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IMPORTANT REVIEW TOPICS IN CALCULUS FUNCTIONS There are two fundamental processes of Calculus, Differentiation and Integration. These processes are applied to functions. To carry them out, one has to be familiar with functions. A function is a Rule. Usually we use small letters f, g, h ...... to denote different functions or different rules, such as f x x 1, g x 10x 2 , h x x 2 3x 2 etc. Definition: Let A and B be two non-empty sets. A function f from A to B is a rule that associates, with each value of x in set A, exactly one value f(x) in set B. The function is indicated by the notation f :A B . f(x) is read “f of x”. We usually consider functions for which sets A and B are sets of real numbers. f(a) is the value of f(x) when x=a. If f x 2x 3 then f 2 2.2 3 1 The set A, which contains all possible values of the variable x is called the Domain, and the set B which contains the corresponding values of f(x) or y is called the Range of the function (Range consists only of those elements of B which are actually paired with elements of A). We call x the independent variable because we choose it first, and from that we calculate y, which is called dependent variable. Example: Find the Domain & Range of the function. (a) f(x) = Sin x. (b) f x 2x 2 8x 2 Solution : a) Here x can be any real number, so the Domain is the set of real numbers R or x R . All values of Sin x lie between -1 and 1. So, Range is 1 y 1 or y R: 1 y 1 . b) It can be rewritten as y f x 2x 2 8x 2 2 x2 4x 2 2 2 x2 4x 4 8 2 2 x 2 6 Here x can be any real number, so the Domain is the set of real numbers R or x R . 2 As the value of square of any real number cannot be negative therefore, x 2 0. Thus y 6 . So, Range is the interval 6, or y R :y 6 . 1 f 1 h f 1 Example : Let f x 2x 2 5x 3 . Evaluate f 2 , f ,f a , 2 h Solution : Substituting the values of x in f(x) we get 2 f 2 2 2 5.2 3 8 10 3 1 2 1 1 1 1 5 f 2 5 3 3 6 2 2 2 2 2 2 f a 2 a 5 a 3 2a 5 a 3 2 2 f 1 h f 1 21 h 5 1 h 3 2 1 5.1 3 h h 2 2 1 2h h 5 1 h 3 2 5 3 h 2 2 4h 2h 5 5h 3 0 h 2 h 2h 1 2h h A function can also be defined differently for different sets of values of x. Example : Find f 1 and f 2 where the function f(x) is defined as follows : 2x 1 for x<0 f x 1 2x 2 for x 0 Solution : Since 1<0, using f x 2x 1 we get f 1 2 1 1 1 2 Since 2 0, using f x 1 2x 2 we get f 2 1 2.2 1 8 7 GRAPHS OF FUNCTIONS The graph of f is the set of all points (x,y) such that y = f(x) Or The graph of f is the graph of the equation y =f(x) To draw the graph of y f x we make a table consisting two columns: one for x and one for y [or f(x)]. We take different values of x and calculate the corresponding values of y [or f(x)]. Then we plot the points on a graph paper and join them with a smooth curve to get the graph of the function . Example : Draw the the graph of f x x3 Solution : We first make a table. We take different values of x and calculate the corresponding values of y [or f(x)]. Then we plot the points on a graph paper and join them with a smooth curve to get the graph x: -2 3 -1 1 0 1 1 3 2 2 2 2 2 y or -8 27 -1 1 0 1 2 27 8 f(x) : 8 8 8 8 ABSOLUTE VALUE x if x 0 The absolute value of a real number x, written as x x if x < 0 Therefore, | 5 | = 5, | -5 | = -(-5) = 5. INTERVAL NOTATIONS The concept of intervals is very useful in calculus. An uninterrupted portion of a number line is called an interval. The interval represents the collection of all the infinite number of points in that portion. We may write an interval like [-3,2] which means all the real numbers on the number line from -3 to 2 or all real numbers x where 3 x 2 The interval (-3,2) means 3<x<2 . (a,b) is called an open interval where a,b x |x R and a<x<b [a,b] is called an closed interval where a,b x |x R and a x b [a,b) is an interval open at one end and closed at the other, where a,b x | x R and a x<b Some more commonly used intervals are : a, x | x R and x a a, x |x R and x>a ,a x |x R and x a ,a x |x R and x<a Example : Interval 2,1 x |x R and 2<x<1 Interval 2, x |x R and 2 x STRAIGHT LINES SLOPE OF A LINE The slope of a line is a number which indicates the direction or the slant of the line. It is generally represented by “m”. For a non-vertical line passing through P x1 , y1 and Q x 2 , y 2 