DOCSLIB.ORG

Math Flow Chart

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Math Flow Chart Updated November 12, 2014 MATH FLOW CHART Grade 6 Grade 7 Grade 8 Grade 9 Grade 10 Grade 11 Grade 12 Calculus Advanced Advanced Advanced Math AB or BC Algebra II Algebra I Geometry Analysis MAP GLA-Grade 8 MAP EOC-Algebra I MAP - ACT AP Statistics College Calculus Algebra/ Algebra I Geometry Algebra II AP Statistics MAP GLA-Grade 8 MAP EOC-Algebra I Trigonometry MAP - ACT Math 6 Pre-Algebra Statistics Extension Extension MAP GLA-Grade 6 MAP GLA-Grade 7 or or Algebra II Math 6 Pre-Algebra Geometry MAP EOC-Algebra I College Algebra/ MAP GLA-Grade 6 MAP GLA-Grade 7 MAP - ACT Foundations Trigonometry of Algebra Algebra I Algebra II Geometry MAP GLA-Grade 8 Concepts Statistics Concepts MAP EOC-Algebra I MAP - ACT Geometry Algebra II MAP EOC-Algebra I MAP - ACT Discovery Geometry Algebra Algebra I Algebra II Concepts Concepts MAP - ACT MAP EOC-Algebra I Additional Full Year Courses: General Math, Applied Technical Mathematics, and AP Computer Science. Calculus III is available for students who complete Calculus BC before grade 12. MAP GLA – Missouri Assessment Program Grade Level Assessment; MAP EOC – Missouri Assessment Program End-of-Course Exam Mathematics Objectives Calculus Mathematical Practices 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Linear and Nonlinear Functions • Write equations of linear and nonlinear functions in various formats. • Graph linear and nonlinear functions. • Solve linear and nonlinear functions using a variety of methods. • Apply functions to solve real world problems. The Derivative • Show that the difference quotient represents the average rate of change, the slope of the secant line, and the average velocity. • Demonstrate that the derivative can represent the instantaneous rate of change, the slope of the tangent line, and the instantaneous velocity. • Calculate the derivative of several types of functions. • Identify critical points, concavity, and increasing and decreasing intervals using graphs and derivatives. • Investigate and use applications of the second derivative to solve problems. • Use the derivative for business applications, implicit differentiation, related rates, and linear approximation. • Find the derivatives of trigonometric functions and use them to analyze a variety of applications. Integration • Use the Fundamental Theorem of Calculus to find the antiderivative. • Use the anitderivative to find the area under a curve. • Find integrals using integration by parts. • Find the volume of a solid using the definite integral. • Find the integrals of trigonometric functions. Multivariable Calculus • Use multivariable calculus to find maxima and minima and to explore three-dimensional space. Honoring Tradition ~ Continuing Excellence Pending Board Approval, May 13,2015 Mathematics Objectives Advanced Placement Calculus AB Mathematical Practices 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Functions, Graphs, and Limits Analysis of Graphs • Construct slope fields and interpret slope fields as visualizations of various equations. • Solve problems in which a rate is integrated to find the net change over time in a variety of applications. • Use integration to calculate areas of regions in a plane. • Use integration to calculate volumes of solids. • Find limits of indeterminate forms using L’Hopital’s Rule. • Use limits to evaluate improper integrals. • Use the direct comparison test and the limit comparison test to determine the convergence or divergence of improper integrals. • Differentiate, integrate, or substitute into a known power series in order to find additional power series representations. Limits of Functions • Use properties of limits to find the limit of a sequence. • Find limits of indeterminate forms using L’Hopital’s Rule. • Evaluate indefinite and definite integrals by the method of substitution. • Use properties of limits to find the limit of a sequence. • Determine whether a sequence converges or diverges. • Find limits of indeterminate forms using L’Hopital’s Rule. • Use limits to evaluate improper integrals. • Use the direct comparison test and the limit comparison test to determine the convergence or divergence of improper integrals. • Differentiate, integrate, or substitute into a known power series in order to find additional power series representations. • Construct slope fields and interpret slope fields as visualizations of various equations. Asymptotic and Unbounded Behavior • Use limits to evaluate improper integrals. • Use the direct comparison test and the limit comparison test to determine the convergence or divergence of improper integrals. • Solve problems involving exponential growth and decay. Continuity as a Property of Functions • Use Euler’s Method for approximating points for a solution to an initial value problem. • Solve problems in which a rate is integrated to find the net change over time in a variety of applications. • Use integration to calculate areas of regions in a