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Math-M119: Brief Survey of Calculus Don Byrd, Instructor (revised version) Study Guide for Text Chapter 2: Rate of Change: The Derivative Note: items in square brackets “[…]” are relatively unimportant. 2.1 Instantaneous Rate of Change Instantaneous Velocity - Use average velocity over an interval of time to estimate instantaneous - Ex: throw grapefruit straight up Generally, the shorter the interval, the better the estimate Defining Instantaneous Velocity Using the Idea of a Limit Instantaneous Rate of Change - Instantaneous = in an infinitesimal (infinitely small) time interval Change in velocity is acceleration Derivative at a Point - Def’n: derivative of f at a, written f ’(a), is instantaneous rate of change of f at a - …it’s the slope of tangent to f at a Visualizing the Derivative: Slope of the Graph, of the Tangent Line - Average rate of change as ∆x decreases => secant line approaches tangent line - Review: value increasing/decreasing vs. slope increasing/decreasing Does the derivative always exist? = Is there always a tangent at every point in domain? (And is the domain all real numbers?) - No! Exx: 1/x ; abs(x) . (For each, WHERE DOES IT NOT EXIST?) 2.2 The Derivative Function The derivative of a function is another function. Finding the Derivative of a Function Given Graphically What Does It Tell Us Graphically? f ’ > 0 on an interval => f is increasing over the interval f ’ < 0 on an interval => f is decreasing over the interval f ’ = 0 on an interval => f is constant over the interval Finding the Derivative of a Function Given Numerically Finding the Derivative of a Function Given by a Formula 2.3 Interpretations of the Derivative Alternative notation: € dy (Liebniz) dx 1 - NOT a fraction, but kind of like one… (“infinitesimal” change in y) / (“infinitesimal” change in x) [ this is interpreting it as a differential ] Using units to interpret the derivative [ unit analysis, a.k.a. dimensional analysis ] dy Notation for value of derivative at a point: x= nnn dx Using the derivative to estimate function values - From the point where value is desired, can go to right, to left, or both ways Local Linear Approximation: for x€“near” a f (x) ≈ f (a) + f ' (a)(x − a) NB: x – a, not a – x. (WHY?) € € Relative rate of change of y = f(t) at t = a dy / dt f ' (a) = y f (a) [ Marginal cost (p. 115) ] - Linear & non-linear cost functions Non-linearity caused by economy of scale, glutted markets, etc. Marginal cost = slope of cost curve Marginal revenue = slope of revenue curve - Average rate of change is approx. = to instantaneous rate of change (derivative)! 2.4 The Second Derivative Def’n: 2nd derivative is the derivative of the (1st) derivative - derivative exists only when function and 1st derivative are both differentiable Notation: for y = f(x), 1st derivative = f ′ = dy 2 dy ; 2nd derivative = f ′′ = 2 dx d x What Does the 2nd Derivative Tell Us? f ’’ > 0 on an interval => f ’ is increasing over the interval f ’’ < 0 on an interval€=> f ’ is decreasing over the € interval What does this behavior of 1st derivative mean for the original function? f ’ increasing => f is bending upward (concave up) (WHY?) f ’ increasing => f is bending upward (concave up) (WHY?) - caveat: “going up” means “becoming more positive”; “going down” means “becoming more negative”! - common real-life example: f = distance, f ’ = velocity, f ’’ = acceleration Interpretation as Rate of Change - actually, rate of change of a rate of change! - Ex: if you accelerate hard, velocity goes up quickly - …and distance changes more and more quickly 2 - Specific ex: the 2011 Nobel Prize in Physics went to three astronomers for showing that, if t = time & s(t) = distance from Earth to far-off supernovas, s’’ is positive = the expansion of the universe is accelerating Focus on Theory [ Def’n: limit ] [ Notation: the limit as x approaches c of f(x) is L is written: lim f (x) = L ] x→c [ Def’n of derivative in terms of limits ] Def’n: a function f(x) is differentiable at any point x at which the derivative function exists € The Textbookʼs built-in Chapter Summary (p. 120) Rate of change • Average, instantaneous Estimating derivatives • Estimate derivatives from a graph, table of values, or formula Interpretation of derivatives • Rate of change, slope, using units, instantaneous velocity Relative rate of change • Calculation and interpretation Marginality • Marginal cost and marginal revenue Second derivative • Concavity Derivatives and graphs • Understand relationship between sign of f’ and whether f is increasing or decreasing. Sketch graph of f’ from graph of f. Marginal analysis 3 Functions and Derivatives and their Graphs Function and Its Sign Graph of f(x) Sign of function value f(x) above x axis + on x axis 0 below x axis – Function vs. Sign of First Derivative Graph of f(x) Rate of change of f(x) Sign of 1st derivative, f ’ going up positive + constant 0 0 negative – going down Function vs. Sign of Second Derivative Graph of f(x) Rate of change of f(x) Sign of 2nd derivative, f ’’ concave up becoming more positive + straight line 0 0 becoming more negative – concave down DAB, rev. 8 Oct. 2011 4