the slope of the line segment PQ is y2 y1 y Change in y coordinate Slope m x2 x1 x Change in x coordinate The slope of a horizontal line is 0, because as we move along the line the x-coordinate y 0 changes, but the y coordinate does not change. y 0 here So, m 0 x x The slope of a vertical line is not defined, because as we move along the line the x- y y coordinate does not change, but the y coordinate changes. x 0 here So, m x 0 which is not defined (a symbol with denominator 0 does not signify a number). For other lines, if the slope is positive, then it slopes up i.e. with increase in x- coordinate, the y-coordinate increases. If the slope is negative, then it slopes down i.e. with increase in x-coordinate, the y-coordinate decreases. Parallel & Perpendicular lines: If two lines l 1 and l 2 have slopes m1 and m 2 : The lines are parallel or l1 || l2 if and only if m1 m2 The lines are perpendicular or l1 l2 if and only if m1 m2 1 In the above picture the lines l 3 l 4 as their slopes are 1 and -1 and m1 m2 1 1 1 Example : Find the slope of the line joining points A (1,-3) and B (-2,1). Solution : Taking x1 1, y1 3, x 2 2, y 2 1 y 2 y1 1 3 4 4 Slope of AB m x 2 x1 2 1 3 3 Example : Find the slope of a line perpendicular to the line joining A(-3,1) and B(2,3). Solution : 3 1 2 Slope of AB m1 2 3 5 If the slope of the line perpendicular to it is m 2 then m1 m2 1 2 5 m2 1 multiply both sides with 5 2 5 5 m2 Slope of a line perpendicular to AB is 2 2 DIFFERENT FORMS OF EQUATIONS FOR STRAIGHT LINES General or Standard form: A linear equation of x and y represents a straight line and is called the general form or the standard form of the equation of the line. This is written as Ax By C 0 where A,B,C are real numbers and both A and B are not 0 at the same time Thus 2x 3y 5 0, 3 x y 0, 3x 2 0, y 4 0 are equations of straight lines. In 2x 3y 5 0, A 2, B 3, C 5 In 3 x y 0, A 3 , B 1, C 0 Many a times we are required to find the general form of equation of a line from its slope, a few known points on the line or its intercepts on the axes. There we take help of the following forms: Point-Slope form: If the line passes through the point x1 , y1 and its slope is m , then the equation of the line can be written in the form y y1 m x x1 Intercept Form : If a line intersects the x-axis at A(a,0) and the y-axis at B(0,b) then a is called the x- intercept and b the y-intercept. Then the equation of the line can be written in the form x y 1 where a 0, b 0 a b Slope-Intercept form : If the slope of a line is m and it intersects the y-axis at B(0,b) ( i.e. the y-intercept is b ), then the equation is y mx b Example : Find the equation of the straight line which (a) passes through the point (3,4) and has a slope -2. 2 (b) has a slope and crosses y–axis at -3. 3 Solution : a Using the point slope form y y1 m x x1 we get y 4 2 x 3 y 4 2x 6 2x y 10 0 this is the required equation b Using the slope intercept form y mx b we get 2 y x 3 3 3y 2x 9 2x 3y 9 0 this is the required equation LIMITS & CONTINUITY LIMIT OF A FUNCTION The concept of the limit of a function is the starting point of calculus. Without limits calculus does not exist. Every notion in calculus can be expressed in some forms of limits. What is limit of a function? To understand it, let’s look at the following examples: Consider the function f x 2x 3 . At x = 3, the value of the function becomes f 3 2.3 5 1. Here the function is defined and real. But, how does the function behave when x gets closer to 3 but not exactly 3? Putting a few values we see that as x approaches 3, f(x) approaches 1. sin 2x Consider another function f x If we take the interval , we can find 3x 4 4 the value of the function at x , , etc as given below, 6 15 9 sin 2 sin 2 6 15 f 0.5513, f 0.6473, 6 15 3 3 6 15 sin 2 9 f 0.6138 9 3 9 sin 0 But what happens to this function at x = 0? Putting x = 0 in f, we get f 0 which 0 is undefined. So, the function f is undefined at x = 0. But we see that it is not undefined at other x values close to 0. There it is defined and has real values. So how does the function behave there, when x goes closer to 0, but not exactly 0? To know the behavior, we take a few values of x closer to 0 (both from x>0 and x<0) and put them in the function as given in the table below and note