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Learn about concavity and inflection points in mathematics. Understand how to identify intervals of concavity, locate inflection points, and use the second derivative test for relative extrema. Includes graphical examples and MATLAB code.
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Chabot Mathematics §3.2 Concavity& Inflection Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu
3.1 Review § • Any QUESTIONS About • §3.1 → Relative Extrema • Any QUESTIONS About HomeWork • §3.1 → HW-13
§3.2 Learning Goals • Introduce Concavity (a.k.a. Curvature) • Use the sign of the second derivative to find intervals of concavity • Locate and examine inflection points • Apply the second derivatives test for relative extrema
ConCavity Described • Concavity quantifies the Slope-Value Trend (Sign & Magnitude) of a fcn when moving Left→Right on the fcn Graph m≈0 m≈−1.4 m≈−4.4 m≈+2.2 m≈+2.2 m≈−4.4 m≈−1.4
MATLAB Code % Bruce Mayer, PE % MTH-15 •11Jul133 % XYfcnGraph6x6BlueGreenBkGndTemplate1306.m % % The data blue =[2.2 0 -1.4 -4.4] red = [-4.4 -1.4 0 2.2] % % the 6x6 Plot axes; set(gca,'FontSize',12); subplot(1,2,1) bar(blue, 'b'), grid, xlabel('\fontsize{14}Position, x'), ylabel('\fontsize{14}m = df/dx'),... title(['\fontsize{16}MTH15 • BLUE',]), axis([0 5 -5,3]) subplot(1,2,2) bar(red, 'r'), grid, xlabel('\fontsize{14}Position, x'), axis([0 5 -5,3]),... title(['\fontsize{16}MTH15 • RED',]) set(get(gco,'BaseLine'),'LineWidth',4,'LineStyle',':')
ConCavity Defined • A differentiable function f on a < x < b is said to be: … concave DOWN (↓) if df/dx is DEcreasingon the interval …concave up if df/dx is INcreasingon the interval.
Example Graphical Concavity • Consider the function f given in the graph and defined on the interval (−4,4). • Approximate all intervals on which the function is INcreasing, DEcreasing, concave up, or concave down
Example Graphical Concavity • SOLUTION • Because we have NO equation for the function, we need to use our best judgment: • around where the graph changes directions (increasing/decreasing) • where the derivative of the graph changes directions (concave up or down).
Example Graphical Concavity • To determine where the function is INcreasing, we look for the graph to “Rise to the Right (RR)” Rising
Example Graphical Concavity • Similarly, the function is DEcreasing where the graph “Falls to the Right (FR)”: Falling
Example Graphical Concavity • Conclude that f is increasing on the interval (0,4) and decreasing on the interval (−4,0) • Now ExamineConcavity. Falling to Rt Rising to Rt
Example Graphical Concavity • A function is concave UP wherever its derivative is INcreasing. Visually, we look for where the graph is“curved upward”, or “Bowl-Shaped”Similarly, A function is concave DOWN wherever its derivative is DEcreasing. Visually, we look for where the graph is “curved downward”, or “Dome-Shaped”
Example Graphical Concavity • The graph is “curved UPward” for values of x near zero, and might guess the curvature to be positive between −1 & 1 f is ConCave UP
Example Graphical Concavity • The graph is “curved DOWNward” for values of x on the outer edges of the domain. f is ConCave DOWN f is ConCave DOWN
Example Graphical Concavity • Thus the function is concave UP approximately on the interval (−1,1) and concave DOWN on the intervals (−4, −1) & (1,4) f is ConCave DOWN f is ConCave DOWN f is ConCave UP
Inflection Point Defined • A function has an inflection pointat x=a if f is continuous and the CONCAVITY of f CHANGES at Pt-a ConCave UP InflectionPoint ConCave DOWN
MATLAB Code % Bruce Mayer, PE % MTH-15 • 10Jul13 % XYfcnGraph6x6BlueGreenBkGndTemplate1306.m % % The Limits xmin = -2; xmax = 9; ymin =-50; ymax = 50; % The FUNCTION x = linspace(xmin,xmax,1000); y =(x-4).^3/4 + (x+5).^2/7; yOf4 = (4-4).^3/4 + (4+5).^2/7 % % The ZERO Lines zxh = [xminxmax]; zyh = [0 0]; zxv = [0 0]; zyv = [yminymax]; % % the 6x6 Plot axes; set(gca,'FontSize',12); whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Green plot(x,y, 'LineWidth', 5),axis([xminxmaxyminymax]),... grid, xlabel('\fontsize{14}x'), ylabel('\fontsize{14}y = f(x)'),... title(['\fontsize{16}MTH15 • Inflection Point',]) hold on plot(4, yOf4, 'd r', 'MarkerSize', 9,'MarkerFaceColor', 'r', 'LineWidth', 2) plot(zxv,zyv, 'k', zxh,zyh, 'k', 'LineWidth', 2) set(gca,'XTick',[xmin:1:xmax]); set(gca,'YTick',[ymin:10:ymax]) hold off
Example Inflection Graphically change from concave down to up change from concave up to down • The function shown above has TWO inflection points.