plane. • Use integration to calculate volumes of solids. • Use limits to evaluate improper integrals. • Use the direct comparison test and the limit comparison test to determine the convergence or divergence of improper integrals. Honoring Tradition ~ Continuing Excellence Pending Board Approval, May 13, 2015 Derivatives Concept of the Derivative • Construct antiderivatives using the Fundamental Theorem of Calculus. • Solve initial value problems. • Construct slope fields and interpret slope fields as visualizations of various equations. Derivative at a Point • Construct slope fields and interpret slope fields as visualizations of various equations. Derivative as a Function • Construct slope fields and interpret slope fields as visualizations of various equations. • Solve initial value problems. Second Derivatives • Find derivative and second derivatives of parametrically defined functions. Applications and Computation of Derivatives • Calculate lengths of parametrically defined curves. • Use vectors to solve problems involving the velocity, acceleration, speed, and distance traveled. • Construct slope fields and interpret slope fields as visualizations of various equations. • Solve problems involving exponential growth and decay. • Evaluate indefinite and definite integrals by the method of substitution. Integrals Interpretations and Properties of Definite Integrals • Solve problems in which a rate is integrated to find the net change over time in a variety of applications. • Use integration to calculate areas of regions in a plane. • Use integration to calculate volumes of solids. Applications of Integrals • Solve problems in which a rate is integrated to find the net change over time in a variety of applications. • Use integration to calculate areas of regions in a plane. • Use integration to calculate volumes of solids. • Calculate lengths of parametrically defined curves. • Use vectors to solve problems involving the velocity, acceleration, speed, and distance traveled. • Calculate slopes and areas of regions in the plane determined by polar curves. Fundamental Theorem of Calculus • Construct antiderivatives using the Fundamental Theorem of Calculus. Techniques and Applications of Antidifferentiation • Construct antiderivatives using the Fundamental Theorem of Calculus. • Solve initial value problems. • Evaluate indefinite and definite integrals by the method of substitution. • Use integration by parts to evaluate indefinite and definite integrals. • Solve problems involving logistic population growth. • Solve problems in which a rate is integrated to find the net change over time in a variety of applications. • Use integration to calculate areas of regions in a plane. • Use integration to calculate volumes of solids. • Solve problems involving exponential growth and decay. Numerical Approximations to Definite Integrals • Solve problems in which a rate is integrated to find the net change over time in a variety of applications. • Use integration to calculate areas of regions in a plane. • Use integration to calculate volumes of solids. Honoring Tradition ~ Continuing Excellence Pending Board Approval, May 13, 2015 Mathematics Objectives Advanced Placement Calculus BC Mathematical Practices 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Functions, Graphs, and Limits Analysis of Graphs • Construct slope fields and interpret slope fields as visualizations of various equations. • Solve problems in which a rate is integrated to find the net change over time in a variety of applications. • Use integration to calculate areas of regions in a plane. • Use integration to calculate volumes of solids. • Find limits of indeterminate forms using L’Hopital’s Rule. • Use limits
Load more

Recommended publications
  • Numerical Differentiation and Integration
    Chapter 4 Numerical Di↵erentiation and Integration 4.1 Numerical Di↵erentiation In this section, we introduce how to numerically calculate the derivative of a function. First, the derivative of the function f at x0 is defined as f x0 h f x0 f 1 x0 : lim p ` q´ p q. p q “ h 0 h Ñ This formula gives an obvious way to generate an approximation to f x0 : simply compute 1p q f x0 h f x0 p ` q´ p q h for small values of h. Although this way may be obvious, it is not very successful, due to our old nemesis round-o↵error. But it is certainly a place to start. 2 To approximate f x0 , suppose that x0 a, b ,wheref C a, b , and that x1 x0 h for 1p q Pp q P r s “ ` some h 0 that is sufficiently small to ensure that x1 a, b . We construct the first Lagrange ‰ Pr s polynomial P0,1 x for f determined by x0 and x1, with its error term: p q x x0 x x1 f x P0,1 x p ´ qp ´ qf 2 ⇠ x p q“ p q` 2! p p qq f x0 x x0 h f x0 h x x0 x x0 x x0 h p qp ´ ´ q p ` qp ´ q p ´ qp ´ ´ qf 2 ⇠ x “ h ` h ` 2 p p qq ´ for some ⇠ x between x0 and x1. Di↵erentiating gives p q f x0 h f x0 x x0 x x0 h f 1 x p ` q´ p q Dx p ´ qp ´ ´ qf 2 ⇠ x p q“ h ` 2 p p qq „ ⇢ f x0 h f x0 2 x x0 h p ` q´ p q p ´ q´ f 2 ⇠ x “ h ` 2 p p qq x x0 x x0 h p ´ qp ´ ´ qDx f 2 ⇠ x .