the behavior : x 0.1 0.05 0.03 0.02 0.01 0.005 f(x) 0.66223 0.66555 0.66626 0.66649 0.66662 0.66665 x -0.1 -0.05 -0.03 -0.02 -0.01 -0.005 f(x) 0.66223 0.66555 0.66626 0.66649 0.66662 0.66665 2 The value of f(x) gets closer and closer to 0.666….. or as x gets closer to 0 from both 3 sides. sin 2x 2 This behavior is written as lim f x lim , which reads “the limit of f(x) as x x 0 x 0 3x 3 2 tends to 0 is ” . It means, as the value of x gets closer and closer to 0 (i.e. x 3 2 2 approaches 0), f(x) is defined there and gets closer to (or it approaches ) . [There is 3 3 an easier limit evaluation method to find this value, which is discussed later in trigonometric limits]. x2 4 Taking another example, lim 4 means as the value of x gets closer and closer to x 2 x 2 x2 4 2, gets closer to 4 (though the function is undefined at x=2). x 2 Thus, the limit of a function f(x) at x = c gives us an idea about how this function behaves, when the value of x goes very near to c (but not equal to c). limc f x x L [read as “the limit of f(x) as x tends to c is L” ] means “as x approaches c, f(x) approaches L” or “as the value of x goes very close to c, but not equal to c, f(x)goes very close to L”. Definition of limit: The limit of a function lima f x x A if and only if, for any chosen number > 0 , however small, there exists a number >0 such that, whenever 0 < |x a| < , then |f x A| < Note that, the limit of a function gives an idea about how the function behaves close to a particular x value. It does not necessarily give the value of the function at x. That means, limc f x and f c may or may not be equal. x Using the definitions we can prove the following limits : x limc |x| |c| x limc x c limc a a x Example : Using the definition show that lim 3x 1 5. x 2 Solution : Finding a : Let > 0 We try to find a number > 0 such that, whenever 0 < |x 2| < , then | 3x 1 5| < Now , | 3x 1 5| |3x 6| 3|x 2| To make | 3x 1 5| < or 3|x 2| < we have to make |x 2| < 3 So we have to choose a such that 0< 3 Showing that the works : If 0 < |x 2| < then 3|x 2| < 3 So, | 3x 1 5| < lim 3x 1 5 x 2 ONE SIDED LIMITS: RIGHT AND LEFT LIMITS To analyze the behavior of the function f(x) when x approaches a, we can separately look at the behavior from one side only i.e. from the left side (when x<a) and from the right side (when x>a ). x lim f x a A is the left hand limit which means “as x approaches a through values less than a, f(x) approaches A.” Definition of Left-Hand Limit: Let f be a function defined at least on an interval c,a , then x lim f x a A if and only if, for any chosen number, > 0 , however small, there exists a number > 0 such that, whenever a <x<a , then |f x A| < lim f x B is the right hand limit which means “as x approaches a through values x a more than a, f(x) approaches B.” Definition of Right-Hand Limit : Let f be a function defined at least on an interval a,d , then lim f x A x a if and only if, for any chosen number, > 0 , however small, there exists a number > 0 such that, whenever a < x < a , then |f x A| < The existence of limit from the left does not imply the existence of limit from the right, and vice versa. EXISTENCE OF LIMIT AT A POINT For a function f(x), when can we say that the limit exists at a point x=a and what would be its value in relation to the left and right limits at that point? x lima f x exists and its value will be equal to A, if and only if, x lim f x and lim f x both exist, and both are equal to A a x a Example : Find lim x and lim x , and comment if lim x exists. x 0 x 0 x 0 Solution : lim x does not exist since x<0 here and x is not defined when x<0 x 0 lim x 0 as we see that x appproaches 0 as x approaches 0 x 0 lim x does not exist as lim x does not exist. x 0 x 0 Example : Evaluate |x 3| (a) lim x 3 x 3 |x 3| (b) lim x 3 x 3 |x 3| (c) lim x 3 x 3 Solution : (a) As x approaches 3 from the left, x < 3 i.e. (x-3) is negative, and |x-3| = -(x-3), hence |x 3| x 3 lim 1 x 3 x 3 x 3 (b) As x approaches 3 from the right, x > 3 i.e. (x-3) is positive, and |x-3| = (x-3), hence |x 3| x 3 lim 1 x 3 x 3 x 3 |x 3| |x 3| |x 3| (c) As lim lim , lim does not exist x 3 x 3 x 3 x 3 x 3 x 3 INFINITE LIMITS Some functions increase or decrease without bounds (that is goes towards or ) near certain values of the independent variable