenite structure, while 1100°C promotes a homogeneous structure. Optimized rolling schedules and minimum entry temperatures were determined to be 1090-1070°C to avoid heterogeneities. Increasing the initial slab thickness was also found to provide higher retained strain without affecting homogeneity.
A current perspectives of corrected operator splitting (os) for systemsAlexander Decker
This document discusses operator splitting methods for solving systems of convection-diffusion equations. It begins by introducing operator splitting, where the time evolution is split into separate steps for convection and diffusion. While efficient, operator splitting can produce significant errors near shocks.
The document then examines the nonlinear error mechanism that causes issues for operator splitting near shocks. When a shock develops in the convection step, it introduces a local linearization that neglects self-sharpening effects. This leads to splitting errors.
To address this, the document discusses corrected operator splitting, which uses the wave structure from the convection step to identify where nonlinear splitting errors occur. Terms are added to the diffusion step to compensate for
The document discusses the Discrete Fourier Transform (DFT). It begins by explaining the limitations of the Discrete Time Fourier Transform (DTFT) and Discrete Fourier Series (DFS) from a numerical computation perspective. It then introduces the DFT as a numerically computable transform obtained by sampling the DTFT in the frequency domain. The DFT represents a periodic discrete-time signal using a sum of complex exponentials. It defines the DFT and inverse DFT equations. The document also discusses properties of the DFT such as linearity and time/frequency shifting. Finally, it notes that the Fast Fourier Transform (FFT) implements the DFT more efficiently by constraining the number of points to powers of two.
This document discusses various feature detectors used in computer vision. It begins by describing classic detectors such as the Harris detector and Hessian detector that search scale space to find distinguished locations. It then discusses detecting features at multiple scales using the Laplacian of Gaussian and determinant of Hessian. The document also covers affine covariant detectors such as maximally stable extremal regions and affine shape adaptation. It discusses approaches for speeding up detection using approximations like those in SURF and learning to emulate detectors. Finally, it outlines new developments in feature detection.
The document provides an overview of key physics equations and concepts for Form 4 students, including equations for relative deviation, prefixes, units for area and volume, equations for average speed, velocity, acceleration, momentum, Newton's laws of motion, and impulse. Key terms are defined for important concepts like displacement, time, mass, force, and velocity. Formulas are presented for calculations involving these fundamental physics quantities and relationships.
WAVELET-PACKET-BASED ADAPTIVE ALGORITHM FOR SPARSE IMPULSE RESPONSE IDENTIFI...bermudez_jcm
Presented at IEEE ICASSP-2007:
This paper proposes a wavelet-packet-based (WPB) algorithm for efficient identification of sparse impulse responses with arbitrary frequency spectra. The discrete wavelet packet transform (DWPT) is adaptively tailored to the energy distribution of the unknown system\'s response spectrum. The new algorithm leads to a reduced number of active coefficients and to a reduced computational complexity, when compared to competing wavelet-based algorithms. Simulation results illustrate the applicability of the proposed algorithm.
The document summarizes a simulation of news and insider influences on stock market price dynamics using a non-linear model. It describes two types of trader strategies (F and N), how aggregate excess demand is calculated, dynamic price adjustment, and how the share of each trader type changes over time based on past relative returns. It also discusses using the model to simulate dollar/ruble exchange rates and analyze patterns in the time series data.
Summary of "A Universally-Truthful Approximation Scheme for Multi-unit Auction"Thatchaphol Saranurak
1) The document summarizes a paper that presents a universally truthful approximation scheme for multi-unit auctions.
2) The paper introduces two key concepts - ∆-perturbed maximizer and consensus function with drop-outs - that are used to develop a randomized polynomial time approximation scheme (PTAS) for multi-unit auctions.
3) The ∆-perturbed maximizer adds random perturbations to bid valuations, which allows the multi-unit allocation problem to be modeled as a multiple-choice knapsack problem that can be solved in polynomial time. The consensus function combines allocation results from different perturbed instances in a way that maintains feasibility and truthfulness properties.
Second Order Perturbations During Inflation Beyond Slow-rollIan Huston
This document outlines research on second-order perturbations during inflation beyond the slow-roll approximation. It discusses perturbation theory at first and second order, and presents results on the source term and second-order perturbations for inflation models with features. The document also describes Pyflation, an open-source Python code for numerically calculating inflationary perturbations up to second order, and outlines future goals for the code including calculating the three-point function and incorporating multi-field models.
call for papers, research paper publishing, where to publish research paper, journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJEI, call for papers 2012,journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, research and review articles, engineering journal, International Journal of Engineering Inventions, hard copy of journal, hard copy of certificates, journal of engineering, online Submission, where to publish research paper, journal publishing, international journal, publishing a paper, hard copy journal, engineering journal
This document discusses Hidden Markov Models (HMMs) and Markov chains. It begins with an introduction to Markov processes and how HMMs are used in various domains like natural language processing. It then describes the properties of a Markov chain, which has a set of states that the system transitions between randomly at discrete time steps based on transition probabilities. The Markov property is explained as the conditional independence of future states from past states given the present state. HMMs extend Markov chains by making the state sequence hidden and only allowing observation of the output states.
