- Slope fields are visual representations of differential equations of the form dy / dx = f (x, y). At each sample point of a slope field, there is a segment having slope equal to the value of dy / dx. Any curve that follows the flow suggested by the directions of the segments is a solution to the differential equation.
- Notes 7.4 Slope Fields.
Version #1
The course below follows CollegeBoard's Course and Exam Description. Lessons will begin to appear starting summer 2020.
BC Topics are listed, but there will be no lessons available for SY 2020-2021Unit 0 - Calc Prerequisites (Summer Work)
0.1 Summer Packet Unit 1 - Limits and Continuity
1.1 Can Change Occur at an Instant?
1.2 Defining Limits and Using Limit Notation
1.3 Estimating Limit Values from Graphs
1.4 Estimating Limit Values from Tables
1.5 Determining Limits Using Algebraic Properties
(1.5 includes piecewise functions involving limits)
1.6 Determining Limits Using Algebraic Manipulation
1.7 Selecting Procedures for Determining Limits
(1.7 includes rationalization, complex fractions, and absolute value)
1.8 Determining Limits Using the Squeeze Theorem
1.9 Connecting Multiple Representations of Limits
Mid-Unit Review - Unit 1
1.10 Exploring Types of Discontinuities
1.11 Defining Continuity at a Point
1.12 Confirming Continuity Over an Interval
1.13 Removing Discontinuities
1.14 Infinite Limits and Vertical Asymptotes
1.15 Limits at Infinity and Horizontal Asymptotes
1.16 Intermediate Value Theorem (IVT)
Review - Unit 1 Unit 2 - Differentiation: Definition and Fundamental Properties
2.1 Defining Average and Instantaneous Rate of
Change at a Point
2.2 Defining the Derivative of a Function and Using
Derivative Notation
(2.2 includes equation of the tangent line)
2.3 Estimating Derivatives of a Function at a Point
2.4 Connecting Differentiability and Continuity
2.5 Applying the Power Rule
2.6 Derivative Rules: Constant, Sum, Difference, and
Constant Multiple
(2.6 includes horizontal tangent lines, equation of the
normal line, and differentiability of piecewise)
2.7 Derivatives of cos(x), sin(x), e^x, and ln(x)
2.8 The Product Rule
2.9 The Quotient Rule
2.10 Derivatives of tan(x), cot(x), sec(x), and csc(x)
Review - Unit 2 Unit 3 - Differentiation: Composite, Implicit, and Inverse Functions
3.1 The Chain Rule
3.2 Implicit Differentiation
3.3 Differentiating Inverse Functions
3.4 Differentiating Inverse Trigonometric Functions
3.5 Selecting Procedures for Calculating Derivatives
3.6 Calculating Higher-Order Derivatives
Review - Unit 3
Unit 4 - Contextual Applications of Differentiation
4.1 Interpreting the Meaning of the Derivative in Context
4.2 Straight-Line Motion: Connecting Position, Velocity,
and Acceleration
4.3 Rates of Change in Applied Contexts Other Than
Motion
4.4 Introduction to Related Rates
4.5 Solving Related Rates Problems
4.6 Approximating Values of a Function Using Local
Linearity and Linearization
4.7 Using L'Hopital's Rule for Determining Limits of
Indeterminate Forms
Review - Unit 4 Unit 5 - Analytical Applications of Differentiation
5.1 Using the Mean Value Theorem
5.2 Extreme Value Theorem, Global Versus Local
Extrema, and Critical Points
5.3 Determining Intervals on Which a Function is
Increasing or Decreasing
5.4 Using the First Derivative Test to Determine Relative
Local Extrema
5.5 Using the Candidates Test to Determine Absolute
(Global) Extrema
5.6 Determining Concavity of Functions over Their
Domains
5.7 Using the Second Derivative Test to Determine
Extrema
Mid-Unit Review - Unit 5
5.8 Sketching Graphs of Functions and Their Derivatives
5.9 Connecting a Function, Its First Derivative, and Its
Second Derivative
(5.9 includes a revisit of particle motion and
determining if a particle is speeding up/down.)
