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True/False practice problems for Exam I
In the exam, you will get +2 for a correct response, and 0 for an incorrect response on a True/False question. You will get +1 for writing “I don’t know.” Remember: In logic, “True” means always true. “False” means “not always true,” meaning the statement may be false for some examples. State whether the statement is true or false:
(a) If lim (^) x æ a f ( x ) exists, then lim (^) x æ a ≠ f ( x ) and lim (^) x æ a + f ( x ) both exist.
(b) If lim (^) x æ a f ( x ) does not exist, then lim (^) x æ a ≠ f ( x ) and lim (^) x æ a + f ( x ) do not exist.
(c) If lim (^) x æ a ≠ f ( x ) and lim (^) x æ a + f ( x ) exist, then lim (^) x æ a f ( x ) exists.
(d) If lim (^) x æ a ≠^ f ( x ) and lim (^) x æ a +^ f ( x ) exist and if they agree, then lim (^) x æ a f ( x ) exists.
(e) If f Õ^ ( a ) = 0, then a is either a local maximum or a local minimum.
(f) If a is an inflection point of f , then f Õ^ ( a ) = 0.
(g) If a is an inflection point of f , then f ÕÕ^ ( a ) = 0.
(h) If a is a critical point of f , then f Õ^ ( a ) = 0.
(i) If a is a critical point of f , then f ÕÕ^ ( a ) = 0.
(j) If lim (^) x æ a +^ f ( x ) = 0 and lim (^) x æ a +^ g ( x ) = 0, and if f and g are di eren- tiable, we can apply L’Hopital’s Rule to compute lim (^) x æ a + f g^ (( xx )).
(k) If lim (^) x æ a f ( x ) = 0 and lim (^) x æ a g ( x ) = Œ, and if f and g are di erentiable, we can apply L’Hopital’s Rule to compute lim (^) x æ a f g^ (( xx )).
(l) If lim (^) x æ a f ( x ) = Œ and lim (^) x æ a g ( x ) = 0, and if f and g are di erentiable, we can apply L’Hopital’s Rule to compute lim (^) x æ a f g^ (( xx )).
(m) If lim (^) x æ a + f ( x ) = Œ and lim (^) x æ a + g ( x ) = Œ, and if f and g are di eren-
tiable, we can apply L’Hopital’s Rule to compute lim (^) x æ a + f g^ (( xx )).
(n) When testing for absolute maxima and minima for f along an interval [ a, b ]: If f is di erentiable, then one need only check the values of f ( x ) when x is a critical point or when x = a or when x = b.
(o) The second derivative test tells us (at least) that if f Õ^ ( x ) = 0 and f ÕÕ^ ( x ) < 0, then x is an absolute maximum.
(p) The second derivative test tells us (at least) that if f Õ^ ( x ) = 0 and f ÕÕ^ ( x ) < 0, then x is a local maximum.
(q) The function f ( x ) = | x | has a derivative at x = 0.
(r) The function f ( x ) = | x | has a derivative at x = 1.
(s) The function f ( x ) = | x | has a derivative at x = ≠1.
(t) The function f ( x ) = | x | has a derivative at every value of x except 0.
(m) If lim (^) x æ a +^ f ( x ) = Œ and lim (^) x æ a +^ g ( x ) = Œ, and if f and g are di eren- tiable, we can apply L’Hopital’s Rule to compute lim (^) x æ a + f g^ (( xx )). True.
(n) When testing for absolute maxima and minima for f along an interval [ a, b ]: If f is di erentiable, then one need only check the values of f ( x ) when x is a critical point or when x = a or when x = b. True.
(o) The second derivative test tells us (at least) that if f Õ^ ( x ) = 0 and f ÕÕ^ ( x ) < 0, then x is an absolute maximum. False.
(p) The second derivative test tells us (at least) that if f Õ^ ( x ) = 0 and f ÕÕ^ ( x ) < 0, then x is a local maximum. True.
(q) The function f ( x ) = | x | has a derivative at x = 0. False. Writing the di erence quotient at x = 0 , we see that lim (^) h æ 0 + f^ (0+ h h )≠ f^ (0) = 1 while lim (^) h æ 0 ≠^ f^ (0+ h h )≠ f^ (0)= ≠ 1. Thus the di erence quotient does not have a limit as h æ 0 , meaning f does not have a derivative at x = 0.
(r) The function f ( x ) = | x | has a derivative at x = 1. True. The deriva- tive is 1.
(s) The function f ( x ) = | x | has a derivative at x = ≠1. True. The derivative is -1.
(t) The function f ( x ) = | x | has a derivative at every value of x except 0. True. The derivative is 1 when x is positive, and -1 when x is negative.