This document summarizes a paper that applies multivariate integer-valued autoregressive (MINAR) models to earthquake count data. Specifically, it generalizes existing bivariate integer-valued autoregressive (BINAR) models and applies the generalized BINAR model to study the joint dynamics of earthquake counts across different tectonic plates over time. The paper derives several theoretical properties of MINAR models, including unconditional mean, variance, autocorrelations, and forecasting moments. It then analyzes earthquake count data from various tectonic plate pairs to investigate space-time clustering of earthquakes and the impact of medium and large earthquakes on counts over time.
1) The document discusses 1D kinematics graphs of position, velocity, and acceleration over time.
2) The slope of a position vs. time graph gives velocity, while the slope of a velocity vs. time graph gives acceleration.
3) Taking the derivative of a position, velocity, or acceleration function provides the next variable (velocity, acceleration, or change in acceleration), just as taking the integral of acceleration or velocity functions provides position or change in velocity.
This presentation explains how the differentiation is applied to identify increasing and decreasing functions,identifying the nature of stationary points and also finding maximum or minimum values.
[shaderx5] 4.2 Multisampling Extension for Gradient Shadow Maps종빈 오
This document discusses techniques for improving gradient shadow maps, including merging gradient shadow maps with percentage-closer filtering (PCF) by calculating gradients over PCF areas and optimizing texture lookups. It introduces using slope-scale depth bias, fuzzy depth comparison, and linearly filtered depth values to reduce acne artifacts. It then describes combining gradient shadow maps and PCF by calculating average gradients over PCF areas to reduce sampling counts.
This document discusses various two-dimensional geometric transformations including translations, rotations, scaling, reflections, shears, and composite transformations. Translations move objects without deformation using a translation vector. Rotations rotate objects around a fixed point or pivot point. Scaling transformations enlarge or shrink objects using scaling factors. Reflections produce a mirror image of an object across an axis. Shearing slants an object along an axis. Composite transformations combine multiple basic transformations using matrix multiplication.
This document introduces the concepts of linear acceleration, computing acceleration from changes in velocity over time, and the differences between average and instantaneous acceleration. It provides examples of calculating acceleration from velocity data and graphs, and reviews the laws of constant acceleration for relating changes in velocity, displacement, and time when acceleration is constant. Key topics covered include computing acceleration from changes in velocity and time, using graphs of velocity over time to determine acceleration, and applying the kinematic equations for constant acceleration.
This document discusses 2D and 3D transformations. It begins with an overview of basic 2D transformations like translation, scaling, rotation, and shearing. It then covers representing transformations with matrices and combining transformations through matrix multiplication. Homogeneous coordinates are introduced as a way to represent translations with matrices. The key transformations can all be represented as 3x3 matrices using homogeneous coordinates.
Integral calculus allows us to calculate quantities like distance traveled, work done, and area under a curve by summing up infinitely many infinitesimally small quantities. The three examples given all involve calculating a quantity that is the product of two factors where one factor varies with respect to the other over an interval. Integral calculus provides a way to find the total of this varying product by breaking it into infinitely many strips and adding them up. Graphically, the definite integral represents the area under a function over an interval, with the area of each strip having a physical meaning relevant to the application.
Condition Monitoring Of Unsteadily Operating EquipmentJordan McBain
The document discusses techniques for condition monitoring of unsteadily operating equipment. It proposes a statistical parameterization approach involving segmenting vibration data based on steady speeds/loads, extracting statistical parameters from segments, and using novelty detection with support vectors to classify patterns as normal or faulted while accounting for changing operating conditions. Experimental results on gearbox data demonstrated superior fault detection performance compared to alternative approaches.
Composition Of Functions & Difference Quotientcpirie0607
The document discusses composition of functions and the difference quotient. It provides an example of using the composition of functions to determine the area of an oil patch given the increasing radius over time. It then defines the difference quotient as the formula used to compute the slope of the secant line through two points on a function graph. An example problem demonstrates finding the difference quotient for the function φ(x)=2x^2-3x.
Composition Of Functions & Difference Quotientcpirie0607
The document discusses composition of functions and the difference quotient. It provides an example of using the composition of functions to determine the area of an oil patch given the increasing radius over time. It then defines the difference quotient as a formula to compute the slope of a secant line and uses an example function to demonstrate finding the difference quotient through simplifying the expression.
