Differential Equations Restricting the Domain of the Solution.

Slides:

Advertisements

Similar presentations

Sketch the derivative of the function given by the following graph:

Advertisements

Equations of Tangent Lines

The Derivative and the Tangent Line Problem. Local Linearity.

Think, Think, Think. Algebra I Seminar. How Much Do you Remember? The Coordinate Plane X-axis, Y-axis Slope Y-intercept Ordered Pairs Slope Intercept.

Don’t need graph paper. Parallel & Perpendicular Lines Two lines are Parallel  slopes are = Two lines are Perpendicular  negative reciprocal slopes.

Objective To be able to recognize Horizontal and Vertical lines on the coordinate plane.

Slope Supplement 2 Find a slope Graphing with Slope.

Writing Parallel and Perpendicular Lines Part 1. Parallel Lines // (symbol) All parallel lines have the same slope. Parallel lines will NEVER CROSS. Since.

Notes Over 4.3 Finding Intercepts Find the x-intercept of the graph of the equation. x-intercept y-intercept The x value when y is equal to 0. Place where.

Slope Intercept Form & Graphing Definition - Intercepts x-intercept y-intercept BACK The x-intercept of a straight line is the x-coordinate of the point.

Rectangular Coordinate System

Do Now Find the slope of the line passing through the given points. 1)( 3, – 2) and (4, 5) 2)(2, – 7) and (– 1, 4)

Slope-Intercept Form Page 22 10/15. Vocabulary y-Intercept: the point at which a function crosses the y-axis (0, y) x-intercept: the point at which a.

4.2 Graphing Linear Functions Use tables and equations to graph linear functions.

Rectangular Coordinate System

M. Pickens 2006 Slope. M. Pickens 2006 Objectives To learn what slope is To learn what a line looks like when it has positive, negative, zero or undefined.

Anthony Poole & Keaton Mashtare 2 nd Period. X and Y intercepts  The points at which the graph crosses or touches the coordinate axes are called intercepts.

W-up Get out note paper Find 12.3 notes

Graphing Using x & y Intercepts

Graphing Rational Functions Example #5 End ShowEnd ShowSlide #1 NextNext We want to graph this rational function showing all relevant characteristics.

What is the slope of a line parallel to the line seen below? m= -1/3

How do I graph and use exponential growth and decay functions?

Thinking Mathematically Algebra: Graphs, Functions and Linear Systems 7.2 Linear Functions and Their Graphs.

Journal Entry Equation of a Line May 1, Slope Slope is a measure of the steepness of a line. Slope is calculated as. Remember rise is the vertical.

Slopes and Parallel Lines Goals: To find slopes of lines To identify parallel lines To write equations of parallel lines.

6-8 Graphing Radical Functions

What is a Line? A line is the set of points forming a straight path on a plane The slant (slope) between any two points on a line is always equal A line.

Linear Equations in Two Variables

What are the characteristics of Lines in the Plane? Section P4 (new text)

Graphing Reciprocal Functions

Graphs with restricted(forbidden) regions!. From this information, we can complete the sketch Note that the graph is symmetrical about the y-axis, so.

X AND Y INTERCEPTS. INTERCEPT  In a function, an intercept is the point at which the graph of the line crosses an axis.  If it crosses the y-axis it.

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)

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.

6.5 Point – Slope Form. Quick Review 1. What is the Slope-Intercept Form? y = mx + b 2. What is the Standard Form? Ax + By = C 3. How does an equation.

Equations of Lines Standard Form: Slope Intercept Form: where m is the slope and b is the y-intercept.

P7 The Cartesian Plane. Quick review of graphing, axes, quadrants, origin, point plotting.

Y-intercept: the point where the graph crosses the y-axis, the value of x must = 0. find by graphing or plugging in 0 for x and solving.

7.1 R eview of Graphs and Slopes of Lines Standard form of a linear equation: The graph of any linear equation in two variables is a straight line. Note:

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

MTH 091 Section 13.3 Graphing with x- and y-intercepts Section 13.4 Slope.