x. When this happens, we say that the function has infinite limit. So we can write, lima f x x or xlima f x . The function has a vertical asymptote at x = a if either of the above limits hold true. f x A function of the form will have an infinite limit at x=a if there the limit of g(x) is g x zero but the limit of f(x) is non –zero. 1 Example : Evaluate lim x 0 x2 Solution : As x approaches 0, the numerator remains 1 which is positive, and the denominator goes closer to 0 but remains positive So, the function increases without bound and is positive 1 lim 2 x 0 x The following graph also shows that f x approaches as x 0 LIMITS AT INFINITY In some cases we may need to observe the behavior of the function when the independent variable x increases or decreases without bound, that is, x or x . If in such a case the function approaches a real number A , then we can write x lim f x A or x lim f x A and f(x) has a horizontal asymptote at y = A. 3x 2 7 Example : Evaluate lim 2 . x 5x 2x 3 Solution : 3x 2 7 lim 2 x 5x 2x 3 7 x2 3 x2 x lim 2 3 x2 5 x x2 7 3 x2 1 lim As x approaches , approaches 0 x 2 3 x 5 x x2 3 0 lim x 5 0 0 3 5 THEOREMS ON LIMITS 1 If f x c, where c is a constant, then xlima f x c If xlima f x A and xlima g x B where A,B < 2 xlima kf x kA where k is a constant 3 xlima f x g x lima f x x lima g x x A B 4 xlima f x g x lima f x x lima g x x A B f x x lima f x A 5 xlima if B 0 g x lima g x x B n n 6 xlima n f x n x lima f x A , if A is a real number The uniqueness of a limit : If lim f x x a A and lim f x x a B then, A B The Pinching Theorem : If x is close to a but different from a, a function f always lies between two functions h and g. If, as x tends to a, both h and g tend to the same limit A, then f(x) also tends to A at x = a. Suppose, h x f x g x for all x where 0 < |x a| < p and p>0, If lima h x x A and lima g x x A, then lima f x x A LIMITS OF TRIGONOMETRIC FUNCTIONS The following trigonometric limits, which can be proven by the definition, is quite useful : x limc sin x sin c x limc cos x cos c sin x 1 cos x lim 1 lim 0 x 0 x x 0 x sin 4x Example : Find lim x 0 x Solution : sin 4x lim x 0 x sin 4x lim 4 x 0 4x sin 4x 4 lim as x 0, we can write 4x 0 4x 0 4x sin x 4 1 using lim 1 x 0 x 4 tan 3x Example : Find lim x 0 2x 2 5x Solution : tan 3x lim x 0 2x 2 5x sin 3x lim x 0 x 2x 5 cos 3x 3 sin 3x lim x 0 3 x 2x 5 cos 3x sin 3x 1 1 3 lim x 0 3x 2x 5 cos 3x sin 3x 1 1 3 lim lim lim x 0 3x x 0 2x 5 x 0 cos 3x 1 1 3 1 5 1 3 5 CONTINUITY While observing the graphs of different functions we note that some functions are discontinuous at some points. For example, the graph of f x tan x is discontinuous at 1 x The graph of f x is discontinuous at x = 0. 2 x x2 2x In the above graph of f x , we see that the graph is discontinuous at x=2, but x 2 continuous at all other points. Is there any other method, by which we know if the graph of a function is continuous or discontinuous at a point? The answer is, we can know it by using limits. Definition of Continuity : A function f x is continuous at a, if and only if xlima f x f a We can also define continuity as : f is continuous at a if and only if, for each > 0 there exists > 0 such that if |x c| < then |f x f c | < thus, f(x) will be continuous at a, 1. if f(a) is defined, 2. if xlima f x exists and 3. if lima f x x f a A function f(x) is discontinuous at x = a, if any of the above conditions of continuity fails there. Geometrically it means that there is no gap, split or missing point in the graph of f(x) at a. A function f is continuous in a closed interval [a, b] if it is continuous at every point in [a,b]. So, the function f x tan x is continuous in the interval , as it is 4 4 continuous at all the points in this interval, whereas it is not continuous in the interval 3 , as it is discontinuous at x 4 4 2 A function is said to be continuous if it is continuous at every point of its domain. So, f x x2 4 is a continuous function as it is continuous at every point in its domain (its domain is the set of real numbers R). ONE-SIDED CONTINUITY: Definition :A function f x is continuous from left at a, if and only if x lim f x a f a A function f x is continuous from right at a, if and only if lim f x f a x a THEOREMS ON CONTINUITY The most important theorems of continuity