5.10 Introduction to Optimization Problems
5.11 Solving Optimization Problems
5.12 Exploring Behaviors of Implicit Relations
Review - Unit 5 Unit 6 - Integration and Accumulation of Change
6.1 Exploring Accumulation of Change
6.2 Approximating Areas with Riemann Sums
6.3 Riemann Sums, Summation Notation, and Definite
Integral Notation
6.4 The Fundamental Theorem of Calculus and
Accumulation Functions
6.5 Interpreting the Behavior of Accumulation Functions
Involving Area
Mid-Unit Review - Unit 6
6.6 Applying Properties of Definite Integrals
6.7 The Fundamental Theorem of Calculus and Definite
Integrals
6.8 Finding Antiderivatives and Indefinite Integrals:
Basic Rules and Notation
6.9 Integrating Using Substitution
6.10 Integrating Functions Using Long Division
and Completing the Square
6.11 Integrating Using Integration by Parts (BC topic)
6.12 Integrating Using Linear Partial Fractions (BC topic)
6.13 Evaluating Improper Integrals (BC topic)
6.14 Selecting Techniques for Antidifferentiation
Review - Unit 6 Unit 7 - Differential Equations
7.1 Modeling Situations with Differential Equations
7.2 Verifying Solutions for Differential Equations
7.3 Sketching Slope Fields
7.4 Reasoning Using Slope Fields
7.5 Euler's Method (BC topic)
7.6 General Solutions Using Separation of Variables
7.7 Particular Solutions using Initial Conditions and
Separation of Variables
7.8 Exponential Models with Differential Equations
7.9 Logistic Models with Differential Equations (BC topic)
Review - Unit 7 Unit 8 - Applications of Integration
8.1 Average Value of a Function on an Interval
8.2 Position, Velocity, and Acceleration Using Integrals
8.3 Using Accumulation Functions and Definite Integrals
in Applied Contexts
8.4 Area Between Curves (with respect to x)
8.5 Area Between Curves (with respect to y)
8.6 Area Between Curves - More than Two Intersections
Mid-Unit Review - Unit 8
8.7 Cross Sections: Squares and Rectangles
8.8 Cross Sections: Triangles and Semicircles
8.9 Disc Method: Revolving Around the x- or y- Axis
8.10 Disc Method: Revolving Around Other Axes
8.11 Washer Method: Revolving Around the x- or y- Axis
8.12 Washer Method: Revolving Around Other Axes
8.13 The Arc Length of a Smooth, Planar Curve and
Distance Traveled (BC topic)
Review - Unit 8 Unit 9 - Parametric Equations, Polar Coordinates, and Vector-Valued Functions (BC topics)
9.1 Defining and Differentiating Parametric Equations
9.2 Second Derivatives of Parametric Equations
9.3 Arc Lengths of Curves (Parametric Equations)
9.4 Defining and Differentiating Vector-Valued Functions
9.5 Integrating Vector-Valued Functions
9.6 Solving Motion Problems Using Parametric and
Vector-Valued Functions
9.7 Defining Polar Coordinates and Differentiating in
Polar Form
9.8 Find the Area of a Polar Region or the Area Bounded
by a Single Polar Curve
9.9 Finding the Area of the Region Bounded by Two
Polar Curves
Review - Unit 9 Unit 10 - Infinite Sequences and Series (BC topics)
10.1 Defining Convergent and Divergent Infinite Series
10.2 Working with Geometric Series
10.3 The nth Term Test for Divergence
10.4 Integral Test for Convergence
10.5 Harmonic Series and p-Series
10.6 Comparison Tests for Convergence
10.7 Alternating Series Test for Convergence
10.8 Ratio Test for Convergence
10.9 Determining Absolute or Conditional Convergence
10.10 Alternating Series Error Bound
10.11 Finding Taylor Polynomial Approximations of
Functions
10.12 Lagrange Error Bound
10.13 Radius and Interval of Convergence of Power
Series
10.14 Finding Taylor Maclaurin Series for a Function
10.15 Representing Functions as a Power Series
Review - Unit 8 | Version #2
The course below covers all topics for the AP Calculus AB exam, but was built for a 90-minute class that meets every other day.
