(a)-(d) Please d. n the axes provided in Hgure 1./9, sketch an accurate graph of

Question

(a)-(d) Please

d. n the axes provided in Hgure 1./9, sketch an accurate graph of 6. For each of the following prompts, give an example of a function that satisfies the stated criteria. A formula or a graph, with reasoning, is sufficient for each. If no such example is possible, explain why a. A function f that is continuous at a = 2 but not differentiable at a-2 b. A function g that is differentiable at a 3 but does not have a limit at a=3. les A function h that has a limit at a--2, is defined at a=-2, but is not continuous at a =-2. c. d. A function p that satisfies all of the following p(-1)-3 and linz , 1 p(z) = 2 p(0)=1and p'(0), 0 lille,iPlz)=p(1)andp'(1) does not exist * · . 7, Let h(z) be a function whose derivative y h,(z) is given by the graph on the

Explanation / Answer

a) For a function to be continuous but not differentiable at a point, the graph must be continuous but have a sharp edge at the required point (which is a= 2 over here). Thus, one such function is f(a) = | a-2 |

b) It is not possible for a function to be differentiable but not have a limit at that point

c) f(a) = { a+ 1 for a < -2 ; 2a + 3 for a > -2 ; 10 for a= -2}

We see that for a = -2; LHL = -2+1 = -1 and RHL = 2*(-2) + 3= -1

Thus LHL = RHL so limit exists at a= -2 ; f(-2)= 10 so it is defined at a= -2 but it is not continuous

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