This document provides an overview of the Fourier transform and discrete Fourier transform (DFT). It introduces complex numbers and periodic sine and cosine functions used as the basis for Fourier analysis. The Fourier transform decomposes a signal into its frequency components, and the DFT does the same for discrete, finite signals. Key properties of the DFT include separability, periodicity, translation, rotation, and how operations in the spatial and frequency domains relate. The magnitude and phase components of the DFT encode important information about the original signal.
The document discusses key concepts in calculus that are applied to physical systems, including:
- Displacement (distance from a point with direction)
- Velocity (rate of change of displacement over time, including direction)
- Acceleration (rate of change of velocity over time)
It provides examples of how to calculate acceleration, velocity, and displacement from equations describing physical phenomena. The relationships between these concepts are illustrated, such as how displacement can be determined by integrating velocity or differentiating acceleration. Calculus allows describing physical systems quantitatively and determining values like speed, acceleration, and position at different points in time.
The document discusses key concepts in calculus that are applied to physical systems, including:
- Displacement (distance from a point with direction)
- Velocity (rate of change of displacement over time, including direction)
- Acceleration (rate of change of velocity over time)
It provides examples of how to calculate acceleration, velocity, and displacement from equations describing physical phenomena. The relationships between these concepts are illustrated, such as how displacement can be determined by integrating velocity or differentiating acceleration. Calculus allows describing physical systems quantitatively and analyzing properties like when an object is stationary.
Similar to 12 x1 t04 06 integrating functions of time (2013) (14)
Slideshare is discontinuing its Slidecast feature as of February 28, 2014. Existing Slidecasts will be converted to static presentations without audio by April 30, 2014. The document informs users that new slidecasts can be found on myPlick.com or the author's blog starting in 2014. However, myPlick proved unreliable, so future slidecasts will instead be hosted on the author's YouTube channel.
The document discusses different methods for factorising expressions:
1) Looking for a common factor and dividing it out of all terms
2) Using the difference of two squares formula (a2 - b2 = (a - b)(a + b))
3) Factorising quadratic trinomials into two binomial factors by identifying the values that multiply to give the constant term and sum to give the coefficient of the linear term.
The document provides information on index laws and the meaning of indices in algebra:
- Index laws state that am × an = am+n, am ÷ an = am-n, and (am)n = amn. Exponents can be added or subtracted when multiplying or dividing terms with the same base.
- Positive exponents indicate a term is raised to a power. Negative exponents indicate a root is being taken. Terms with exponents are evaluated from left to right.
- Examples demonstrate how to simplify expressions using index laws and interpret different types of indices.
12 x1 t01 03 integrating derivative on function (2013)Nigel Simmons
The document discusses integrating derivatives of functions. It states that the integral of the derivative of a function f(x) is equal to the natural log of f(x) plus a constant. It then provides examples of integrating several derivatives: (i) ∫(1/(7-3x)) dx = -1/3 log(7-3x) + c, (ii) ∫(1/(8x+5)) dx = 1/8 log(8x+5) + c, and (iii) ∫(x5/(x-2)) dx = 1/6 log(x6-2) + c. It also discusses techniques for integrating fractions by polynomial long division and finds
The document discusses logarithms and their properties. Logarithms are defined as the inverse of exponentials. If y = ax, then x = loga y. The natural logarithm is log base e, written as ln. Properties of logarithms include: loga m + loga n = loga mn; loga m - loga n = loga(m/n); loga mn = n loga m; loga 1 = 0; loga a = 1. Examples of evaluating logarithmic expressions are provided.
The document discusses relationships between the coefficients and roots of polynomials. It states that for a polynomial P(x) = axn + bxn-1 + cxn-2 + ..., the sum of the roots equals -b/a, the sum of the roots taken two at a time equals c/a, and so on for higher order terms. It also provides examples of using these relationships to find the sums of roots for a given polynomial.
P
4
3
2
The document discusses properties of polynomials with multiple roots. It first proves that if a polynomial P(x) has a root x = a of multiplicity m, then the derivative of P(x), P'(x), will have a root x = a of multiplicity m-1. It then provides an example of solving a cubic equation given it has a double root. Finally, it examines a quartic polynomial and shows that its root α cannot be 0, 1, or -1, and that 1/
The document discusses factorizing complex expressions. The main points are:
- If a polynomial's coefficients are real, its roots will appear in complex conjugate pairs.
- Any polynomial of degree n can be factorized into a mixture of quadratic and linear factors over real numbers, or into n linear factors over complex numbers.