    [Show full text]
  • MPI - Lecture 11
    MPI - Lecture 11 Outline • Smooth optimization – Optimization methods overview – Smooth optimization methods • Numerical differentiation – Introduction and motivation – Newton’s difference quotient Smooth optimization Optimization methods overview Examples of op- timization in IT • Clustering • Classification • Model fitting • Recommender systems • ... Optimization methods Optimization methods can be: 1 2 1. discrete, when the support is made of several disconnected pieces (usu- ally finite); 2. smooth, when the support is connected (we have a derivative). They are further distinguished based on how the method calculates a so- lution: 1. direct, a finite numeber of steps; 2. iterative, the solution is the limit of some approximate results; 3. heuristic, methods quickly producing a solution that may not be opti- mal. Methods are also classified based on randomness: 1. deterministic; 2. stochastic, e.g., evolution, genetic algorithms, . 3 Smooth optimization methods Gradient de- scent methods n Goal: find local minima of f : Df → R, with Df ⊂ R . We assume that f, its first and second derivatives exist and are continuous on Df . We shall describe an iterative deterministic method from the family of descent methods. Descent method - general idea (1) Let x ∈ Df . We shall construct a sequence x(k), with k = 1, 2,..., such that x(k+1) = x(k) + t(k)∆x(k), where ∆x(k) is a suitable vector (in the direction of the descent) and t(k) is the length of the so-called step. Our goal is to have fx(k+1) < fx(k), except when x(k) is already a point of local minimum. Descent method - algorithm overview Let x ∈ Df .
    [Show full text]
  • Math 142 – Quiz 6 – Solutions 1. (A) by Direct Calculation, Lim 1+3N 2
    Math 142 – Quiz 6 – Solutions 1. (a) By direct calculation, 1 + 3n 1 + 3 0 + 3 3 lim = lim n = = − . n→∞ n→∞ 2 2 − 5n n − 5 0 − 5 5 3 So it is convergent with limit − 5 . (b) By using f(x), where an = f(n), x2 2x 2 lim = lim = lim = 0. x→∞ ex x→∞ ex x→∞ ex (Obtained using L’Hˆopital’s Rule twice). Thus the sequence is convergent with limit 0. (c) By comparison, n − 1 n + cos(n) n − 1 ≤ ≤ 1 and lim = 1 n + 1 n + 1 n→∞ n + 1 n n+cos(n) o so by the Squeeze Theorem, the sequence n+1 converges to 1. 2. (a) By the Ratio Test (or as a geometric series), 32n+2 32n 3232n 7n 9 L = lim (16 )/(16 ) = lim = . n→∞ 7n+1 7n n→∞ 32n 7 7n 7 The ratio is bigger than 1, so the series is divergent. Can also show using the Divergence Test since limn→∞ an = ∞. (b) By the integral test Z ∞ Z t 1 1 t dx = lim dx = lim ln(ln x)|2 = lim ln(ln t) − ln(ln 2) = ∞. 2 x ln x t→∞ 2 x ln x t→∞ t→∞ So the integral diverges and thus by the Integral Test, the series also diverges. (c) By comparison, (n3 − 3n + 1)/(n5 + 2n3) n5 − 3n3 + n2 1 − 3n−2 + n−3 lim = lim = lim = 1. n→∞ 1/n2 n→∞ n5 + 2n3 n→∞ 1 + 2n−2 Since P 1/n2 converges (p-series, p = 2 > 1), by the Limit Comparison Test, the series converges.