2nd Derivative Test • Consider a function for Which is Defined on some interval containing a critical Point (Recall that ) Then: • If , then is Concave UP at so is a Relative MIN • If , then is Concave DOWN at so is a Relative MAX
Example Apply 2ndDeriv Test • Use the 2nd Derivative Test to Find and classify all critical points for the Function • SOLUTION • Find the critical points by solving:
Example Apply 2ndDeriv Test • By Zero-Products: • Also need to check for values of x that make the derivative undefined. • ReCall the 1st Derivative: • Thus df/dx is UNdefined for x = −1, But the ORIGINAL function is ALSO Undefined at the this value • Thus there is NO Critical Point at x = −1
Example Apply 2ndDeriv Test • Thus the only critical points are at −2 & 0 • Now use the second derivative test to determine whether each is a MAXimum or MINimum (or if the test is InConclusive):
Example Apply 2ndDeriv Test • Before expanding the BiNomials, note that the numerator and denominator can be simplified by removing a common factor of (x+1) from all terms:
Example Apply 2ndDeriv Test • Now expand BiNomials: • Now Check Value of f’’’(0) & f’’’(−2)
Example Apply 2ndDeriv Test • The 2nd Derivative is NEGATIVE at x = −2 • Thus the orginalfcn is ConCaveDOWN at x = −2, and aRelative MAX exists at this Pt • Conversely, 2nd Derivative is POSITIVE at x = 0 • Thus the orginalfcn is ConCave UP at x= 0 anda Relative MIN exists at this Pt
Example Apply 2ndDeriv Test • Confirm by Plot → • Note the relative MINimum at 0, relative MAXimumat −2, and a vertical asymptote where the function is undefined at x=−1 (although the vertical line is not part of the graph of the function)
ConCavity Sign Chart • A form of the df/dx (Slope) Sign Chart (Direction-Diagram) Analysis Can be Applied to d2f/dx2 (ConCavity) • Call the ConCavity Sign-Charts “Dome-Diagrams” for INFLECTION Analysis ConCavityForm ++++++ −−−−−− −−−−−− ++++++ d2f/dx2 Sign x Critical (Break)Points a b c Inflection NOInflection Inflection
Example Dome-Diagram • Find All Inflection Points for • Notes on this (and all other) PolyNomial Function exists for ALL x • Use the ENGR25 Computer Algebra System, MuPAD, to find • Derivatives • Critical Points
Example Dome-Diagram • The Derivatives • The Critical Points • The ConCavity Values Between Break Pts • At x = −1 • At x = ½ • At x = ½
Example Dome-Diagram • Draw Dome-Diagram • The ConCavity Does NOT change at 0, but it DOES at 1 • Since Inflection requires Change, the only Inflection-Pt occurs at x = 1 ConCavityForm −−−−−− −−−−−− ++++++ d2f/dx2 Sign x Critical (Break)Points 0 1 NOInflection Inflection
Example Dome-Diagram • TheFcnPlotShowingInflectionPoint at(1,y(1))= (1,−3) (1,−3)
MATLAB Code % Bruce Mayer, PE % MTH-15 • 11Jul13 % XYfcnGraph6x6BlueGreenBkGndTemplate1306.m % % The Limits xmin = -1.5; xmax = 2.5; ymin =-15; ymax = 15; % The FUNCTION x = linspace(xmin,xmax,1000); y =3*x.^5 - 5*x.^4 - 1; % % The ZERO Lines zxh = [xminxmax]; zyh = [0 0]; zxv = [0 0]; zyv = [yminymax]; % % the 6x6 Plot axes; set(gca,'FontSize',12); whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Green plot(x,y, 'LineWidth', 5),axis([xminxmaxyminymax]),... grid, xlabel('\fontsize{14}x'), ylabel('\fontsize{14}y = f(x) = 3x^5 - 5x^4 - 1'),... title(['\fontsize{16}MTH15 • Dome-Diagram',]) hold on plot(1,-3, 'd r', 'MarkerSize', 10,'MarkerFaceColor', 'r', 'LineWidth', 2) plot(zxv,zyv, 'k', zxh,zyh, 'k', 'LineWidth', 2) set(gca,'XTick',[xmin:0.5:xmax]); set(gca,'YTick',[ymin:5:ymax]) hold off
Example Population Growth • A population model finds that the number of people, P, living in a city, in kPeople, t years after the beginning of 2010 will be: • Questions • In what year will the population be decreasing most rapidly? • What will be the population at that time?
Example Population Growth • SOLUTION: • “Decreasing most rapidly” is a phrase that requires some examination. “Decreasing” suggests a negative derivative. • “Decreasing most rapidly” means a value for which the negative derivative is as negative as possible. In other words, where the derivative is a MIN
Example Population Growth • Need to find relative minima of functions (derivative functions are no exception) where the rate of change is equal to 0. • “Rate of change in the population derivative, set equal to zero” TRANSLATES mathematically to
Example Population Growth • The only time at which the second derivative of P is equal to zero is the beginning of 2013. • Need to verify that the derivative is, in fact, negative at that point:
Example Population Growth • Thus the function is decreasing most rapidly at the inflection point at the beginning of 2013: • The Model Predicts 2013 Population: • x
WhiteBoard Work • Problems From §3.2 • P45 → Sketch Graph using General Description • P66 → Spreading a Rumor
All Done for Today RememgeringConCavity:cUP & frOWN
Chabot Mathematics Appendix Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu –
ConCavity Sign Chart ConCavityForm ++++++ −−−−−− −−−−−− ++++++ d2f/dx2 Sign x Critical (Break)Points a b c Inflection NOInflection Inflection
Max/Min Sign Chart Slope ++++++ −−−−−− −−−−−− ++++++ df/dx Sign x Critical (Break)Points a b c Max NOMax/Min Min