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CONTINUITY AND DIFFERENTIABILITY

CONTINUITY AND DIFFERENTIABILITY. Consider the function f(x)= 1, if x 0 =2, if x > 0 Graph of this function is Y This function is defined at every

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CONTINUITY AND DIFFERENTIABILITY

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  1. CONTINUITY AND DIFFERENTIABILITY

  2. Consider the function f(x)= 1, if x 0 =2, if x > 0 Graph of this function is Y This function is defined at every Points of real line.The value of the Function at nearby points on the (0,2) y=f(x) X-axis remain close to each other Except at x=0. At the points near and to the left (0,1) Of 0 , the value of the function is 1.At X’ X The points near and to the right of 0 O The value of the function is 2.L.H.L. of the Function at 0 is 1 and R.H.L. of the function At 0 is 2, so L.H.L. IS NOT EQUAL TO R.H.L Also the value of the function at x=0 is 1=L.H.L. Y The graph of this function cannot be drawn in one stroke This function is not continuous at x=0 Y’ CONTINUITY

  3. Suppose f is a real function on a subset of the real number and let c be a point in the domain of f . Then f is continuous at c if f(x)=f(c) If the left hand limit , right hand limit and the value of the function at x=c exist and equal to each other, then f is said to be continuous at x=c, if f is not continuous at c , then f is discontinuous at c and c is called a point of discontinuity of f. Definition

  4. Q1) Discuss the continuity of the function f defined by f(x)=x+2, if x 1 =x-2, if x>1 Solu:- The function f is defined at all the points of the real line. Case 1:- If c is less then 1 , then f(c)=c+2 Therefore f(x)= (x+2)=c+2 , thus , f is continuous at all real numbers less than 1. QUESTION

  5. Case2 If c is greater than 1, then f(c)=c-2 Therefore f(x)= (x-2)=c-2=f(c ) Thus f is continuous at all points greater then 1 Case3 if c=1, then L.H.L. of f at x=1 is y f(x)= (x+2)=1+2=3 X’ X R.H.L. of f at x=1 is f(x)= (x-2)=-1 Since L.H.L. Is not equal to R.H.L.of f at x=1 Y’z So , f is not continuous at x=1 continued

  6. Suppose f is a real function and c is a point in its domain. The derivative of f at c is defined by [ f(c+h)-f(c)]/h provided this limit exists.Derivative of f at c is denoted by f’(c). The function defined by f’(x)= wherever the limit exists is defined to be the derivative of f . If this limit does not exist the function is not differenciable,i.e. if and are finite and equal, then f is differentiable in the interval[a,b] DIFFERERENTIABILITY

  7. If a function f is differentiable at a point c,then it is also continuous at that point. Every differentiable function is continuous. CHAIN RULE:-Let f be a real valued function which is a composite of two functions u and v ; i.e. f=vou. Suppose t=u(x) and if both dt/dx and dv/dt exist, then If f is a real valued function and composite of three functions, u,v,and w , then f=(wou)ov. If t=v(x) and s=u(t), then Theorem

  8. Provided all the derivatives in the statement exist. Consider the function x+sinxy-y=0 , in this y cannot be expressed as a function of x, such type of functions are called implicit functions. We differentiate these types of function w.r.t.x and then find dy/dx Derivatives of implicit functions

  9. Q1 Find the derivative of the function xy+y.²=tanx+y Solu:- Differentiating both sides w.r.t.x 1.y+xdy/dx+2ydy/dx= (x+2y-1) = question

  10. Consider the function of the form y=f(x)= By taking logarithm on both sides , rewriting log y=v(x) log[u(x)] Using chain rule , we may differentiate this Logarithmic differentiation

  11. Q1 DIFFERENTIAT THE FUNCTION W.R.T. X • Solu:-Let u= and v= taking log on both sides logu=xloglogx and logv=logxlogx • Differentiating both sides w.r.t.x • =1.loglogx+x.1/xlogx =2logx/x • =u(loglogx+1/logx) =v(2logx/x) • Let y= , y=v+u, QUESTION

  12. = continued

  13. The relation expressed between two variables x and y in the form x=f(t), y=g(t) is said to be parametric form with t as a parameter. Derivative of the functions of such form is provided f’(t) 0 Derivatives of functions in parametric forms

  14. Q1 If x and y are connected parametrically by equations , without eliminating the parameter , find dy/dx Of x=a( • Solu:- x= • = = questions

  15. THANK YOU

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