5 Differentiability and Continuity.notebook Differentiability and Continuity 1. Understand the relationship between differentiability and continuity 22. Recognize that a function that is differentiable at a point must be continuous at that point 3. Show that a continuous function may fail to be differentiable at a point in its domain Lesson objectives Teachers notes Differentiability and Continuity The alternative limit form is useful in investigating the relationship between differentiability and continuity. The derivative of f at c (provided the limit exists) is.. S(e)=lim-S) x-c The existence of the limit in this alternate form requiresthat: 1. The one-sided limits exist, and 2. They are equal We say that f is differentiable on the closed interval [a, b] if it is differentiable on (a, b) and if the derivative from the right at a, and the derivative from the left at b, both exist. 1 5 Differentiability and Continuity.notebook How f(a) May Fail to Exist.. The next examples will show us how a function whose graph is smooth will fail to have a derivative at a point where there is a corner, a cusp, a vertical tangent line or a discontinuity. EX # 1: Sketch the slope graph (derivative) for the function y =x-1; then, discuss and explain whether or not the function is differentiable at x = 1 based on the concepts of limits. 123 4 = }* the alternate EX # 2: Take a look at the graph of f(x) . Using form of the derivative, can you show that the function has a vertical tangent line at x =0? 418'x) 5- 43 2 2 5 Differentiability and Continuity.notebook Differentiability Implies Continuity x = C If f is differentiable at x = C, then f is continuous at x = c,then it is 1. If a function is differentiable at continuous at x = c. So, differentiability implies continuity. continuous at x = c and 2 It is possible for a function to be not be differentiable at x = C. Continuity does not imply differentiability. VERTICAL TANGENT LINE SHARP ORNER, CUSP, or differentiable. which the function is EX # 3: Describe the x-values at 2 A) f(r) =(x+ 3): B.) &(r)=k*-5| 8g() 3 98 5 3? 1 2 3 5 Differentiability and Continuity.notebook EX #4: Find all values x of for which the function is differentiable. (x+1), xs0 Given: h(r) ={ 2x+1, 0<x<3 (4-x. x23 EX # 5: Find the value(s) of the constant a where the function is continuous over all reals. ax-x x<3 sx) 2a x23 A 5 Differentiability and Continuity.notebook EX # 6: Show algebraically, using the alternate form of derivative, why g(x ) is not differentiable. g8(r)=xSI x;x>l Xx>1 5 .