please solve these problems and show me your work. sorry please just ignore my text and works on the problems ...

Math 180 Worksheets W3 3 Continuity and Definition of the Derivative Keywords: continuity, graphing, intermediate value theorem, limit definition of the deriva- tive, secant lines 1. (a) On the grid below, draw a function f that is not continuous at x = 2, but lim f (x) = 1. 20 - 2 ' (b) On the grid below, draw a function f that is defined everywhere and whose limit lim f(x) is undefined. x - 2 " (c) Are the functions you drew above continuous at x = 2 i. from the left? ii. from the right?Math 180 Worksheets W3 2 2 ( xth ) 6. Consider the function f (x) = 2. x+h ( (4th ) (a) Use the limit definition to compute f'(x). (b) Use your answer from above to evaluate f'(-2). -2h ( xth)xon (c) Complete the following table of difference quotients. - 2 2= f (x ) h 1.5 0.5 0.1 f ( - 2 + h) - f (- 2) ~ -0.66 ~ -0.52 h Do the quotients get closer to f'(-2) as h gets smaller? (d) For h = 1 in the table above, draw a line through (-2, f(-2)) and (-2th, f(-2+ h)) on the figure below. Then try to sketch the tangent line to the function at the point x = -2. - 4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0.5 1.5 - 2 -2.5 -3 3.5 -4 -4.5 -5 The slopes of these lines should correspond to the values of the difference quotients in the table. ! 7. Find the following limit by identifying it with the definition of a derivative: lim ( 1 + h ) 8+ ( 1+ 1 ) 3 - 2 13Math 180 Worksheets W3 4. Recall the intermediate value theorem (IVT). (a) What are the conditions of the IVT? If a function is continous on [a, b), anditL is any number between fla) and fob), then there must be a vive for where acccb, such that flo) = L (b) What is the conclusion of the IVT? J (X ) [a , b ] J ( a) 70 , g ch ) 70 (c) Use the IVT to determine if x3 + 2x + 1 has a root on the interval (-1, 0). ( - 1 8 + 2 ( - 1 ) +1 = ( 2 ) 5. (a) Write down the limit definition of the derivative of a function f. (b) Using the definition you gave above, compute f'(x) for f(x) = x2 + 1. lim ( ( x th) - fcx ) ( th ) + 1 - x 2 + 1 h - 70 * + 2 *h th = 4 2 - zxth 12 = 24- f ( x )Math 180 Worksheets W3 2. On the grid below, draw the function 5 if x 1. On what intervals is f(x) continuous? Explain your reasoning! 01-2 2 4 2 3 ( * * 1 ) ( x-2) 3. Let f(x) = 20 2 - 4 ac + 2 * + 2 (a) Evaluate lim f(x) X - 2 = - 4 (b) Determine whether or not f(x) is continuous at a = -2. If not, explain why. di's continuous because Qual r? 11