1. Engineering Mathematics: Differential Calculus

2. Contents Concepts of Limits and Continuity Derivatives of functions Differentiation rules and Higher Derivatives Applications

3. Differential Calculus Concepts of Limits and Continuity

4. The idea of limits Consider a function The function is well-defined for all real values of x The following table shows some of the values: x2.92.99 33.0013.013.1 F(x)8.418.948.99499.0069.069.61

5. The idea of limits

6. Concept of Continuity E.g. is continuous at x=3? The following table shows some of the values: exists as and => f(x) is continuous at x=3! x2.92.99 33.0013.013.1 F(x)8.418.948.99499.0069.069.61

7. Differential Calculus Derivatives of functions

8. Derivative ( 導數 ) Given y=f(x), if variable x is given an increment  x from x=x 0, then y would change to f(x 0 +  x)  y= f(x 0 +  x) – f(x)  y  x is the slope ( 斜率 ) of triangular ABC x f(x 0 +  x) f(x 0 ) x0+xx0+x x0x0 y xx yy Y=f(x) A B C

9. Derivative What happen with  y  x as  x tends to 0? It seems that  y  x will be close to the slope of the curve y=f(x) at x 0. We defined a new quantity as follows If the limit exists, we called this new quantity as the derivative ( 導數 ) of f(x). The process of finding derivative of f(x) is called differentiation ( 微分法 ).

10. Derivative y f(x 0 +  x) f(x 0 ) x0+xx0+x x0x0 x xx yy Y=f(x) A B C Derivative of f(x) at Xo = slope of f(x) at Xo

11. Differentiation from first principle Find the derivative of with respect to (w.r.t.) x To obtain the derivative of a function by its definition is called differentiation of the function from first principles

12. Differential Calculus Differentiation rules and Higher Derivatives

13. Fundamental formulas for differentiation I Let f(x) and g(x) be differentiable functions and c be a constant. for any real number n

14. Examples Differentiate and w.r.t. x

15. Table of derivative (1) FunctionDerivative Constant k0 x1 kxk

16. Table of Derivatives (2) Function Derivative Angles in radians

17. Differential Calculus Product Rule, Quotient Rule and Chain Rule

18. - The product rule and the quotient rule Form of product function Form of quotient function e.g.1 This is a _________ function with and

19. e.g.2 This is a _________ function with and e.g.3 e.g.4 e.g.5

20. The product rule Consider the function Using the product rule, e.g.1 Find where Solution:

21. e.g.2 Find where Solution: e.g.3 Find where Solution:

22. The quotient rule Consider the function. Applying the quotient rule, e.g.1 Find where Solution:

23. e.g.2 Find where Solution: e.g.3 Find where Solution:

24. More Example (1) Differentiate w.r.t. x

25. More Example (2) Differentiate w.r.t. x

26. Fundamental formulas for differentiation II Let f(x) and g(x) be differentiable functions

27. Fundamental formulas for differentiation III ln(x) is called natural logarithm ( 自然對數 )

28. Differentiation of composite functions To differentiate w.r.t. x, we may have problems as we don’t have a formula to do so. The problem can be simplified by considering composite function: Let so and we know derivative of y w.r.t. u (by formula): but still don ’ t know

29. Chain Rule ( 鏈式法則 ) Chain Rule states that : given y=g(u), and u=f(x) So our problem and

30. Example 1 Differentiate w.r.t. x Simplify y by letting so now By chain rule

31. Example 2 Differentiate w.r.t. x Simplify y by letting so now By chain rule

32. Example 3 Differentiate w.r.t. x Simplify y by letting so now By chain rule

33. Higher Derivatives ( 高階導數 ) If the derivatives of y=f(x) is differentiable function of x, its derivative is called the second derivative ( 二階導數 ) of y=f(x) and is denoted by or f ’’(x). That is Similarly, the third derivative = the n-th derivative =

34. Example Find if

35. Differential Calculus Applications

36. Slope of a curve Recall that the derivative of a curve evaluate at a point is the slope of the curve at that point. f(x 0 +  x) f(x 0 ) x0+xx0+x x0x0 x xx yy Y=f(x) A B C Derivative of f(x) at Xo = slope of f(x) at Xo

37. Slope of a curve Find the slope of y=2x+3 at x=0 To find the slope of a curve, we have to compute the derivative of y and then evaluate at a point The slope of y at x=0 equals 2 (y=mx+c now m=2)

38. Slope of a curve Find the slope of at x=0, 2, -2 The slope of y = 2x The slope of y (at x=0) = 2(0) = 0 The slope of y (at x=-2) =2(-2) = -4 The slope of y (at x=2) =2(2) = 4 X=0 X=2X=-2

39. Local maximum and minimum point For a continuous function, the point at which is called a stationary point. This gives the point local maximum or local minimum of the curve A B C D X1X1 X2X2

40. First derivative test (Max pt.) Given a continuous function y=f(x) If dy/dx = 0 at x=x o & dy/dx changes from +ve to –ve through x 0, x=x 0 is a local maximum point x=x 0 local maximum point

41. First derivative test (Min pt.) Given a continuous function y=f(x) If dy/dx = 0 at x=x o & dy/dx changes from -ve to +ve through x 0, x=x 0 is a local minimum point x=x 0 local minimum point

42. Example 1 Determine the position of any local maximum and minimum of the function First, find all stationary point (i.e. find x such that dy/dx = 0), so when x=0 By first derivative test x=0 is a local minimum point

43. Example 2 Find the local maximum and minimum of Find all stationary points first: x0x<1x=11<x<3x=3x>3 dy/dx++0-0+

44. Example 2 (con’d) By first derivative test, x=1 is the local maximum (+ve -> 0 -> -ve) x=3 is the local minimum (-ve -> 0 -> +ve)

45. Second derivative test Second derivative test states: There is a local maximum point in y=f(x) at x=x0, if at x=x 0 and < 0 at x=x 0. There is a local minimum point in y=f(x) at x=x0, if at x=x 0 and > 0 at x=x 0. If dy/dx = 0 and =0 both at x=x 0, the second derivative test fails and we must return to the first derivative test.

46. Example Find the local maximum and minimum point of Solution Find all stationary points first: By second derivative test, x=1 is max and x=3 is min. So, (1,2) is max. point and (3,-2) in min. point.

47. Practical Examples 1 e.g.1 A rectangular block, with square base of side x mm, has a total surface area of 150 mm 2. Show that the volume of the block is given by. Hence find the maximum volume of the block. Solution:

48. Practical Examples 2 A window frame is made in the shape of a rectangle with a semicircle on top. Given that the area is to be 8, show that the perimeter of the frame is. Find the minimum cost of producing the frame if 1 metre costs $75. Solution: