AP Calculus BC : 2.4 Connecting Differentiability and Continuity- Exam Style questions with Answer- MCQ

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

Scroll to Top