Question
Which of the following statements, if true, can be used to conclude that f(3) exists?
I.\(\lim_{x\rightarrow 3}f(x)\) exists.
II. f is continuous at x=3.
III. f is differentiable at x=3.
A I only
B II only
C II and III only
D I, II, and III
Answer/Explanation
Ans:C
Question
\(\lim_{h\rightarrow 0}\frac{ln(e+h)-1}{h} \) is
(A) \( f ′(e )\), where \(f(x) = lnx \)
(B)\( f ′(e ) \), where \(f(x)= \frac{lnx}{x}\)
(C) \(f'(1)\), where \(f(x) = lnx \)
(D) \(f ′(1)\), where \( f(x) = ln(x+e)\)
(E) \( f'(0)\), where \(f(x) =lnx\)
Answer/Explanation
Ans:A
Question
Let f be the function defined above. Which of the following statements is true?
A f is neither continuous nor differentiable at x=2 .
B f is continuous but not differentiable at x=2.
C f is differentiable but not continuous at x=2.
D f is both continuous and differentiable at x=2.
Answer/Explanation
Ans:B
Question
The graph of the function f, shown above, has a vertical tangent at x=3 and horizontal tangents at x=2 and x=4. Which of the following statements is false?
A f is not differentiable at x=3 because the graph of f has a vertical tangent at x=3 .
B f is not differentiable at x=−2 and x=0 because f is not continuous at x=−2 and x=0.
C f is not differentiable at x=−1 and x=1 because the graph of f has sharp corners at x=−1 and x=1.
D f is not differentiable at x=2 and x=4 because the graph of f has horizontal tangents at x=2 and x=4.
Answer/Explanation
Ans:D