are: If the functions f and g are continuous at a, then 1 f g is continuous at a, 2 f is continuous at a for each real , 3 f g is continuous at a, f 4 is continuous at a provided g a 0 g 5 If g is continuous at a and f is continuous at g a , then the composition of functions f g is continuous at a Two more theorems on continuity are: The Intermediate Value Theorem : 6 If f is continuous on a,b and C is a number between f a and f b , then there is at least one number c between a and b for which f c C The Maximun Minimum Theorem : 7 If f is continuous on a,b , then f takes on both a maximum value M and a minimum value N on a,b Example : Determine the continuity of a) f x 3x 5 at x = -3 x2 9 b) f x at x = -3 x 3 Solution : a lim f x lim 3x 5 3 3 5 14 x 3 x 3 lim f x lim 3x 5 3 3 5 14 x 3 x 3 f 3 3 3 5 14 Hence, f is continuous at x 3 x2 9 b For f x we find that f 3 is undefined and does not exist x 3 Hence, f is discontinuous at x 3 Example : Discuss the continuity of x2 9 f x , x 3 at x = 3. x 3 6, x 3 Solution : x 3 x 3 lim f x lim lim x 3 3 3 6 x 3 x 3 x 3 x 3 similarly lim f x 6 x 3 lim f x 6 x 3 and f 3 6 As lim f x f 3 ; f is continuous at x 3 x 3 Example : Discuss the continuity of 6 3x, x< 2 f x at x = -2. x 2 4, x 2 Solution : 2 1 f 2 2 4 4 4 0 2 lim f x lim 6 3x 6 3 2 0 x 2 x 2 2 lim f x lim x2 4 2 4 0 x 2 x 2 Hence lim f x 0 as lim f x lim f x x 2 x 2 x 2 As lim f x f 2 ; f is continuous at x 3 x 2 DIFFERENTIATION Let us take the graph of a function f . We know that the point ( x,f(x) ) is a point on the graph. What line, if any, should be the tangent to the graph at this point? We can take another point on ( x+h, f(x+h) ) on the graph and draw a secant line through these two points. Keeping ( x,f(x) ) fixed, we move the other point closer to it (h tends to zero from the left or from the right). The secant line tends to a limiting position. This is “the tangent to the graph at the point ( x,f(x) )”. If this limit exists, that is defined to be the slope of the tangent line at that fixed point ( x,f(x) ) on the graph of y = f (x). DIFFERENTIABILITY & THE DERIVATIVE OF A FUNCTION Differentiation is the process of finding the derivative of a function. A function f is said to be differentiable at x if and only if f x h f x lim exists. h 0 h If this limit exists, it is called the derivative of f at x and is denoted by f x . f x h f x Therefore, f x lim . h 0 h If we draw a tangent to the graph at this point, the slope of the tangent is equal to f x . The derivative of a function y = f(x) with respect to x at a point x x 0 is given by f x0 h f x0 f x0 lim provided that this limit exists. h 0 h This limit is also called the instantaneous rate of change of y with respect to x at x x0 . d dy The d/dx notation : If y = f(x), then we can write f x f x dx dx The derivative of the function y =f(x) with respect to x may be indicated by any one of d dy df x the symbols y , , y , f x , Dx , Dx f, . We can use any of these, dx dx dx while solving problems. A function is said to be differentiable at a point x x 0 if the derivative of the function exists at that point. Also, if a function is differentiable at every point in an interval, the function is said to be differentiable on that interval. If we say “a function (of x) is differentiable” without mentioning the interval, we mean that it is differentiable for every value of x in its domain. Example : 1. The function f x x3 3x is differentiable on the interval (-1,1) as it is differentiable at every point in that interval. 2. The function f x |x 2| is not differentiable on the interval (1,3) as it is not differentiable at x = 2 which lies in that interval. Example : Find f’(x) for the function f x x2 4. Solution : We formthe difference quotient 2 f x h f x x h 4 x2 4 h h 2 2 x 2 2xh h 4 x2 4 2xh h 2x h h h f x h f x Therefore f x lim lim 2x h 2x h 0 h h 0 Example : Find f 3 and f 1 given that f x x2 Solution : f 3 h f 3 By definition f 3 lim h 0 h 2 2 3 h 3 lim h 0 h 2 9 6h h 9 lim h 0 h 2 6h h lim h 0 h lim 6 h h 0 6 Similarly f 1 h f 3 f 1 lim h 0 h 2 2 1 h 1 lim h 0 h lim 2 h h 0 2 Alternative method: Find f x from f(x) first, and to put x=-3 and x=-1 in f x to get f 3 and f 1 as shown below f x h f x f x lim h 0 h 2 h x2 x
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