Differential calculus

1. Pertemuan 9. Differential calculus Kalkulus I

2. Summary meliputi: Coordinate geometry Slope of a curve Rapid differentiation Derivatives of • sum and products; • of a ’function of a function’ ; • a ratio. Higher derivatives Notations Maxima and minima Points of inflection Sketching curves Partial differentiation Kalkulus I

3. Relationships between lengths and angles Trigonometry Use of geometrical insights and understanding for studying algebraic problems Coordinate geometry Geometrical model The point is 2 units RIGHT from y-axis and 27 units UP from x-axis Algebraic equation ordinate y abscissa x Geometry The point is 2 units LEFT from y-axis and 14 units DOWN from x-axis origin (0,0) Coordinate geometry ´ Kalkulus I

4. Intercept on y-axis y = a y2 - y1 x2 - x1 Slope The quantity b has a meaning very similar to that of the gradient of a hill in everyday life; the steeper the hill the greater gradient, and the more rapidly the height increases with the horizontal distance travelled. Intercept on x-axis x = –b / a Coordinate geometry Geometrical model Algebraic equation Kalkulus I

5. Notation for a change x2-x1 in a variable x: SymbolMeans Dx Change of any magnitude dxSmall change dx infinitesimal change (approaching zero) x infinitesimal change under specified condition A positive change of x Increase in x Any change of x, but zero If Dx is an increment in x, then Dy is the corresponding increment in y, i.e. change in y that occurs as a result of change in x Incrementin x Kemiringan curva Algebraic equation Geometrical model Kalkulus I

6. Slope of a curve Algebraic equation Geometrical model Kalkulus I

7. Values of the increment ratio for x = 3 dxdy 1 10 10 0.5 4.75 9.5 0.2 1.84 9.2 0.1 0.91 9.1 0.01 0.0901 9.01 0.001 0.009001 9.001 0.0001 0.00090001 9.0001 0.00001 0.0000900001 9.00001 tangent at x = 3 has slope 9. The derivative of the function x2+3x+2 with respect to x is 3 + 2x Derivative of y with respect to x Slope of a curve Algebraic equation Geometrical model ´ Kalkulus I

8. Rapid differentiation Some useful relations Kalkulus I

9. Derivative of a sum Kalkulus I

10. du dv u dv u v deacreases slower du v u v deacreases very fast u v Derivative dari Perkalian Kalkulus I

11. ??? Chain rule Derivative of a 'function of a function' Michaelis function Kalkulus I

12. Function of a function Derivative of a ratio Kalkulus I

13. Rate of change of slope (curvature) First derivative Second derivative Third derivative Derivative of n-th order Higher derivatives Kalkulus I

14. Leibnitz notations: Function: y 1st derivative: or Dy 2nd derivative: or D2y n-th derivative: or Dny Compact notations: Function: y or f(x) 1st derivative: y' or f '(x) 2nd derivative: y'' or f ''(x) n-th derivative: y(n) or f (n)(x) Concerns derivative with respect to x. Newton’s notations: Function: x(t) 1st derivative: 2nd derivative: Concerns derivatives of time-dependent quantities. Notasi Kalkulus I

15. Substrate inhibition in an enzyme-catalyzed reaction Slope = 0 at the maximum A derivative shows the slope ! Where a maximim occurs ??? Not always true Maximum of a function Slope = 0 Always true Maxima Kalkulus I

16. Slope = 0 at the maximum Maxima Substrate inhibition in an enzyme-catalyzed reaction Kalkulus I

17. If I want to plot y against log x, would a maximum appear at the x value as in normal plot ??? Chain rule The maximum will be at the same place !! Maxima Kalkulus I

18. Maximum slope = 0 Function increases when x increases, slope > 0 Function decreases when x increases, slope < 0 '+ 0 –'is the sequence of signs of first derivative around a maximum. '– 0 +'is the sequence of signs of first derivative around a minimum. A stationary point Not always true Maximum of a function Slope = 0 Function decreases when x increases, slope < 0 Minimum slope = 0 A stationary point embraces both maxima and minima. Always true Minima Kalkulus I

19. A maximum corresponds to a zero in first derivative and negative second derivative. A minimumcorresponds to a zero in first derivative and positive second derivative. A inflection pointcorresponds to a stacionary pointin first derivative and zeroin second derivative. Points of inflection Kalkulus I

20. Buffer x mol L-1 of NaOH Amol L-1 acetic acid, HOAc Points of inflection in biochemistry define conditions in which a response (e.g., rate of reaction) is most (or least) sensitive to an influence (e.g. the concentration of a metabolite). Henderson-Hasselbalch equation Concent. of a salt of the acid Concent. of a weak acid Negative logarithm of the acid dissiciacion constant The first derivative of pH with respect to x is a measure of the sensitivity of the pH to addition of base. If it is small it means that pH of the buffer will be not changed much with adding a trace of alkali (an effective buffer). So, the buffer is most efficient at pHwhere the first derivative has a minimum or the second derivative is zero. Points of inflection Kalkulus I

21. Sketching curves Useful rekomendations Find where an unfamiliar function f(x) crosses axis: • a value of f(x) at x = 0; • a value of x at which f(x)= 0; Find location of stacionary points (i.e. x value(s) where the first derivative f '(x)=0 ) Check the sign of the second derivative at x values of the stacionary points (i.e. determine whether stacionary point is maximum or minimum ) Find the limiting behaviour of the function (i.e. when values of x are extremely large) Kalkulus I

22. Wouuu Kalkulus I