Unit #5 - Implicit Differentiation, Related Rates
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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, -
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 -
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. -
CALCULUS I §2.2: Differentiability, Graphs, and Higher Derivatives
MATH 12002 - CALCULUS I x2.2: Differentiability, Graphs, and Higher Derivatives Professor Donald L. White Department of Mathematical Sciences Kent State University D.L. White (Kent State University) 1 / 10 Differentiability The process of finding a derivative is called differentiation, and we define: Definition Let y = f (x) be a function and let a be a number. We say f is differentiable at x = a if f 0(a) exists. What does this mean in terms of the graph of f ? If f 0(a) = lim f (a+h)−f (a) exists, then f (a) must be defined. h!0 h Since the denominator is approaching 0, in order for the limit to exist, the numerator must also approach 0; that is, lim (f (a + h) − f (a)) = 0: h!0 Hence lim f (a + h) = f (a), and so lim f (x) = f (a), h!0 x!a meaning f must be continuous at x = a. D.L. White (Kent State University) 2 / 10 Differentiability But being continuous at a is not enough to make f differentiable at a. Differentiability is \continuity plus." The \plus" is smoothness: the graph cannot have a sharp \corner" at a. The graph also cannot have a vertical tangent line at x = a: the slope of a vertical line is not a real number. Hence, in order for f to be differentiable at a, the graph of f must 1 be continuous at a, 2 be smooth at a, i.e., no sharp corners, and 3 not have a vertical tangent line at x = a. -
Example: Using the Grid Provided, Graph the Function
3.2 Differentiability Calculus 3.2 DIFFERENTIABILITY Notecards from Section 3.2: Where does a derivative NOT exist, Definition of a derivative (3rd way). The focus on this section is to determine when a function fails to have a derivative. For all you non-English majors, the word differentiable means you are able to take a derivative, or the derivative exists. Example 1: Using the grid provided, graph the function fx( ) = x − 3 . y a) What is fx'( ) as x → 3− ? b) What is fx'( ) as x → 3+ ? c) Is f continuous at x = 3? d) Is f differentiable at x = 3? x 2 Example 2: Graph fx x3 y a) Describe the derivative of f (x) as x approaches 0 from the left and the right. 2 b) Suppose you found fx' . 33 x x What is the value of the derivative when x = 0? 3 Example 3: Graph fx x y a) Describe the derivative of f (x) as x approaches 0 from the left and the right. 1 b) Suppose you found fx' . 33 x2 What is the value of the derivative when x = 0? x These last three examples (along with any graph that is not continuous) are NOT differentiable. The first graph had a “corner” or a sharp turn and the derivatives from the left and right did not match. The second graph had a “cusp” where secant line slope approach positive infinity from one side and negative infinity from the other. The third graph had a “vertical tangent line” where the secant line slopes approach positive or negative infinity from both sides. -
A Gallery of Visualization DEMOS for Related Rate Problems The
A Gallery of Visualization DEMOS for Related Rate Problems The following is a gallery of demos for visualizing common related rate situations. These animations can be used by instructors in a classroom setting or by students to aid in acquiring a visualization background relating to the steps for solving related problems. Two file formats, gif and mov (QuickTime) are available. 1. The gif animations should run on most systems and the file sizes are relatively small. 2. The mov animations require the QuickTime Player which is a free download available by clicking here; these file are also small. (The mov files may not execute properly in old versions of QuickTime.) 3. The collection of animations in gif and mov format can be downloaded; see the 'bulk' zipped download category at the bottom of the following table. General problem Click to view the Animation Sample Click to view the gif file. description. QuickTime file. Click to see gif Click to see mov Filling a conical tank. animation animation Searchlight rotates to Click to see gif Click to see mov follow a walker. animation animation Filling a cylindrical tank, Click to see gif Click to see mov two ways. animation animation Airplane & observer. Click to see gif Click to see mov animation animation Click to see gif Click to see mov Conical sand pile. animation animation Click to see gif Click to see mov Sliding ladder. animation animation Click to see gif Click to see mov Raising ladder. animation animation Shadow of a walking Click to see gif Click to see mov figure. -
Figure 1: Finding a Tangent Plane to a Graph
Figure 1: Finding a tangent plane to a graph Math 8 Winter 2020 Tangent Approximations and Differentiability We have seen how to find partial derivatives of functions from R2 to R, and seen how to use them to find tangent planes to graphs. Figure 1 shows the graph of the function f(x; y) = x2 + y2. The two red curves are the intersections of the graph with planes x = x0 and y = y0, and the yellow lines are tangent lines to the red curves, also lying in those planes. The slopes of the yellow lines (vertical rise over horizontal run, where the z- axis is vertical) are the partial derivatives of f at the point (x; y) = (x0; y0). The plane containing those yellow lines should be tangent to the graph of f. The phrase should be is important here. So far, we have pretty much been assuming this works. There is a very good argument that if there is any plane tangent to this graph at this point, it must be the plane containing these yellow lines, because those lines are tangent to the graph. But how do we know there is any tangent plane at all? 1 2xy Figure 2: Graph of f(x; y) = . px2 + y2 If (x; y) = (r sin θ; r cos θ), then f(x; y) = r sin(2θ). It turns out that a function can have partial derivatives without its graph having a tangent plane. Here is an example. Figure 2 shows two different pictures of a portion of the graph of the function 2xy f(x; y) = : px2 + y2 The x- and y-axes are drawn in red, and the intersection of the graph with the vertical plane y = x is drawn in yellow. -
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. -
Math 31A Review