Graphs & Models (Precalculus Review 1) September 8th, 2015.

IFDOES F(X) HAVE AN INVERSE THAT IS A FUNCTION? Find the inverse of f(x) and state its domain.

This screen shows two lines which have exactly one point in common. The common point when substituted into the equation of each line makes that equation.

5.3 Slope-Intercept Form I can write linear equations using slope-intercept form and graph linear equations in slope-intercept form.

Writing Equations of Lines. Find the equation of a line that passes through (2, -1) and (-4, 5).

Properties of Functions. First derivative test. 1.Differentiate 2.Set derivative equal to zero 3.Use nature table to determine the behaviour of.

Lesson 3-4: Graphs of Linear Equations Objective: Students will: recognize linear equations graph linear equations graph horizontal and vertical lines.

Chapter 7 Graphing Linear Equations REVIEW. Section 7.1 Cartesian Coordinate System is formed by two axes drawn perpendicular to each other. Origin is.

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.

SECT 3-8B RELATING GRAPHS Handout: Relating Graphs.

Creating a foldable for Equations of Lines Equations of Lines Slope Slope-Intercept Form Standard Form Horizontal & Vertical Lines Graphing Lines X-intercept.

Section 3.8. Derivatives of Inverse Functions Theorem: If is differentiable at every point of an interval I and is never zero on I, then has an inverse.

6.1: DIFFERENTIAL EQUATIONS AND SLOPE FIELDS. DEFINITION: DIFFERENTIAL EQUATION An equation involving a derivative is called a differential equation.

1 6.1 Slope Fields and Euler's Method Objective: Solve differential equations graphically and numerically.

Parallel and Perpendicular Lines. Overview This set of tutorials provides 32 examples that involve finding the equation of a line parallel or perpendicular.

6.1 – 6.3 Differential Equations

Slope Slope is the steepness of a straight line..

STANDARD FORM OF A LINEAR EQUATION

The Derivative and the Tangent Line Problem (2.1)

Sketching the Derivative

The Derivative Chapter 3.1 Continued.

26. Graphing Rational Functions

The Derivative and the Tangent Line Problems

Domain and Range From a Graph

What is a system of equations?

Warm Up – 1/27 - Monday Sketch a graph of the piecewise function:

Slope Fields (6.1) January 10th, 2017.

More with Rules for Differentiation

Point-Slope Form y – y1 = m(x – x1)

Presentation transcript:

Differential Equations Restricting the Domain of the Solution

Sketch the solution of the differential equation on the given slope field:  Note that while there are slopes to the left of the y-axis, our sketch can’t ever get that far because of the vertical slope at x = 0

Because the domain can only be one interval that contains the initial value. Find the solution of the differential equation: …and the domain is… Plug in initial value (1, 1)

Because the domain can only be one interval that contains the initial value. Sketch the solution of the differential equation on the given slope field:  Note that while there are slopes to the left of the y-axis, our sketch can’t ever get that far because of the vertical slope at x = 0

Sketch the solution of the differential equation on the given slope field:  Note that because we are dealing with positive slopes only, the sketch must stop just before the x-axis

One way to see why this is the domain would be to graph our solution Find the solution of the differential equation: …and the domain is… Plug in initial value (1, 4)

One way to see why this is the domain would be to graph our solution Sketch the solution of the differential equation on the given slope field: Note that to the left of –7, the solution graph crosses the slopes instead of following them.  And the derivative is undefined for y = 0 so we must exclude x = –7

Sketch the solution of the differential equation on the given slope field: Note that to the left of –7, the solution graph crosses the slopes instead of following them.  …and we would get negative slopes for x < –7 which contradicts the slope field Because derivative is undefined for y = 0, we must exclude x = –7

The domain to any solution to a differential equation is the largest open interval containing the initial value.  It also needs to satisfy any restrictions on the derivative as well When sketching the solution, the graph should never go against the slope field. Like a continuous function, you should never have to pick your pencil up off the paper when sketching the solution