== Informal discussion of continuity == The simplest graphs studied so far have consisted almost exclusively of unbroken curves. This is a sufficient basis for the intuitive idea of continuity. The behaviour of 1/x and | x | /x near the origin should be demonstrated, but discontinuities should not be further stressed. == The notion of the limit of a function and the definition of continuity in terms of this notion. Continuity of == f + g {\displaystyle f+g} , f − g {\displaystyle f-g} , f g {\displaystyle fg} in terms of continuity of f {\displaystyle f} and g {\displaystyle g} Intuitively a function is ‘continuous’ at a given point x = c {\displaystyle \left.x=c\right.} if the function value f ( c ) {\displaystyle f(c)} is ‘approached continuously’ from ‘neighbouring’ values of x, that is, if the ‘limit of f(x) as x approaches c’ agrees with the actual function value f(c) when x is precisely equal to c {\displaystyle c} . Otherwise, f(x) has a ‘jump’ at x = c {\displaystyle \left.x=c\right.} . We use the notations lim x → c f ( x ) {\displaystyle \lim _{x\to c}f(x)} and lim h → 0 f ( c + h ) {\displaystyle \lim _{h\to 0}f(c+h)} to mean the limit of the function as x → c. If f(x) is ‘continuous’ at x = c, then lim x → c f ( x ) = f ( c ) {\displaystyle \lim _{x\rightarrow c}f(x)=f(c)} , and lim h → 0 f ( c + h ) = f ( c ) {\displaystyle \lim _{h\to 0}f(c+h)=f(c)} for negative and positive values of h. We use this intuitive notion to define continuity precisely as follows. A function f(x) is said to be continuous at x = c if: f(x) is defined at c; the limit of f(x) as x approaches c exists; f(c) is equal to this limit.A function f(x) is called continuous or a continuous function if it is continuous at each point in its domain, i.e., if f(x) is continuous at x = c for every choice of c in the domain of the function. There should be light treatment of the formal proofs of the limits of the sum, difference and product of two functions, and of the corollaries that, given f is continuous and g is continuous, then f + g, f – g, fg are continuous. == Gradient of a secant to the curve y = f(x) == A secant is defined as the straight line passing through two given points on the curve. The gradients of secants for particular cases should be calculated. The general expression for the gradient of the secant through the two points P(c, f(c)) and Q(x, f(x)) on the curve y = f(x) should be derived. == Tangent as the limiting position of a secant. The gradient of the tangent. Equations of tangent and normal at a given point of the curve y = f(x) == By drawing secants through a given point P on the curve y = f(x), and through a succession of points Q1,Q2,Q3, …, on the curve, first on one side of P, then on the other, the idea of the tangent line at P as the limiting position of the secant is illustrated. The geometric picture strongly suggests that there is a limiting value for the gradient of the secant through P(c, f(c)) and Q(x, f(x)) as Q approaches P, although the formula derived in 8.3 becomes meaningless if x = c. This limiting value of the gradient of the secant is defined to be the gradient of the tangent line (or tangent) at P and is called simply the gradient of the curve at P. The gradient of the tangent at a specified point on a given curve should be calculated as above in a few simple cases and verified graphically. == Formal definition of the gradient of y = f(x) at the point where x = c == Notations f ′ ( c ) , d y d x {\displaystyle f'(c),{\frac {dy}{dx}}} at x = c {\displaystyle \left.x=c\right.} The intuitive notion of tangent to a curve, as described above, leads to a way of defining tangent in terms of a limit. Formally, the gradient of the curve y = f(x) at the point P(c, f(c)) is defined as the limiting value lim x → c f ( x ) − f ( c ) x − c = f ′ ( c ) {\displaystyle \lim _{x\rightarrow c}{\frac {f(x)-f(c)}{x-c}}=f'(c)} provided this limit exists. Thus f'(c) is the slope of the tangent line to the curve y = f(x) at the point x = c. The gradient of y = f(x) at x = c is also called the derivative or differential coefficient of f(x) at x = c. Note that the limit may fail to exist even for a continuous curve: the curve may have a vertical tangent (infinite slope), or a sharp bend, as in y = | x | {\displaystyle y=\left|x\right|} at x = 0. By putting x = c + h, where h may be positive or negative, in the definition, we get the alternative and equivalent forms f ′ ( c ) = lim h → 0 f ( c + h ) − f ( c ) h = lim δ x → 0 δ y δ x {\displaystyle f'(c)=\lim _{h\rightarrow 0}{\frac {f(c+h)-f(c)}{h}}=\lim _{\delta x\rightarrow 0}{\frac {\delta y}{\delta x}}} where Δ x = h {\displaystyle \left.\Delta x=h\right.} , Δ y = f ( c + h ) − f ( c ) {\displaystyle \left.\Delta y=f(c+h)-f(c)\right.} , y = f ( x ) {\displaystyle \left.y=f(x)\right.