2: The Derivative
- Page ID
- 216155
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 2.1: Prelude to the Derivative
- This page explains the concept of slope as a measure of change in lines and curves. It highlights that the slope of a straight line reflects its rise or fall, representing velocity in contexts like distance over time. To find the slope of a curve at a point, one must determine the slope of the tangent line, distinguishing it from the secant line that connects two points. The accuracy of the slope calculation improves as points get closer together, transitioning from secant to tangent.
- 2.2: Limits and Continuity
- This page explains limits in mathematics, detailing how they describe function behavior as they approach specific values without necessarily equaling them. It provides definitions, contrasts limits with actual function values, and shows how to determine limits through graphs, tables, and algebra. The page also discusses continuity and one-sided limits, illustrating through examples that a function is continuous at a point if the limit matches the function value there.
- 2.3: The Derivative
- This page covers the concepts of derivatives and their applications in analyzing average and instantaneous rates of change, particularly in relation to functions, velocity, and marginal cost. It explains how to compute average rates using secant lines and instantaneous rates using tangent lines. The discussions include formal definitions using limits, visual and algebraic methods for finding derivatives, and their relevance in real-life contexts like motion and business dynamics.
- 2.4: Power and Sum Rules for Derivatives
- This page covers essential derivative rules for calculating derivatives, such as the Constant Multiple Rule, Sum and Difference Rule, and Power Rule, with examples for polynomial functions. It introduces tangent lines and derivatives in the context of business, explaining terms like costs and how derivatives help approximate marginal costs.
- 2.5: Product and Quotient Rules
- This page covers key concepts in calculus and business math. It introduces the product and quotient rules for differentiating functions, essential for handling products and ratios of functions. Examples clarify their application. Additionally, it explores average costs (AC) and average revenue (AR), detailing their calculations and the relationship with marginal costs/revenues.
- 2.6: Chain Rule
- This page covers crucial differentiation techniques including the Chain Rule, Product Rule, and Quotient Rule for finding derivatives of complex functions. It provides examples and common pitfalls, emphasizing when to apply these rules. The text also discusses situations where derivatives may not exist, such as discontinuities or sharp corners, highlighting that a derivative can represent the slope of a tangent line which may be undefined under certain conditions.
- 2.7: Linearization and Differentials
- This page provides an overview of linear approximation in calculus, emphasizing how derivatives are used to estimate function values through tangent lines. It includes methods for approximating functions like \(\frac{1}{x}\) and \(\sqrt{x}\), and introduces differentials to calculate small changes and measurement errors. The discussion covers the impact of measurement inaccuracies on calculated values and highlights the importance of differentiating to assess error propagation.
- 2.8: Second Derivative and Concavity
- This page explains concavity, the second derivative, and inflection points in functions, highlighting their importance in interpreting real-world scenarios. It elaborates that concavity indicates the curvature of a graph, while the second derivative reflects this curvature (positive for concave up, negative for concave down). Inflection points denote changes in concavity, identified through the second derivative.
- 2.9: Optimization
- This page covers the optimization of functions by identifying and classifying local and global extrema through critical points where derivatives are zero or undefined. It explains the use of the First and Second Derivative Tests to classify these critical points and the importance of evaluating endpoints on closed intervals to find global maxima or minima.
- 2.10: Curve Sketching
- This page explores the relationship between a function and its first and second derivatives, detailing how to determine intervals of increase, decrease, and concavity. It explains that the first derivative indicates the function's trend, while the second derivative reveals concavity.
- 2.11: Applied Optimization
- This page covers optimization in calculus, focusing on maximizing and minimizing functions in practical applications. It discusses formulating problems with objective functions and constraints, illustrated by examples such as minimizing costs for garden enclosures and maximizing revenue through ticket pricing. It highlights profit maximization when Marginal Revenue equals Marginal Cost and explores average cost relations using derivative tests.
- 2.12: Other Applications
- This page covers two main topics: tangent line approximation (TLA) for estimating function values near a point and the concept of elasticity of demand, which assesses how demand changes with price variations. It provides examples for applying TLA and calculating elasticity, including revenue implications for different demand types.
- 2.13: Implicit Differentiation and Related Rates
- This page covers implicit differentiation, a method for differentiating complex functions where \(y\) isn't explicitly defined in terms of \(x\), and how it applies to analyzing slopes and related rates. Additionally, it addresses the relationship between demand and costs in production scenarios, demonstrating how to compute total daily costs using a specific cost function and assessing the rate of cost increase with production changes.
- 2.E: The Derivative (Exercises)
- This page consists of various exercises in calculus, focusing on limits, continuity, derivatives, and optimization. It includes graph analysis, calculating slopes of tangents, identifying critical points, and applying the Second Derivative Test. Practical applications are emphasized through problems related to real-world scenarios like temperature changes, production costs, and maximizing volumes.