- Odd degree polynomials must have at least one real root.
- Examples of factorizing polynomials over both real and complex numbers are provided.
The document describes the Trapezoidal Rule for approximating the area under a curve between two points. It shows that the area A is estimated by dividing the region into trapezoids with height equal to the function values at the interval endpoints and bases equal to the intervals. In general, the area is approximated as the sum of the areas of each trapezoid, which is equal to the average of the endpoint function values multiplied by the interval length.
The document discusses methods for calculating the volumes of solids of revolution. It provides formulas for finding volumes when an area is revolved around either the x-axis or y-axis. Examples are given for finding volumes of common solids like cones, spheres, and others. Steps are shown for using the formulas to calculate volumes based on given functions and limits of revolution.
The document discusses different methods for calculating the area under a curve or between curves.
(1) The area below the x-axis is given by the integral of the function between the bounds, which can be positive or negative depending on whether the area is above or below the x-axis.
(2) To calculate the area on the y-axis, the function is solved for x in terms of y, then the bounds are substituted into the integral of this new function with respect to y.
(3) The area between two curves is calculated by taking the integral of the upper curve minus the integral of the lower curve, both between the same bounds on the x-axis.
The document discusses 8 properties of definite integrals:
1) Integrating polynomials results in a fraction.
2) Constants can be factored out of integrals.
3) Integrals of sums are equal to the sum of integrals.
4) Splitting an integral range results in the sum of the integrals.
5) Integrals of positive functions over a range are positive, and negative if the function is negative.
6) Integrals can be compared based on the relative values of the integrands.
7) Changing the limits of integration flips the sign of the integral.
8) Integrals of odd functions over a symmetric range are zero, and integrals of even functions are twice the integral over
Philippine Edukasyong Pantahanan at Pangkabuhayan (EPP) CurriculumMJDuyan
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𝐄𝐱𝐩𝐥𝐚𝐢𝐧 𝐭𝐡𝐞 𝐍𝐚𝐭𝐮𝐫𝐞 𝐚𝐧𝐝 𝐒𝐜𝐨𝐩𝐞 𝐨𝐟 𝐚𝐧 𝐄𝐧𝐭𝐫𝐞𝐩𝐫𝐞𝐧𝐞𝐮𝐫:
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Temple of Asclepius in Thrace. Excavation resultsKrassimira Luka
The temple and the sanctuary around were dedicated to Asklepios Zmidrenus. This name has been known since 1875 when an inscription dedicated to him was discovered in Rome. The inscription is dated in 227 AD and was left by soldiers originating from the city of Philippopolis (modern Plovdiv).
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8. Derivative Graphs
Function 1st derivative 2nd derivative
displacement velocity acceleration
stationary point x intercept
9. Derivative Graphs
Function 1st derivative 2nd derivative
displacement velocity acceleration
stationary point x intercept
inflection point stationary point x intercept
10. Derivative Graphs
Function 1st derivative 2nd derivative
displacement velocity acceleration
stationary point x intercept
inflection point stationary point x intercept
increasing positive
11. Derivative Graphs
Function 1st derivative 2nd derivative
displacement velocity acceleration
stationary point x intercept
inflection point stationary point x intercept
increasing positive
decreasing negative
12. Derivative Graphs
Function 1st derivative 2nd derivative
displacement velocity acceleration
stationary point x intercept
inflection point stationary point x intercept
increasing positive
decreasing negative
concave up increasing positive
13. Derivative Graphs
Function 1st derivative 2nd derivative
displacement velocity acceleration
stationary point x intercept
inflection point stationary point x intercept
increasing positive
decreasing negative
concave up increasing positive
concave down decreasing negative
15. graph type integrate differentiate
horizontal line oblique line x axis
16. graph type integrate differentiate
horizontal line oblique line x axis
oblique line parabola horizontal line
17. graph type integrate differentiate
horizontal line oblique line x axis
oblique line parabola horizontal line
parabola cubic oblique line
inflects at turning pt
18. graph type integrate differentiate
horizontal line oblique line x axis
oblique line parabola horizontal line
parabola cubic oblique line
inflects at turning pt
Remember:
• integration = area
19. graph type integrate differentiate
horizontal line oblique line x axis
oblique line parabola horizontal line
parabola cubic oblique line
inflects at turning pt
Remember:
• integration = area
• on a velocity graph, total area = distance
total integral = displacement
20. graph type integrate differentiate
horizontal line oblique line x axis
oblique line parabola horizontal line
parabola cubic oblique line
inflects at turning pt
Remember:
• integration = area
• on a velocity graph, total area = distance
total integral = displacement
• on an acceleration graph, total area = speed
total integral = velocity
21. (ii) 2003 HSC Question 7b)
The velocity of a particle is given by v 2 4 cos t for 0 t 2 ,
where v is measured in metres per second and t is measured in seconds
(i) At what times during this period is the particle at rest?
