Advanced Calculus

1. Advanced Calculus Lecture for first year by Nguyen Xuan Hung

2. Sets Concepts Union of sets Intersections Complementary sets

3. Definition of Sets Definition Sets are to consist of objects with common properties Notation A, B, C, D,E, etc. Each element belongs to A : x  A Example A={0,1, 2, 3, 4, 5, 6, 7} B={x| -x3-x +2 =0}

4. Definition of Sets - Set of Natural number N={1, 2, 3, 4, …} - Set of Integer number Z={0, ±1, ± 2, ± 3, ± 4, …} - Set of Rational number Q={1/2, 1, 3/2, 4, 5/7,…} - Set of Real number R={1/2, 1, 1.3333,1.41, …} Z ∩ N R ∩ Q ∩

5. B = {x  A or x  B } A ∩ Operators on sets Union of sets Intersection of sets B = {x  A and x  B } A ∩

6. U\A = {x U and x  A } Operators on sets Complement of sets U A

7. Functions Functions and their GraphsInjectivity and Surjectivity

8. Definition of Functions Definition Given sets X R and Y R. A function f : X  Y is a rule which assigns an element f(x) of the set Y for every x in X. ∩ ∩ f : X  Y or y = f(x) Let f : X  Ybe a function. The set X is the domain of definition D(f) of the function f. The set Y is the target domain R(f) of the function f. The set f(X) = { f(a) | aX }  Y is the range of the function f.

9. Graphs of Functions

10. Graphs of Functions Examples

11. Curves and Graphs Which of the following curves in the plane are graphs of functions? Problem Answer The first two curves are not graphs of functions since they do not correspond to a rule which associates a unique y-value to any given x-value. Graphically this means that there are vertical lines which intersect the first two curves at more than 1 point.

12. X Y Injective Functions Definition A one-to-one function associates at most one point in the set X to any given point in the set Y. Problem Which of the following graphs are graphs of one-to-one functions? None of the above graphs are graphs of one-to-one functions since they correspond to rules which associate several x-values to some y-values. This follows since there are horizontal lines intersecting the graphs at more than 1 point. Answer

13. Surjective Functions Definition Definition Observe that the property of being surjective or onto depends on how the set B in the above is defined. Possibly reducing the set B any mapping f: A B can always be made surjective.

14. x-axis y-axis w-axis Composed Functions (1) Definition Observe that the composed function f og can be defined by the above formula whenever the range of the function g is contained in the domain of definition of the function f. Example There are infinitely many other ways to represent the above function as a composed function. This is never unique. The composition used depends on the computation to be performed.

15. Composed Functions (2) Observations • Assume that f and g are functions for which the composed function h = f og is defined. • If both f and g are increasing, then also h is increasing. • If f is increasing and g decreasing, then h is decreasing. • If f is decreasing and g increasing, then h is decreasing. • If both f and g are decreasing, then h is increasing.

16. Here the operation “-1” is applied to the function f rather than the values of the function. Inverse Functions If a function f: A  B is injective, then one can solve x in terms of y from the equation y = f(x) provided that y is in the range of f. This defines the inverse function of the function f. Definition Notation Warning

17. y=x f f-1 Finding Inverse Functions To find the inverse function of a given function f: A  B one can simply solve x in terms of y from the equation y = f(x). If solving is possible and the solution is unique, then the function f has an inverse function, and the solution defines the inverse function. Example

18. The Logarithm Let a > 0. We know that the exponential function ax is increasing if a > 1 and decreasing if a < 1. In both cases the function ax is injective. Hence the exponential function has an inverse function. Definition Notation Definition Notation

19. Properties of the Logarithm Proof The formulae 1 and 2 follow directly from the properties of the exponential function.

20. y=sin(x) y=arcsin(x) y=sin(x) The Inverse Function of the Sine Function The sine function is not injective since there are horizontal lines intersecting the curve at infinitely many points. Hence one cannot solve x in terms of y uniquely from the equation y=sin(x). In fact, there are no solutions if y > 1 or y < -1. If -1  y  1, there are infinitely many solutions. The solution becomes unique, if we require it to be between -/2 and /2. This is equivalent to restricting the domain of definition of the sine function to the interval [-/2, /2]. Definition

21. tan(x) arccos(x) arctan(x) cos(x) The Inverse Function of the Cosine and the Tangent Functions Definition

22. New Functions from Old Piecewise Defined FunctionsDeformations of FunctionsComposed FunctionsInverse FunctionsInverses of Exponential FunctionsInverses of Trigonometric Functions

23. Piecewise Defined Functions (1) Sometimes it is necessary to define a function by giving several expressions, for the function, which are valid on certain specified intervals. Such a function is a piecewise defined function. Definition The absolute value |x| is an example of a piecewise defined function. We have |x| = x if x0 and |x| = -x otherwise. Computations with the absolute value have to be done using its definition as a piecewise defined function. Problem Solution We have to strip the absolute values from the expression by starting with the innermost absolute values.

24. f(x) Piecewise Defined Functions (2) Problem Solution

25. 1.5 f(x) f(x) 0.5f(x) Simple Deformations (1) Let f be a given function, and let a be a real number. The following picture illustrates how the graph of the function f gets deformed as we replace the values f(x) by a f(x). By multiplying the function by a positive constant a the graph gets stretched in the vertical direction if a>1 and squeezed if a<1. By multiplying the function by a negative constant a the graph gets first reflected about the x-axis and then stretched in the vertical direction if a<-1 and squeezed if 0>a>-1.

26. Simple Deformations (2) The effect, on the graph, of multiplying a function with a constant is either stretching, squeezing or, if the constant is negative, then first reflecting and then stretching or squeezing. 1.5 f(x) f(x)+1.7 f(x) f(x)-1.7 Adding a constant to a function means a vertical translation in the graph. The picture on the right illustrates this situation. 0.5f(x)

27. f(x-1) f(x+1) f(x) Simple Deformations (3) Let f be a given function, and let b be a real number. The following problem illustrates how the graph of the function f gets deformed as we replace the values f(x) by f(x+b). Problem The picture on the right shows the graphs of functions f(x-1), f(x) and f(x+1). Which is which? x-1 takes a value x0 when x= x0 +1. Solution Similarly x+1 takes a value x0 when x= x0 -1. We conclude that the black graph must be the graph of the function f(x), and that the other graphs are as labeled in the picture.