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Limits and continuity. Start from an example. If the voltage apply to the resistor can be express by the function Determine the maximum voltage across the resistor. We can calculate the maximum or minimum by use derivative. Contents. Concepts of Limits and Continuity
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Start from an example • If the voltage apply to the resistor can be express by the function Determine the maximum voltage across the resistor. • We can calculate the maximum or minimum by use derivative
Contents • Concepts of Limits and Continuity • Derivatives of functions • Differentiation rules and Higher Derivatives • Applications
Differential Calculus Concepts of Limits and Continuity
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:
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!
Differential Calculus Derivatives of functions
y Y=f(x) f(x0+Dx) B Dy A C f(x0) Dx x0 x0+Dx Derivative (導數) • Given y=f(x), if variable x is given an increment Dx from x=x0, then y would change to f(x0+Dx) • Dy= f(x0+Dx) – f(x) • Dy/Dx is the slope of triangular ABC x
Derivative • What happen with Dy/Dx as Dx tends to 0? • It seems that Dy/Dx will be close to the slope of the curve y=f(x) at x0. • We defined a new quantity as follows • If the limit exists, we called this new quantity as the derivative of f(x). • The process of finding derivative of f(x)is called differentiation.
Y=f(x) f(x0+Dx) B Dy A C f(x0) Dx x x0 x0+Dx Derivative y Derivative of f(x) at Xo = slope of f(x) at Xo
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
Differential Calculus Differentiation rules and Higher Derivatives
Fundamental formulas for differentiation I • Let f(x) and g(x) be differentiable functions and c be a constant. for any real number n
Examples • Differentiate and w.r.t. x
Fundamental formulas for differentiation II • Let f(x) and g(x) be differentiable functions
Example 1 Differentiate w.r.t. x
Example 2 Differentiate w.r.t. x
Fundamental formulas for differentiation III ln(x) is called natural logarithm
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
Chain Rule Chain Rule states that : given y=g(u), and u=f(x) So our problem and
Example 1 Differentiate w.r.t. x Simplify y by letting so now By chain rule
Example 2 Differentiate w.r.t. x Simplify y by letting so now By chain rule
Example 3 Differentiate w.r.t. x Simplify y by letting so now By chain rule
Higher Derivatives) • If the derivatives of y=f(x) is differentiable function of x, its derivative is called the second derivative of y=f(x) and is denoted by or f ’’(x). That is • Similarly, the third derivative = • the n-th derivative =
Example • Find if
Differential Calculus Applications
Y=f(x) f(x0+Dx) B Dy A C f(x0) Dx x x0 x0+Dx Slope of a curve • Recall that the derivative of a curve evaluate at a point is the slope of the curve at that point. Derivative of f(x) at Xo = slope of f(x) at Xo
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)
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=-2 X=2 X=0
D B X1 C X2 A 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
First derivative test (Max) Given a continuous function y=f(x) If dy/dx = 0 at x=xo & dy/dx changes from +ve to –ve through x0, x=x0 is a local maximum point local maximum point x=x0
First derivative test (Min) Given a continuous function y=f(x) If dy/dx = 0 at x=xo & dy/dx changes from -ve to +ve through x0, x=x0 is a local minimum point x=x0 local minimum point
Example • 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
Example 2 Find the local maximum and minimum of Find all stationary points first:
Example 2 By first derivative test, • x=1 is the local maximum (+ve -> 0 -> -ve) • x=3 is the local minimum (-ve -> 0 -> +ve)
Second derivative test Second derivative test states: • There is a local maximum point in y=f(x) at x=x0, if at x=x0 and < 0 at x=x0. • There is a local minimum point in y=f(x) at x=x0, if at x=x0 and > 0 at x=x0. • If dy/dx = 0 and =0 both at x=x0, the second derivative test fails and we must return to the first derivative test.
Example Find the local maximum and minimum of Find all stationary points first: By second derivative test, x=1 is max and x=3 is min