22. (ii) 2003 HSC Question 7b)
The velocity of a particle is given by v 2 4 cos t for 0 t 2 ,
where v is measured in metres per second and t is measured in seconds
(i) At what times during this period is the particle at rest?
v0
2 4cos t 0
1
cos t
2
23. (ii) 2003 HSC Question 7b)
The velocity of a particle is given by v 2 4 cos t for 0 t 2 ,
where v is measured in metres per second and t is measured in seconds
(i) At what times during this period is the particle at rest?
v0 Q1, 4
2 4cos t 0 1
1 cos
cos t 2
2
3
24. (ii) 2003 HSC Question 7b)
The velocity of a particle is given by v 2 4 cos t for 0 t 2 ,
where v is measured in metres per second and t is measured in seconds
(i) At what times during this period is the particle at rest?
v0 Q1, 4 t , 2
2 4cos t 0 1 5
1 cos t ,
cos t 2 3 3
2
3
5
particle is at rest after seconds and again after seconds
3 3
25. (ii) 2003 HSC Question 7b)
The velocity of a particle is given by v 2 4 cos t for 0 t 2 ,
where v is measured in metres per second and t is measured in seconds
(i) At what times during this period is the particle at rest?
v0 Q1, 4 t , 2
2 4cos t 0 1 5
1 cos t ,
cos t 2 3 3
2
3
5
particle is at rest after seconds and again after seconds
3 3
(ii) What is the maximum velocity of the particle during this period?
26. (ii) 2003 HSC Question 7b)
The velocity of a particle is given by v 2 4 cos t for 0 t 2 ,
where v is measured in metres per second and t is measured in seconds
(i) At what times during this period is the particle at rest?
v0 Q1, 4 t , 2
2 4cos t 0 1 5
1 cos t ,
cos t 2 3 3
2
3
5
particle is at rest after seconds and again after seconds
3 3
(ii) What is the maximum velocity of the particle during this period?
4 4 cos t 4
2 2 4 cos t 6
maximum velocity is 6 m/s
28. (iii) Sketch the graph of v as a function of t for 0 t 2
2 2
amplitude 4 units period divisions
shift 2 units 1 4
2
flip upside down
2
29. (iii) Sketch the graph of v as a function of t for 0 t 2
2 2
amplitude 4 units period divisions
shift 2 units 1 4
2
flip upside down
2
v
6
5
4
3
2
1
-1
3 2 t
-2
2 2
30. (iii) Sketch the graph of v as a function of t for 0 t 2
2 2
amplitude 4 units period divisions
shift 2 units 1 4
2
flip upside down
2
v
6
5
4
3
2
1
-1
3 2 t
-2
2 2
31. (iii) Sketch the graph of v as a function of t for 0 t 2
2 2
amplitude 4 units period divisions
shift 2 units 1 4
2
flip upside down
2
v
6
5
4
3
2
1
-1
3 2 t
-2
2 2
32. (iii) Sketch the graph of v as a function of t for 0 t 2
2 2
amplitude 4 units period divisions
shift 2 units 1 4
2
flip upside down
2
v
6
5
4
3
2
1
-1
3 2 t
-2
2 2
33. (iii) Sketch the graph of v as a function of t for 0 t 2
2 2
amplitude 4 units period divisions
shift 2 units 1 4
2
flip upside down
2
v
6
5 v 2 4 cos t
4
3
2
1
-1
3 2 t
-2
2 2
34. (iv) Calculate the total distance travelled by the particle between t = 0
and t =
35. (iv) Calculate the total distance travelled by the particle between t = 0
and t =
3
distance = 2 4 cos t dt 2 4 cos t dt
0
3
36. (iv) Calculate the total distance travelled by the particle between t = 0
and t =
3
distance = 2 4 cos t dt 2 4 cos t dt
0
= 2t 4sin t 2t 3 4sin t
0
3 3
37. (iv) Calculate the total distance travelled by the particle between t = 0
and t =
3
distance = 2 4 cos t dt 2 4 cos t dt
0
= 2t 4sin t 2t 3 4sin t
0
3 3
2 4sin
= 0 0 2 4sin 2
3 3
2 4 3
=2 2
3 2
2
=4 3 metres
3
38. (iii) 2004 HSC Question 9b)
A particle moves along the x-axis. Initially it is at rest at the origin.
The graph shows the acceleration, a, of the particle as a function of
time t for 0 t 5
(i) Write down the time at which the velocity of the particle is a maximum
39. (iii) 2004 HSC Question 9b)
A particle moves along the x-axis. Initially it is at rest at the origin.