Lessons and packets are longer because they cover more material.Unit 0 - Calc Prerequisites (Summer Work)
0.1 Things to Know for Calc
0.2 Summer Packet
0.3 Calculator Skillz Unit 1 - Limits
1.1 Limits Graphically
1.2 Limits Analytically
1.3 Asymptotes
1.4 Continuity
Review - Unit 1 Unit 2 - The Derivative
2.1 Average Rate of Change
2.2 Definition of the Derivative
2.3 Differentiability [Calculator Required]
Review - Unit 2 Unit 3 - Basic Differentiation
3.1 Power Rule
3.2 Product and Quotient Rules
3.3 Velocity and other Rates of Change
3.4 Chain Rule
3.5 Trig Derivatives
Review - Unit 3 Unit 4 - More Deriviatvies
4.1 Derivatives of Exp. and Logs
4.2 Inverse Trig Derivatives
4.3 L'Hopital's Rule
Review - Unit 4 Unit 5 - Curve Sketching
5.1 Extrema on an Interval
5.2 First Derivative Test
5.3 Second Derivative Test
Review - Unit 5 Unit 6 - Implicit Differentiation
6.1 Implicit Differentiation
6.2 Related Rates
6.3 Optimization
Review - Unit 6 Unit 7 - Approximation Methods
7.1 Rectangular Approximation Method
7.2 Trapezoidal Approximation Method
Review - Unit 7 Unit 8 - Integration
8.1 Definite Integral
8.2 Fundamental Theorem of Calculus (part 1)
8.3 Antiderivatives (and specific solutions)
Review - Unit 8 Unit 9 - The 2nd Fundamental Theorem of Calculus
9.1 The 2nd FTC
9.2 Trig Integrals
9.3 Average Value (of a function)
9.4 Net Change
Review - Unit 9 Unit 10 - More Integrals
10.1 Slope Fields
10.2 u-Substitution (indefinite integrals)
10.3 u-Substitution (definite integrals)
10.4 Separation of Variables
Review - Unit 10 Unit 11 - Area and Volume
11.1 Area Between Two Curves
11.2 Volume - Disc Method
11.3 Volume - Washer Method
11.4 Perpendicular Cross Sections
Review - Unit 11 |
Asked. 07/26/19 Write the equation of the line that passes through 7, -4 and -1, 2 in slope intercept form. Coordinate grid, through the point they used to calculate their slope. This is a slope field. Technological Introduction: At the end of this article is a program for generating slope fields on the TI-83 and TI-83+. These may be used to produce the slope field of a differential equation. This lesson contains the following Essential Knowledge (EK) concepts for the.AP Calculus course.Click here for an overview of all the EK's in this course. EK 1.1A1. AP® is a trademark registered and owned by the College Board, which was not involved in the production of, and does not endorse, this site.
Calculus AB and Calculus BC
CHAPTER 9 Differential Equations
B. SLOPE FIELDS
In this section we solve differential equations by obtaining a slope field or calculator picture that approximates the general solution. We call the graph of a solution of a d.e. a solution curve.
The slope field of a d.e. is based on the fact that the d.e. can be interpreted as a statement about the slopes of its solution curves.
EXAMPLE 1
The d.e. tells us that at any point (x, y) on a solution curve the slope of the curve is equal to its y-coordinate. Since the d.e. says that y is a function whose derivative is also y, we know that
y = ex
is a solution. In fact, y = Cex is a solution of the d.e. for every constant C, since y′ = Cex = y.
