Correct Answer (a) Since (x) 10, if > 0 then 10x. Thus by the Squeeze Theorem . Next, if < 0, then f x xf(x) x −5 ≤ ≤ −5x ≤ ≤ 10x. Applying the Squeeze Theorem again, . Thus . Checking the functional value, we xf(x) −5x ≥ ≥ have (0) = 0 · 3 = 0. Thus , so is continuous at = 0. g g x (b) No. , which does not exist. Let . (a) Use the definition to show is continuous at = 4. f x (b) What is the average rate of change of on the interval [0, 0.004]? f (c) Is there an instantaneous rate of change of when = 0? Explain f x
Correct Answer (a) Since −5 ≤ f (x) ≤ 10, if x > 0 then −5x ≤ xf(x) ≤ 10x. Thus by the Squeeze Theorem . Next, if x < 0, then −5x ≥ xf(x) ≥ 10x. Applying the Squeeze Theorem again, . Thus . Checking the functional value, we have g (0) = 0 · 3 = 0. Thus , so g is continuous at x = 0. (b) No. , which does not exist. Let . (a) Use the definition to show f is continuous at x = 4. (b) What is the average rate of change of f on the interval [0, 0.004]? (c) Is there an instantaneous rate of change of f when x = 0? Explain
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