The graph shows the acceleration, a, of the particle as a function of
time t for 0 t 5
(i) Write down the time at which the velocity of the particle is a maximum
v adt
adt is a maximum when t 2
40. (iii) 2004 HSC Question 9b)
A particle moves along the x-axis. Initially it is at rest at the origin.
The graph shows the acceleration, a, of the particle as a function of
time t for 0 t 5
(i) Write down the time at which the velocity of the particle is a maximum
dv
v adt OR v is a maximum when 0
dt
adt is a maximum when t 2
41. (iii) 2004 HSC Question 9b)
A particle moves along the x-axis. Initially it is at rest at the origin.
The graph shows the acceleration, a, of the particle as a function of
time t for 0 t 5
(i) Write down the time at which the velocity of the particle is a maximum
dv
v adt OR v is a maximum when 0
dt
adt is a maximum when t 2
velocity is a maximum when t 2 seconds
42. (ii) At what time during the interval 0 t 5 is the particle furthest
from the origin? Give reasons for your answer.
43. (ii) At what time during the interval 0 t 5 is the particle furthest
from the origin? Give reasons for your answer.
Question is asking, “when is displacement a maximum?”
dx
x is a maximum when 0
dt
44. (ii) At what time during the interval 0 t 5 is the particle furthest
from the origin? Give reasons for your answer.
Question is asking, “when is displacement a maximum?”
dx
x is a maximum when 0
dt
But v adt
We must solve adt 0
45. (ii) At what time during the interval 0 t 5 is the particle furthest
from the origin? Give reasons for your answer.
Question is asking, “when is displacement a maximum?”
dx
x is a maximum when 0
dt
But v adt
We must solve adt 0
i.e. when is area above the axis = area below
By symmetry this would be at t = 4
particle is furthest from the origin at t 4 seconds
46. (iv) 2007 HSC Question 10a) dx
An object is moving on the x-axis. The graph shows the velocity, ,
dt
of the object, as a function of t.
The coordinates of the points shown on the graph are A(2,1), B(4,5),
C(5,0) and D(6,–5). The velocity is constant for t 6
(i) Using Simpson’s rule, estimate the distance travelled between t = 0
and t = 4
47. (iv) 2007 HSC Question 10a) dx
An object is moving on the x-axis. The graph shows the velocity, ,
dt
of the object, as a function of t.
The coordinates of the points shown on the graph are A(2,1), B(4,5),
C(5,0) and D(6,–5). The velocity is constant for t 6
(i) Using Simpson’s rule, estimate the distance travelled between t = 0
and t = 4 h
distance y0 4 yodd 2 yeven yn
3
48. (iv) 2007 HSC Question 10a) dx
An object is moving on the x-axis. The graph shows the velocity, ,
dt
of the object, as a function of t.
The coordinates of the points shown on the graph are A(2,1), B(4,5),
C(5,0) and D(6,–5). The velocity is constant for t 6
(i) Using Simpson’s rule, estimate the distance travelled between t = 0
and t = 4 h
distance y0 4 yodd 2 yeven yn
3
t 0 2 4
v 0 1 5
49. (iv) 2007 HSC Question 10a) dx
An object is moving on the x-axis. The graph shows the velocity, ,
dt
of the object, as a function of t.
The coordinates of the points shown on the graph are A(2,1), B(4,5),
C(5,0) and D(6,–5). The velocity is constant for t 6
(i) Using Simpson’s rule, estimate the distance travelled between t = 0
and t = 4 h
distance y0 4 yodd 2 yeven yn
3
1 4 1
t 0 2 4
v 0 1 5
50. (iv) 2007 HSC Question 10a) dx
An object is moving on the x-axis. The graph shows the velocity, ,
dt
of the object, as a function of t.
The coordinates of the points shown on the graph are A(2,1), B(4,5),
C(5,0) and D(6,–5). The velocity is constant for t 6
(i) Using Simpson’s rule, estimate the distance travelled between t = 0
and t = 4 h
distance y0 4 yodd 2 yeven yn
3
1 4 1 2
0 4 1 5
t 0 2 4 3
v 0 1 5
6 metres
51. (ii) The object is initially at the origin. During which time(s) is the
displacement decreasing?
52. (ii) The object is initially at the origin. During which time(s) is the
displacement decreasing?
dx
x is decreasing when 0
dt
displacement is decreasing when t 5 seconds
53. (ii) The object is initially at the origin. During which time(s) is the
displacement decreasing?
dx
x is decreasing when 0
dt
displacement is decreasing when t 5 seconds
(iii) Estimate the time at which the object returns to the origin. Justify
your answer.