} . A number of simple numerical examples of the following type should be given in order to facilitate understanding of the above definitions. Example. Find the derivative of the function f(x) = x3 + 5x at x = 1 and hence find the equation of the tangent line to the curve y = f(x) at the point (1, 6). By the definition, f ′ ( 1 ) = lim h → 0 f ( 1 + h ) − f ( 1 ) h {\displaystyle f'(1)=\lim _{h\to 0}{\frac {f(1+h)-f(1)}{h}}} f ′ ( 1 ) = lim h → 0 ( 1 + h ) 3 + 5 ( 1 + h ) − ( 1 + 5 ) h {\displaystyle f'(1)=\lim _{h\to 0}{\frac {(1+h)^{3}+5(1+h)-(1+5)}{h}}} f ′ ( 1 ) = lim h → 0 6 + 8 h + 3 h 2 + h 3 − 6 h {\displaystyle f'(1)=\lim _{h\to 0}{\frac {6+8h+3h^{2}+h^{3}-6}{h}}} f ′ ( 1 ) = lim h → 0 ( 8 + 3 h + h 2 ) {\displaystyle f'(1)=\lim _{h\to 0}(8+3h+h^{2})} f ′ ( 1 ) = 8 {\displaystyle f'(1)=8} The equation of the required tangent line is therefore y − 6 = 8 ( x − 1 ) {\displaystyle y-6=8(x-1)} Examples should be chosen using the same function with several different (numerical) values of x, as a lead-in to the definition of the gradient function. For the special case of the straight line y = mx + b, it should be verified that the definition of gradient yields the correct value m at any point P(c, mc + b) on the line. == The gradient or derivative as a function == Notations f ′ ( x ) , d y d x , d d x f ( x ) , y ′ {\displaystyle f'(x),{\frac {dy}{dx}},{\frac {d}{dx}}f(x),y'} Given the function y = f(x), with domain D say, we may use the definition of gradient at each point (c, f(c)), c ε D, to find out if the function has a derivative at that point. Examples should be given (e.g. x2, x3 – x, |x |) leading to the understanding that for common functions, a derivative exists at all points in the domain or at all points with isolated exceptions which can be identified from a sketch of the function. That f '(c) is given by an expression involving c and related to the expression for f(c) should also be illustrated by examples. After first showing on a sketch of y = f(x) the gradient f '(c) at various points c (illustrated as the slope of the tangent to the curve at (c, f(c))), the transition should then be made to sketching the gradient function or derivative y = f '(x), which is the function whose domain is the set of points x for which f '(x) exists, and whose value at x = c is f '(c). This function should be identified and drawn for a number of functions f(x) (for example, y = f(x) and y = f '(x) could be drawn on the same graph or one below the other, using the same line for the y-axis and different x-axes). In adopting the usual x, y notation for the derivative function, care must be taken to avoid any confusion of notation and certainly to avoid nonsense such as f ′ ( x ) = lim x → x f ( x ) − f ( x ) x − x {\displaystyle f'(x)=\lim {x\to x}{\frac {f(x)-f(x)}{x-x}}} (i.e. the replacing of c by x in the definition of f '(c)). Right from the beginning of work involving f(x) and f '(x) as functions, students must be encouraged to remember the geometrical significance of f'(x) in relation to the graph of f(x), and to relate properties of one graph to properties of the other. This relationship is further developed in Topic 10. The notations f ′ ( x ) , d y d x , d d x f ( x ) , y ′ {\displaystyle f'(x),{\frac {dy}{dx}},{\frac {d}{dx}}f(x),y'} , and the use of different variables in place of x or y should be discussed and used in examples. If f(x) possesses a derivative f'(x) for each x belonging to the domain of f, then f(x) is called a differentiable function. The statement ‘f(x) is differentiable’ means ‘f(x) has a derivative at each point of its domain’. == Differentiation of xn for positive integral n == the tangent to y = x n {\displaystyle \left.y=x^{n}\right.} The geometric series xn–1 + cxn–2 + c2xn–3 + … + cn–2x + cn–1 has first term xn–1, and ratio c/x, so that the series has the sum (xn – cn)/(x–c). This identity and theorems on limits of sums and products may be used to find the derivative of xn for positive integral values of n. This result should be used to find the equation of the tangent to the curve y = xn at the point (c, cn), for positive integral values of n. Students should be asked to verify this result graphically for particular values of n and c, comparing the tangent drawn by eye with the straight line whose equation has been found analytically. The derivative of a constant function y = c is the constant function y = 0. == Differentiation of x1/2 and x–1 from first principles. For the two functions u and v, differentiation of Cu (C constant), u + v, u – v, uv. The composite function rule. Differentiation of u/v == Although a few simple functions can be differentiated by straightforward use of the definition given in 8.5 i.e., differentiated ‘from first principles’, this procedure is in general far from easy. Even for such simple functions as √x and 1/x, we resort to the devices lim x → c x 1 2 − c 1 2 x − c = lim x → c x 1 2 − c 1 2 ( x 1 2 + c 1 2 ) ( x 1 2 − c 1 2 ) = lim x → c 1 x 1 2 + c 1 2 = 1 2 c 1 2 {\displaystyle \lim _{x\to c}{\frac {x^{\frac {1}{2}}-c^{\frac {1}{2}}}{x-c}}=\lim _{x\to c}{\frac {x^{\frac {1}{2}}-c^{\frac {1}{2}}}{(x^{\frac {1}{2}}+c^{\frac {1}{2}})(x^{\frac {1}{2}}-c^{\frac {1}{2}})}}=\lim _{x\to c}{\frac {1}{x^{\frac {1}{2}}+c^{\frac {1}{2}}}}={\frac {1}{2c^{\frac {1}{2}}}}} Quite apart from the frequent need for ingenuity, differentiation from first principles is in general a tedious procedure. Fortunately, there are theorems which allow us to find the derivatives of complicated functions, starting from derivatives of simpler functions. These theorems are: if u,v are differentiable functions of x, then == Differentiation of: general polynomial, xn for n rational, and functions of the form {f(x)}n or f(x)/g(x), where f(x), g(x) are polynomials ==