    [Show full text]
  • 3.3 Convergence Tests for Infinite Series
    3.3 Convergence Tests for Infinite Series 3.3.1 The integral test We may plot the sequence an in the Cartesian plane, with independent variable n and dependent variable a: n X The sum an can then be represented geometrically as the area of a collection of rectangles with n=1 height an and width 1. This geometric viewpoint suggests that we compare this sum to an integral. If an can be represented as a continuous function of n, for real numbers n, not just integers, and if the m X sequence an is decreasing, then an looks a bit like area under the curve a = a(n). n=1 In particular, m m+2 X Z m+1 X an > an dn > an n=1 n=1 n=2 For example, let us examine the first 10 terms of the harmonic series 10 X 1 1 1 1 1 1 1 1 1 1 = 1 + + + + + + + + + : n 2 3 4 5 6 7 8 9 10 1 1 1 If we draw the curve y = x (or a = n ) we see that 10 11 10 X 1 Z 11 dx X 1 X 1 1 > > = − 1 + : n x n n 11 1 1 2 1 (See Figure 1, copied from Wikipedia) Z 11 dx Now = ln(11) − ln(1) = ln(11) so 1 x 10 X 1 1 1 1 1 1 1 1 1 1 = 1 + + + + + + + + + > ln(11) n 2 3 4 5 6 7 8 9 10 1 and 1 1 1 1 1 1 1 1 1 1 1 + + + + + + + + + < ln(11) + (1 − ): 2 3 4 5 6 7 8 9 10 11 Z dx So we may bound our series, above and below, with some version of the integral : x If we allow the sum to turn into an infinite series, we turn the integral into an improper integral.
    [Show full text]
  • 1 Improper Integrals
    July 14, 2019 MAT136 { Week 6 Justin Ko 1 Improper Integrals In this section, we will introduce the notion of integrals over intervals of infinite length or integrals of functions with an infinite discontinuity. These definite integrals are called improper integrals, and are understood as the limits of the integrals we introduced in Week 1. Definition 1. We define two types of improper integrals: 1. Infinite Region: If f is continuous on [a; 1) or (−∞; b], the integral over an infinite domain is defined as the respective limit of integrals over finite intervals, Z 1 Z t Z b Z b f(x) dx = lim f(x) dx; f(x) dx = lim f(x) dx: a t!1 a −∞ t→−∞ t R 1 R a If both a f(x) dx < 1 and −∞ f(x) dx < 1, then Z 1 Z a Z 1 f(x) dx = f(x) dx + f(x) dx: −∞ −∞ a R 1 If one of the limits do not exist or is infinite, then −∞ f(x) dx diverges. 2. Infinite Discontinuity: If f is continuous on [a; b) or (a; b], the improper integral for a discon- tinuous function is defined as the respective limit of integrals over finite intervals, Z b Z t Z b Z b f(x) dx = lim f(x) dx f(x) dx = lim f(x) dx: a t!b− a a t!a+ t R c R b If f has a discontinuity at c 2 (a; b) and both a f(x) dx < 1 and c f(x) dx < 1, then Z b Z c Z b f(x) dx = f(x) dx + f(x) dx: a a c R b If one of the limits do not exist or is infinite, then a f(x) dx diverges.
    [Show full text]
  • Series: Convergence and Divergence Comparison Tests
    Series: Convergence and Divergence Here is a compilation of what we have done so far (up to the end of October) in terms of convergence and divergence. • Series that we know about: P∞ n Geometric Series: A geometric series is a series of the form n=0 ar . The series converges if |r| < 1 and 1 a1 diverges otherwise . If |r| < 1, the sum of the entire series is 1−r where a is the first term of the series and r is the common ratio. P∞ 1 2 p-Series Test: The series n=1 np converges if p1 and diverges otherwise . P∞ • Nth Term Test for Divergence: If limn→∞ an 6= 0, then the series n=1 an diverges. Note: If limn→∞ an = 0 we know nothing. It is possible that the series converges but it is possible that the series diverges. Comparison Tests: P∞ • Direct Comparison Test: If a series n=1 an has all positive terms, and all of its terms are eventually bigger than those in a series that is known to be divergent, then it is also divergent. The reverse is also true–if all the terms are eventually smaller than those of some convergent series, then the series is convergent. P P P That is, if an, bn and cn are all series with positive terms and an ≤ bn ≤ cn for all n sufficiently large, then P P if cn converges, then bn does as well P P if an diverges, then bn does as well. (This is a good test to use with rational functions.