Suggestions Differential calculus Integral Calculus Parting thoughts (from Piet Hein) The road to wisdom? Well it’s plain and simple to express: err and err and err again but less and less and less. Problems worthy of attack prove their worth by fighting back. Suggestions Differential calculus Integral Calculus Math 31A Review Steve 12 March 2010 Suggestions Differential calculus Integral Calculus About the final Thursday March 18, 8am-11am, Haines 39 No notes, no book, no calculator Ten questions ≈Four review questions (Chapters 3,4) ≈Five new questions (Chapters 5,6) ≈One essay question No limits, curve sketching, Riemann sums, or density Office hours: Monday 8:30am - 1:30pm Wednesday 10:30am - 3:30pm Suggestions Differential calculus Integral Calculus Start studying well in advance. If you haven’t started yet, start today! The best way to remember anything is through repetition. One of the best ways to help remember things is through regular short review sessions; as compared to one last minute cram session1. Study what ideas and techniques we have gone over this quarter. It is one thing to be able to know an identity or a procedure; it is an entirely different thing to be able to use that identity at the right time and in the right way. 1Also make sure to get plenty of sleep the night before the final. It will help you to focus on the arithmetic and keep you from falling asleep during the final! Suggestions Differential calculus Integral Calculus Learn from your mistakes, otherwise you will repeat them! Look back over your midterms, particularly those where you were not able to complete the problem, and ask “where did I get stuck and how can I avoid getting stuck like that again?” It is helpful to look at the solutions and ask “how would I have thought of that?” or “what about the problem would have told me to do such and such a technique?” Fool me once, shame on you. -
AP Calculus BC Scope & Sequence
AP Calculus BC Scope & Sequence Grading Period Unit Title Learning Targets Throughout the *Apply mathematics to problems in everyday life School Year *Use a problem-solving model that incorporates analyzing information, formulating a plan, determining a solution, justifying the solution and evaluating the reasonableness of the solution *Select tools to solve problems *Communicate mathematical ideas, reasoning and their implications using multiple representations *Create and use representations to organize, record and communicate mathematical ideas *Analyze mathematical relationships to connect and communicate mathematical ideas *Display, explain and justify mathematical ideas and arguments First Grading Preparation for Pre Calculus Review Period Calculus Limits and Their ● An introduction to limits, including an intuitive understanding of the limit process and the Properties formal definition for limits of functions ● Using graphs and tables of data to determine limits of functions and sequences ● Properties of limits ● Algebraic techniques for evaluating limits of functions ● Comparing relative magnitudes of functions and their rates of change (comparing exponential growth, polynomial growth and logarithmic growth) ● Continuity and one-sided limits ● Geometric understanding of the graphs of continuous functions ● Intermediate Value Theorem ● Infinite limits ● Understanding asymptotes in terms of graphical behavior ● Using limits to find the asymptotes of a function, vertical and horizontal Differentiation ● Tangent line to a curve and -
Calculus BC Syllabus
Calculus BC Syllabus Unit Information Unit Name (Timeframe): I. Limits and Continuity (3 weeks) Content and /or Skills Taught: -Intuitive understanding of limits - Using graphs and tables to understand limits - Finding limits analytically - Use limits to justify horizontal and vertical asymptotes - Connect the limits to justify horizontal and vertical asymptotes - Connect the limit as x approaches infinity to global behavior of function - Intuitive understanding of continuity - Definition of continuity - Types of continuity (removable, jump, and infinite) - Prove functions continuous or discontinuous - Intermediate Value Theorem II. Differentiation (3 weeks) Content and /or Skills Taught: - Definition of derivative - Derivative as slope of tangent line - Find derivatives using definition - Use the definition to discover the power rule for finding derivatives - Power rule, sum/difference rule, product rule, quotient rule, chain rule - Location linearization (see curves as locally linear) - Use local linearization to estimate functional values - Derivative as a rate of change - Relationship between differentiability and continuity - Prove functions differentiable or not differentiable - Three cases where the derivative does not exist (discontinuous, vertical tangent, cusp) - Derivatives of the trig functions III. Applications of Derivative (3 weeks) Content and /or Skills Taught: - First derivative test for relative extrema - Justify increasing, decreasing, and relative extrema - justify concavity and inflection points with second derivative -
Is Differentiable at X=A Means F '(A) Exists. If the Derivative
The Derivative as a Function Differentiable: A function, f(x), is differentiable at x=a means f '(a) exists. If the derivative exists on an interval, that is , if f is differentiable at every point in the interval, then the derivative is a function on that interval. f ( x h ) f ( x ) Definition: f ' ( x ) lim . h 0 h ( x h ) 2 x 2 x 2 2 xh h 2 x 2 Example: f ( x ) x 2 f ' ( x ) lim lim lim (2 x h ) 2 x h 0 h h 0 h h 0 Exercise: Show the derivative of a line is the slope of the line. That is, show (mx + b)' = m The derivative of a constant is 0. Linear Rule: If A and B are numbers and if f '(x) and g '(x) both exist then the derivative of Af+Bg at x is Af '(x) + Bg '(x). That is (Af + Bg)'(x)= Af '(x) + Bg '(x) Applying the linear rule for derivatives, we can differentiate any quadratic. Example: g ( x ) 4 x 2 6 x 7 g ' ( x ) 4(2 x ) 6 8 x 6 We can find the tangent line to g at a given point. Example: For g(x) as above, find the equation of the tangent line at (2, g(2)). g(2)=4(4)-12+7= 11 so the point of tangency is (2, 11). To get the slope, we plug x=2 into g '(x)=8x-6 the slope is g'(2)= 8(2)-6=10 so the tangent line is y = 10(x - 2) + 11 Notations for the derivative: df df f ' ( x ) f ' (a ) dx dx x a If a function is differentiable at x=a then it must be continuous at a.