The d.e. y′ = y says that, at any point where y = 1, say (0, 1) or (1, 1) or (5, 1), the slope of the solution curve is 1; at any point where y = 3, say (0, 3), (ln 3,3), or (π, 3), the slope equals 3; and so on.
In Figure N9–1a we see some small line segments of slope 1 at several points where y = 1, and some segments of slope 3 at several points where y = 3. In Figure N9–1b we see the curve of y = ex with slope segments drawn in as follows:
FIGURE N9–1a
FIGURE N9–1b
Figure N9–1c is the slope field for the d.e. Slopes at many points are represented by small segments of the tangents at those points. The small segments approximate the solution curves. If we start at any point in the slope field and move so that the slope segments are always tangent to our motion, we will trace a solution curve. The slope field, as mentioned above, closely approximates the family of solutions.
FIGURE N9–1c
EXAMPLE 2
The slope field for the d.e. is shown in Figure N9–2.
(a) Carefully draw the solution curve that passes through the point (1, 0.5).
(b) Find the general solution for the equation.
FIGURE N9–2
SOLUTIONS:
(a) In Figure N9–2a we started at the point (1, 0.5), then moved from segment to segment drawing the curve to which these segments were tangent. The particular solution curve shown is the member of the family of solution curves
y = ln x + C
that goes through the point (1, 0.5).
FIGURE N9–2a
7 4 Slope Field Sap Calculus Equation
FIGURE N9–2b
(b) Since we already know that, if then we are assured of having found the correct general solution in (a).
In Figure N9–2b we have drawn several particular solution curves of the given d.e. Note that the vertical distance between any pair of curves is constant.
EXAMPLE 3
Match each slope field in Figure N9–3 with the proper d.e. from the following set. Find the general solution for each d.e. The particular solution that goes through (0,0) has been sketched in.
(A) y′ = cos x
(B)
7 4 Slope Field Sap Calculus Solver
(C)
(D)
FIGURE N9–3a
FIGURE N9–3b
FIGURE N9–3c
FIGURE N9–3d
SOLUTIONS:
(A) goes with Figure N9–3c. The solution curves in the family y = sin x + C are quite obvious.
(B) goes with Figure N9–3a. The general solution is the family of parabolas y = x2 + C.
For (C) the slope field is shown in Figure N9–3b. The general solution is the family of cubics y = x3 − 3x + C.
(D) goes with Figure N9–3d; the general solution is the family of lines y =
EXAMPLE 4
(a) Verify that relations of the form x2 + y2 = r2 are solutions of the d.e.
(b) Using the slope field in Figure N9–4 and your answer to (a), find the particular solution to the d.e. given in (a) that contains point (4, −3).
FIGURE N9–4
SOLUTIONS:
(a) By differentiating equation x2 + y2 = r2 implicitly, we get 2x + 2y from which which is the given d.e.
(b) x2 + y2 = r2 describes circles centered at the origin. For initial point (4,−3), (4)2 + (−3)2 = 25. So x2 + y2 = 25. However, this is not the particular solution.
A particular solution must be differentiable on an interval containing the initial point. This circle is not differentiable at (−5,0) and (5,0). (The d.e. shows undefined when y = 0, and the slope field shows vertical tangents along the x-axis.) Hence, the particular solution includes only the semicircle in quadrants III and IV.
Solving x2 + y2 = 25 for y yields The particular solution through point (4,−3) is with domain −5 < x< 5.
Derivatives of Implicitly Defined Functions
In Examples 2 and 3 above, each d.e. was of the form = f (x) or y′ = f (x). We were able to find the general solution in each case very easily by finding the antiderivative
We now consider d.e.’s of the form where f (x,y) is an expression in x and y; that is, is an implicitly defined function. Example 4 illustrates such a differential equation. Here is another example.
EXAMPLE 5
Figure N9–5 shows the slope field for
At each point (x,y) the slope is the sum of its coordinates. Three particular solutions have been added, through the points
FIGURE N9–5