    [Show full text]
  • APPM 1350 - Calculus 1
    APPM 1350 - Calculus 1 Course Objectives: This class will form the basis for many of the standard skills required in all of Engineering, the Sciences, and Mathematics. Specically, students will: • Understand the concepts, techniques and applications of dierential and integral calculus including limits, rate of change of functions, and derivatives and integrals of algebraic and transcendental functions. • Improve problem solving and critical thinking Textbook: Essential Calculus, 2nd Edition by James Stewart. We will cover Chapters 1-5. You will also need an access code for WebAssign’s online homework system. The access code can also be purchased separately. Schedule and Topics Covered Day Section Topics 1 Appendix A Trigonometry 2 1.1 Functions and Their Representation 3 1.2 Essential Functions 4 1.3 The Limit of a Function 5 1.4 Calculating Limits 6 1.5 Continuity 7 1.5/1.6 Continuity/Limits Involving Innity 8 1.6 Limits Involving Innity 9 2.1 Derivatives and Rates of Change 10 Exam 1 Review Exam 1 Topics 11 2.2 The Derivative as a Function 12 2.3 Basic Dierentiation Formulas 13 2.4 Product and Quotient Rules 14 2.5 Chain Rule 15 2.6 Implicit Dierentiation 16 2.7 Related Rates 17 2.8 Linear Approximation and Dierentials 18 3.1 Max and Min Values 19 3.2 The Mean Value Theorem 20 3.3 Derivatives and Shapes of Graphs 21 3.4 Curve Sketching 22 Exam 2 Review Exam 2 Topics 23 3.4/3.5 Curve Sketching/Optimization Problems 24 3.5 Optimization Problems 25 3.6/3.7 Newton’s Method/Antiderivatives 26 3.7/Appendix B Sigma Notation 27 Appendix B,
    [Show full text]
  • Calculus II Chapter 9 Review Name______
    Calculus II Chapter 9 Review Name___________ ________________________ SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. A recursion formula and the initial term(s) of a sequence are given. Write out the first five terms of the sequence. an 1) a = 1, a = 1) 1 n+1 n + 2 Find a formula for the nth term of the sequence. 1 1 1 1 2) 1, - , , - , 2) 4 9 16 25 Find the limit of the sequence if it converges; otherwise indicate divergence. 9 + 8n 3) a = 3) n 9 + 5n 3 4) (-1)n 1 - 4) n Determine if the sequence is bounded. Δn 5) 5) 4n Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges. 3 3 3 3 6) 3 - + - + ... + (-1)n-1 + ... 6) 8 64 512 8n-1 7 7 7 7 7) + + + ... + + ... 7) 1·3 2·4 3·5 n(n + 2) Find the sum of the series. Q n 9 8) _ (-1) 8) 4n n=0 Use partial fractions to find the sum of the series. 3 9) 9) _ (4n - 1)(4n + 3) 1 Determine if the series converges or diverges; if the series converges, find its sum. 3n+1 10) _ 10) 7n-1 1 cos nΔ 11) _ 11) 7n Find the values of x for which the geometric series converges. n n 12) _ -3 x 12) Find the sum of the geometric series for those x for which the series converges.
    [Show full text]
  • Calculus Lab 4—Difference Quotients and Derivatives (Edited from U. Of
    Calculus Lab 4—Difference Quotients and Derivatives (edited from U. of Alberta) Objective: To compute difference quotients and derivatives of expressions and functions. Recall Plotting Commands: plot({expr1,expr2},x=a..b); Plots two Maple expressions on one set of axes. plot({f,g},a..b); Plots two Maple functions on one set of axes. plot({f(x),g(x)},x=a..b); This allows us to plot the Maple functions f and g using the form of plot() command appropriate to Maple expressions. If f and g are Maple functions, then f(x) and g(x) are the corresponding Maple expressions. The output of this plot() command is precisely the same as that of the preceding (function version) plot() command. 1. We begin by using Maple to compute difference quotients and, from them, derivatives. Try the following sequence of commands: 1 f:=x->1/(x^2-2*x+2); This defines the function f (x) = . x 2 − 2x + 2 (f(2+h)-f(2))/h; This is the difference quotient of f at the point x = 2. simplify(%); Simplifies the last expression. limit(%,h=0); This gives the derivative of f at the point where x = 2. Exercise 1: Find the difference quotient and derivative of this function at a general point x (hint: make a simple modification of the above steps). This f (x + h) − f (x) means find and f’(x). Record your answers below. h Use this to evaluate the derivative at the points x = -1 and x = 4. (It may help to remember the subs() command here; for example, subs(x=1,e1); means substitute x = 1 into the expression e1).
