(a) What do the closed circles on the number line for f(x) correspond to on the graph of f(x)?

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Theodore Ramsey
7 years ago
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In te grap of f(x) below, number lines are used to mark were f(x) is zero/positive/negative/undefined, were f (x) is zero/positive/negative/undefined, and were f (x) is

1 In te grap of f(x) below, number lines are used to mark were f(x) is zero/positive/negative/undefined, were f (x) is zero/positive/negative/undefined, and were f (x) is zero/positive/negative/undefined Closed circles on te number lines indicate a zero-value, and open-circles indicate an undefined value (a) Wat do te closed circles on te number line for f(x) correspond to on te grap Tis is were f(x) as x-intercepts (zeroes) (b) How does eac + or sign on te number line for f(x) relate to te grap? Te + and signs indicate were f(x) is positive or negative (c) Wat does te closed circle at x = on te number line for f (x) correspond to on te grap Tis is were f(x) as a orizontal tangent line (d) Wat does te open circle at x = on te number line for f (x) correspond to on te grap ere Tis is were f(x) as a vertical tangent line, so its slope, and f (x), is undefined (e) How does eac + or sign on te number line for f (x) relate to te grap? Tey indicate were f (x) is positive or negative f (x) is negative were f(x) is decreasing and positive were f(x) is increasing (f) Wat does te open circle at x = on te number line for f (x) correspond to on te grap Since f (x) is undefined ere, so is f (x) (g) How does eac + or sign on te number line for f (x) relate to te grap? Tey indicate were f (x) is positive or negative f (x) is negative were f(x) is concave down and positive were f(x) is concave up

2 For te grap of f(x) sown below, fill in te number lines for f(x), f (x) and f (x), marking closed circles were tere is a zero, marking open circles for undefined points, and marking + and signs on eac interval to sow positive/negativeness 3 Draw a grap of f(x) tat fits te information sown in te number lines

3 4 Te middle grap drawn below sows f (x) Using te principles you learned in te previous problem, draw a possible grap of f(x) above it, and a grap of f (x) below it (If you are stuck try drawing te grap of f (x) first) f(x) : f (x) : f (x) : 3

4 5 Tis problem investigates te derivative of te absolute value function Recall tat we define te absolute value as: { x if x 0, x = x if x < 0 (a) In te space provided, draw a grap of te function x (b) Using your grap from part (a), and your understanding of te derivative as te rate of cange/slope of te tangent line, find te derivative function f (x) of te above function x, for x not equal to 0 (fill in te blanks): if x > 0, if x > 0 f (x) = if x < 0 - if x < 0 (c) But wat about f (0)? It is not so clear from te picture even ow to draw a tangent line to te function at te origin So let s try to compute f (0) by first looking at te corresponding leftand and rigtand limits of te difference quotient i Compute lim f(0+) f(0) 0 + Hint: use te piecewise definition of f(x) given above f(0 + ) f(0) 0 0 lim = lim = lim = lim = 0 + (since 0 + means is positive, so = ) ii Compute lim 0 f(0+) f(0) f(0 + ) f(0) 0 0 lim = lim = lim (since 0 means is negative, so = ) = lim = 0 iii Wat do your answers to parts (i) and (ii) tell you about f (0)? Please explain Since te rigtand limit lim f(0+) f(0) 0 + does NOT equal te leftand limit lim f(0+) f(0) f(0+) f(0) 0, te (two-sided) limit lim 0 does not exist But tis (two-sided) limit is f (0), so f (0) does not exist 4

5 6 Using wat you ve learned above, sketc te grap of a continuous function g(x) suc tat g(x) is not differentiable at x =, x =, nor x = Create a piecewise function were one piece is a quadratic function and te oter piece is a linear function wic is continuous everywere but not differentiable at x = 0 if x 0, if x < 0 x if x 0 x if x < 0 8 Create a piecewise function were one piece is a quadratic function and te oter piece is a linear function wic is continuous and differentiable everywere if x > 0, if x 0 (x + ) if x 0 x if x < 0 Note, because d dx (x + ) = (0 + ) = and d x=0 dx x =, so te rigt and left difference x=0 quotient limits agree at x = 0 and terefore te function is differentiable at x = 0 5

f(a + h) f(a) f (a) = lim

f(a + h) f(a) f (a) = lim Lecture 7 : Derivative AS a Function In te previous section we defined te derivative of a function f at a number a (wen te function f is defined in an open interval containing a) to be f (a) 0 f(a + )