54. (ii) The object is initially at the origin. During which time(s) is the
displacement decreasing?
dx
x is decreasing when 0
dt
displacement is decreasing when t 5 seconds
(iii) Estimate the time at which the object returns to the origin. Justify
your answer.
Question is asking, “when is displacement = 0?”
55. (ii) The object is initially at the origin. During which time(s) is the
displacement decreasing?
dx
x is decreasing when 0
dt
displacement is decreasing when t 5 seconds
(iii) Estimate the time at which the object returns to the origin. Justify
your answer.
Question is asking, “when is displacement = 0?”
But x vdt
We must solve vdt 0
56. (ii) The object is initially at the origin. During which time(s) is the
displacement decreasing?
dx
x is decreasing when 0
dt
displacement is decreasing when t 5 seconds
(iii) Estimate the time at which the object returns to the origin. Justify
your answer.
Question is asking, “when is displacement = 0?”
But x vdt
We must solve vdt 0
i.e. when is area above the axis = area below
57. (ii) The object is initially at the origin. During which time(s) is the
displacement decreasing?
dx
x is decreasing when 0
dt
displacement is decreasing when t 5 seconds
(iii) Estimate the time at which the object returns to the origin. Justify
your answer.
Question is asking, “when is displacement = 0?”
But x vdt
We must solve vdt 0
i.e. when is area above the axis = area below
By symmetry, area from t = 4 to 5 equals area
from t = 5 to 6
58. (ii) The object is initially at the origin. During which time(s) is the
displacement decreasing?
dx
x is decreasing when 0
dt
displacement is decreasing when t 5 seconds
(iii) Estimate the time at which the object returns to the origin. Justify
your answer.
Question is asking, “when is displacement = 0?”
But x vdt
We must solve vdt 0
i.e. when is area above the axis = area below
By symmetry, area from t = 4 to 5 equals area
from t = 5 to 6
In part (i) we estimated area from t = 0 to 4 to be 6,
59. (ii) The object is initially at the origin. During which time(s) is the
displacement decreasing?
dx
x is decreasing when 0
dt
displacement is decreasing when t 5 seconds
(iii) Estimate the time at which the object returns to the origin. Justify
your answer.
Question is asking, “when is displacement = 0?”
But x vdt
We must solve vdt 0
i.e. when is area above the axis = area below
By symmetry, area from t = 4 to 5 equals area A4
from t = 5 to 6
In part (i) we estimated area from t = 0 to 4 to be 6,
A4 6
60. (ii) The object is initially at the origin. During which time(s) is the
displacement decreasing?
dx
x is decreasing when 0
dt
displacement is decreasing when t 5 seconds
(iii) Estimate the time at which the object returns to the origin. Justify
your answer.
Question is asking, “when is displacement = 0?”
But x vdt
We must solve vdt 0
a
i.e. when is area above the axis = area below
By symmetry, area from t = 4 to 5 equals area A4 5
from t = 5 to 6
In part (i) we estimated area from t = 0 to 4 to be 6,
A4 6 a 1.2
5a 6
61. (ii) The object is initially at the origin. During which time(s) is the
displacement decreasing?
dx
x is decreasing when 0
dt
displacement is decreasing when t 5 seconds
(iii) Estimate the time at which the object returns to the origin. Justify
your answer.
Question is asking, “when is displacement = 0?”
But x vdt
We must solve vdt 0
a
i.e. when is area above the axis = area below
By symmetry, area from t = 4 to 5 equals area A4 5
from t = 5 to 6
In part (i) we estimated area from t = 0 to 4 to be 6,
A4 6 a 1.2
5a 6 particle returns to the origin when t 7.2 seconds
63. (iv) Sketch the displacement, x, as a function of time.
x
8.5
6
2 4 6 8 t
64. (iv) Sketch the displacement, x, as a function of time.
object is initially at the origin
x
8.5
6
2 4 6 8 t
65. (iv) Sketch the displacement, x, as a function of time.
object is initially at the origin
when t = 4, x = 6
x
8.5
6
2 4 6 8 t
66. (iv) Sketch the displacement, x, as a function of time.
object is initially at the origin
when t = 4, x = 6
by symmetry of areas t = 6, x = 6
x
8.5
6
2 4 6 8 t
67. (iv) Sketch the displacement, x, as a function of time.
object is initially at the origin
when t = 4, x = 6
by symmetry of areas t = 6, x = 6
Area of triangle = 2.5
when t 5, x 8.5
x
8.5
6
2 4 6 8 t
68. (iv) Sketch the displacement, x, as a function of time.
