Ah yeaahhhh!!!! Calculus of V.V.F’s Derivatives and Integration of Vectors This will lead to applications of vectors involving calculus…which is the heart.

Slides:

Advertisements

Similar presentations

Chapter 11 Differentiation.

Advertisements

Differential Equations

1 Chapter 2 Differentiation: Tangent Lines. tangent In plane geometry, we say that a line is tangent to a circle if it intersects the circle in one point.

A graph of the instantaneous velocity of an object over a specified period of time Time is independent (x-axis) Velocity is dependent (y-axis) Remember,

The Calculus of Parametric Equations

3.1 Derivatives. Derivative A derivative of a function is the instantaneous rate of change of the function at any point in its domain. We say this is.

DERIVATIVE OF A FUNCTION 1.5. DEFINITION OF A DERIVATIVE OTHER FORMS: OPERATOR:,,,

Linear Kinematics Chapter 3. Definition of Kinematics Kinematics is the description of motion. Motion is described using position, velocity and acceleration.

The Derivative and the Tangent Line Problem. Local Linearity.

Rate of change / Differentiation (3)

Tangent lines Recall: tangent line is the limit of secant line The tangent line to the curve y=f(x) at the point P(a,f(a)) is the line through P with slope.

{ Semester Exam Review AP Calculus. Exam Topics Trig function derivatives.

Limits Section 15-1.

Basic Derivatives The Math Center Tutorial Services Brought To You By:

1. Use a position vs. time graph to interpret an object’s position or displacement. 2. Define velocity. 3. Differentiate between speed and velocity.

Motion Vectors. Displacement Vector  The position vector is often designated by.  A change in position is a displacement.  The displacement vector.

Business Calculus More Derivative Applications.  2.6 Differentials We often use Δx to indicate a small change in x, and Δy for a small change in y. It.

10.5 Basic Differentiation Properties. Instead of finding the limit of the different quotient to obtain the derivative of a function, we can use the rules.

Every slope is a derivative. Velocity = slope of the tangent line to a position vs. time graph Acceleration = slope of the velocity vs. time graph How.

Chapter 3 Limits and the Derivative

Graphing Linear Inequalities in Two Variables Section 6.5 Algebra I.

3.1 –Tangents and the Derivative at a Point

Constructing the Antiderivative Solving (Simple) Differential Equations The Fundamental Theorem of Calculus (Part 2) Chapter 6: Calculus~ Hughes-Hallett.

Calculus in Physics II x 0 = 0 x axis Knowing the car’s displacement history, find its velocity and acceleration. 1.

3.4 Velocity, Speed, and Rates of Change

Position, Velocity, and Acceleration. Position x.

Displacement vs Time, Velocity vs Time, and Acceleration vs Time Graphs.

Definition of the Natural Exponential Function

Slope Fields and Euler’s Method

Slope Fields. Quiz 1) Find the average value of the velocity function on the given interval: [ 3, 6 ] 2) Find the derivative of 3) 4) 5)

GOAL: USE DEFINITION OF DERIVATIVE TO FIND SLOPE, RATE OF CHANGE, INSTANTANEOUS VELOCITY AT A POINT. 3.1 Definition of Derivative.

11/10/2015Dr. Sasho MacKenzie - HK Kinematic Graphs Graphically exploring derivatives and integrals.

First, a little review: Consider: then: or It doesn’t matter whether the constant was 3 or -5, since when we take the derivative the constant disappears.

Differential Equations and Slope Fields 6.1. Differential Equations  An equation involving a derivative is called a differential equation.  The order.

Stanford University Department of Aeronautics and Astronautics Introduction to Symmetry Analysis Brian Cantwell Department of Aeronautics and Astronautics.

Review of Chapters 1, 2, and 3 Motion diagrams – both physical and pictorial representations Difference between instantaneous and average velocities Meaning.

AP Calculus AB Chapter 4, Section 1 Integration

Exponential Growth and Decay 6.4. Separation of Variables When we have a first order differential equation which is implicitly defined, we can try to.

4.1 Antiderivatives and Indefinite Integration Definition of Antiderivative: A function F is called an antiderivative of the function f if for every x.

Chapter 3 Limits and the Derivative

Separable Differential Equations

Slide 6- 1 What you’ll learn about Differential Equations Slope Fields Euler’s Method … and why Differential equations have been a prime motivation for.

Vector Valued Functions

3.1 Derivatives of a Function, p. 98 AP Calculus AB/BC.

§3.2 – The Derivative Function October 2, 2015.

2.1 Position, Velocity, and Speed 2.1 Displacement  x  x f - x i 2.2 Average velocity 2.3 Average speed  

position time position time tangent!  Derivatives are the slope of a function at a point  Slope of x vs. t  velocity - describes how position changes.

Slope Fields (6.1) March 12th, I. General & Particular Solutions A function y=f(x) is a solution to a differential equation if the equation is true.

AP CALCULUS AB Chapter 6: Differential Equations and Mathematical Modeling Section 6.1: Slope Fields and Euler’s Met hod.

Clicker Question 1 What is the derivative function f '(x ) of the function ? (Hint: Algebra first, calculus second!) A. 12x 2 – (5/2)x -1/2 B. 12x 2 –

DO NOW:  Find the equation of the line tangent to the curve f(x) = 3x 2 + 4x at x = -2.

7.2 Areas in the Plane. How can we find the area between these two curves?

Business Calculus Derivative Definition. 1.4 The Derivative The mathematical name of the formula is the derivative of f with respect to x. This is the.

What is tested is the calculus of parametric equation and vectors. No dot product, no cross product. Books often go directly to 3D vectors and do not have.

Chapter 4 Integration 4.1 Antidifferentiation and Indefinate Integrals.

Basic Derivatives Brought To You By: Tutorial Services The Math Center.

6.1 – 6.3 Differential Equations

Vectors in the Plane Section 10.2.

Position vs. time graphs Review (x vs. t)

The Derivative and the Tangent Line Problems

Physics 13 General Physics 1

Part (a) Keep in mind that dy/dx is the SLOPE! We simply need to substitute x and y into the differential equation and represent each answer as a slope.

Section 6.1 Slope Fields.

More Index cards for AB.

What Information Can We Take from Graphs?

2.7/2.8 Tangent Lines & Derivatives

Acceleration Lesson 1C Unit 1 Motion Conceptual Physics.

In the study of kinematics, we consider a moving object as a particle.

Vectors in the Plane.

Presentation transcript:

Ah yeaahhhh!!!! Calculus of V.V.F’s Derivatives and Integration of Vectors This will lead to applications of vectors involving calculus…which is the heart of all this vector stuff.

Sketch the plane curve represented by: Sketch the vector r(1).

Move the r’(1) vector to the terminal point of r(1) to interpret what it is. What does the slope of this vector represent? How is the slope of this vector just like p-metric dy/dx?

Hmmmm….the magnitude of the vector representing instantaneous rate of change. Lock this away for a lil’ bit.

If you can find the first derivative, then you can find the second derivative! What does each value mean? Do they jive with the graph?

If we can differentiate we can integrate! COMPONENT INTEGRATION We will apply this practically when we discuss pos.,vel, accel., and bounce between all three.

Integrate the following. Remember C! …Wait…what does C represent in this case?

Definite Integral (Like going from velocity curve to finding displacement)

Since we will be solving problems that give a velocity and initial height, it is important we can find particular solutions w/ an initial condition!!!

Try this beast!