    [Show full text]
  • 3.2 the Derivative As a Function 201
    SECT ION 3.2 The Derivative as a Function 201 SOLUTION Figure (A) satisfies the inequality f .a h/ f .a h/ f .a h/ f .a/ C C 2h h since in this graph the symmetric difference quotient has a larger negative slope than the ordinary right difference quotient. [In figure (B), the symmetric difference quotient has a larger positive slope than the ordinary right difference quotient and therefore does not satisfy the stated inequality.] 75. Show that if f .x/ is a quadratic polynomial, then the SDQ at x a (for any h 0) is equal to f 0.a/ . Explain the graphical meaning of this result. D ¤ SOLUTION Let f .x/ px 2 qx r be a quadratic polynomial. We compute the SDQ at x a. D C C D f .a h/ f .a h/ p.a h/ 2 q.a h/ r .p.a h/ 2 q.a h/ r/ C C C C C C C 2h D 2h pa2 2pah ph 2 qa qh r pa 2 2pah ph 2 qa qh r C C C C C C C D 2h 4pah 2qh 2h.2pa q/ C C 2pa q D 2h D 2h D C Since this doesn’t depend on h, the limit, which is equal to f 0.a/ , is also 2pa q. Graphically, this result tells us that the secant line to a parabola passing through points chosen symmetrically about x a is alwaysC parallel to the tangent line at x a. D D 76. Let f .x/ x 2.
    [Show full text]
  • Calculus Terminology
    AP Calculus BC Calculus Terminology Absolute Convergence Asymptote Continued Sum Absolute Maximum Average Rate of Change Continuous Function Absolute Minimum Average Value of a Function Continuously Differentiable Function Absolutely Convergent Axis of Rotation Converge Acceleration Boundary Value Problem Converge Absolutely Alternating Series Bounded Function Converge Conditionally Alternating Series Remainder Bounded Sequence Convergence Tests Alternating Series Test Bounds of Integration Convergent Sequence Analytic Methods Calculus Convergent Series Annulus Cartesian Form Critical Number Antiderivative of a Function Cavalieri’s Principle Critical Point Approximation by Differentials Center of Mass Formula Critical Value Arc Length of a Curve Centroid Curly d Area below a Curve Chain Rule Curve Area between Curves Comparison Test Curve Sketching Area of an Ellipse Concave Cusp Area of a Parabolic Segment Concave Down Cylindrical Shell Method Area under a Curve Concave Up Decreasing Function Area Using Parametric Equations Conditional Convergence Definite Integral Area Using Polar Coordinates Constant Term Definite Integral Rules Degenerate Divergent Series Function Operations Del Operator e Fundamental Theorem of Calculus Deleted Neighborhood Ellipsoid GLB Derivative End Behavior Global Maximum Derivative of a Power Series Essential Discontinuity Global Minimum Derivative Rules Explicit Differentiation Golden Spiral Difference Quotient Explicit Function Graphic Methods Differentiable Exponential Decay Greatest Lower Bound Differential
    [Show full text]
  • CHAPTER 3: Derivatives
    CHAPTER 3: Derivatives 3.1: Derivatives, Tangent Lines, and Rates of Change 3.2: Derivative Functions and Differentiability 3.3: Techniques of Differentiation 3.4: Derivatives of Trigonometric Functions 3.5: Differentials and Linearization of Functions 3.6: Chain Rule 3.7: Implicit Differentiation 3.8: Related Rates • Derivatives represent slopes of tangent lines and rates of change (such as velocity). • In this chapter, we will define derivatives and derivative functions using limits. • We will develop short cut techniques for finding derivatives. • Tangent lines correspond to local linear approximations of functions. • Implicit differentiation is a technique used in applied related rates problems. (Section 3.1: Derivatives, Tangent Lines, and Rates of Change) 3.1.1 SECTION 3.1: DERIVATIVES, TANGENT LINES, AND RATES OF CHANGE LEARNING OBJECTIVES • Relate difference quotients to slopes of secant lines and average rates of change. • Know, understand, and apply the Limit Definition of the Derivative at a Point. • Relate derivatives to slopes of tangent lines and instantaneous rates of change. • Relate opposite reciprocals of derivatives to slopes of normal lines. PART A: SECANT LINES • For now, assume that f is a polynomial function of x. (We will relax this assumption in Part B.) Assume that a is a constant. • Temporarily fix an arbitrary real value of x. (By “arbitrary,” we mean that any real value will do). Later, instead of thinking of x as a fixed (or single) value, we will think of it as a “moving” or “varying” variable that can take on different values. The secant line to the graph of f on the interval []a, x , where a < x , is the line that passes through the points a, fa and x, fx.
    [Show full text]