object is initially at the origin
when t = 4, x = 6
by symmetry of areas t = 6, x = 6
Area of triangle = 2.5
when t 5, x 8.5
returns to x = 0 when t = 7.2
x
8.5
6
2 4 6 7.2 8 t
69. (iv) Sketch the displacement, x, as a function of time.
object is initially at the origin
when t = 4, x = 6
by symmetry of areas t = 6, x = 6
Area of triangle = 2.5
when t 5, x 8.5
returns to x = 0 when t = 7.2
x
v is steeper between t = 2 and 4
8.5
than between t = 0 and 2
6 particle covers more distance
between t 2 and 4
2 4 6 7.2 8 t
70. (iv) Sketch the displacement, x, as a function of time.
object is initially at the origin
when t = 4, x = 6
by symmetry of areas t = 6, x = 6
Area of triangle = 2.5
when t 5, x 8.5
returns to x = 0 when t = 7.2
x
v is steeper between t = 2 and 4
8.5
than between t = 0 and 2
6 particle covers more distance
between t 2 and 4
when t > 6, v is constant
t when t 6, x is a straight line
2 4 6 7.2 8
71. (iv) Sketch the displacement, x, as a function of time.
object is initially at the origin
when t = 4, x = 6
by symmetry of areas t = 6, x = 6
Area of triangle = 2.5
when t 5, x 8.5
returns to x = 0 when t = 7.2
x
v is steeper between t = 2 and 4
8.5
than between t = 0 and 2
6 particle covers more distance
between t 2 and 4
when t > 6, v is constant
t when t 6, x is a straight line
2 4 6 7.2 8
72. (v) 2005 HSC Question 7b)
dx
The graph shows the velocity, dt , of a particle as a function of time.
Initially the particle is at the origin.
(i) At what time is the displacement, x, from the origin a maximum?
73. (v) 2005 HSC Question 7b)
dx
The graph shows the velocity, dt , of a particle as a function of time.
Initially the particle is at the origin.
(i) At what time is the displacement, x, from the origin a maximum?
Displacement is a maximum when area is most positive, also when
velocity is zero
i.e. when t = 2
74. (ii) At what time does the particle return to the origin? Justify
your answer
75. (ii) At what time does the particle return to the origin? Justify
your answer
Question is asking, “when is displacement = 0?”
i.e. when is area above the axis = area below?
76. (ii) At what time does the particle return to the origin? Justify
your answer
2 a w
a 2
Question is asking, “when is displacement = 0?”
i.e. when is area above the axis = area below?
77. (ii) At what time does the particle return to the origin? Justify
your answer
2 a w
a 2
Question is asking, “when is displacement = 0?”
i.e. when is area above the axis = area below?
2w = 2
w=1
78. (ii) At what time does the particle return to the origin? Justify
your answer
2 a w
a 2
Question is asking, “when is displacement = 0?”
i.e. when is area above the axis = area below?
2w = 2
w=1
Returns to the origin after 4 seconds
79. d 2x
(iii) Draw a sketch of the acceleration, 2 , as afunction of
dt
time for 0 t 6
d 2x
dt 2
1 2 3 5 6 t
80. d 2x
(iii) Draw a sketch of the acceleration, 2 , as afunction of
dt
time for 0 t 6 differentiate a horizontal line
you get the xaxis
d 2x
dt 2
1 2 3 5 6 t
81. d 2x
(iii) Draw a sketch of the acceleration, 2 , as afunction of
dt
time for 0 t 6 differentiate a horizontal line
you get the xaxis
from 1 to 3 we have a cubic,
inflects at 2, and is decreasing
differentiate, you get a parabola,
stationary at 2, it is below the x axis
d 2x
dt 2
1 2 3 5 6 t
82. d 2x
(iii) Draw a sketch of the acceleration, 2 , as afunction of
dt
time for 0 t 6 differentiate a horizontal line
you get the xaxis
from 1 to 3 we have a cubic,
inflects at 2, and is decreasing
differentiate, you get a parabola,
stationary at 2, it is below the x axis
d 2x
dt 2
from 5 to 6 is a cubic, inflects at 6
and is increasing (using symmetry)
differentiate, you get a parabola
1 2 3 5 6 t
stationary at 6, it is above the x axis
83. d 2x
(iii) Draw a sketch of the acceleration, 2 , as afunction of
dt
time for 0 t 6 differentiate a horizontal line
you get the xaxis
from 1 to 3 we have a cubic,
inflects at 2, and is decreasing
differentiate, you get a parabola,
stationary at 2, it is below the x axis
d 2x
dt 2
from 5 to 6 is a cubic, inflects at 6
and is increasing (using symmetry)
differentiate, you get a parabola
1 2 3 5 6 t
stationary at 6, it is above the x axis