Home / AP Calculus AB: 2.4 Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist – Exam Style questions with Answer- MCQ

AP Calculus AB: 2.4 Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist – Exam Style questions with Answer- MCQ

AP Calculus AB-Questions and Answers - All Topics

Question

A particle moves along the \( y \)-axis so that at time \( t \geq 0 \) its position is given by: \( y(t) = t^{3} – 4t^{2} + 4t + 3 \) Which of the following statements describes the motion of the particle at time \( t = 1 \)?

A) The particle is moving down the \( y \)-axis with decreasing velocity.
B) The particle is moving down the \( y \)-axis with increasing velocity.
C) The particle is moving up the \( y \)-axis with decreasing velocity.
D) The particle is moving up the \( y \)-axis with increasing velocity.

▶️ Answer/Explanation

Solution

Correct Answer: A

The velocity of the particle is: \( v(t) = \frac{dy}{dt} = 3t^{2} – 8t + 4 \) At \( t = 1 \): \( v(1) = 3(1)^2 – 8(1) + 4 = -1 \) Since \( v(1) \) is negative, the particle is moving down the \( y \)-axis.

The acceleration (rate of change of velocity) is: \( v'(t) = 6t – 8 \) At \( t = 1 \): \( v'(1) = 6(1) – 8 = -2 \) Since \( v'(1) \) is negative, the particle’s velocity is decreasing at \( t = 1 \).

Question

For any function \( f \), which of the following statements must be true?

I If \( f \) is defined at \( x = a \), then \( \lim_{x \to a}f(x) = f(a) \).
II If \( f \) is continuous at \( x = a \), then \( \lim_{x \to a}f(x) = f(a) \).
III If \( f \) is differentiable at \( x = a \), then \( \lim_{x \to a}f(x) = f(a) \).

A) III only
B) I and II only
C) II and III only
D) I, II, and III

▶️ Answer/Explanation

Solution

Correct Answer: C

Analysis:
• Statement I is false (a function can be defined at a point but not continuous there).
• Statement II is true (this is the definition of continuity).
• Statement III is true (differentiability implies continuity).

Therefore, only statements II and III must be true.

Question

The graph of the function f is shown below.

Graph of f

Of the following intervals, on which is f continuous but not differentiable?

A) (0,1)
B) (1,2)
C) (2,3)
D) (3,4)

▶️ Answer/Explanation

Solution

Correct Answer: C

Analysis:
• In interval (2,3), the function transitions from a straight line to a curve in (2,3), forming a corner.
• This corner means the function is not differentiable in (2,3), but the graph is continuous across the interval.
• Therefore, the function is continuous but not differentiable on (2,3).

Question

At \( x = 3 \), the function given by \[ f(x) =\begin{cases} x^2, & x < 3 \\ 6x-9, & x \geq 3 \end{cases} \] is

A) undefined.
B) continuous but not differentiable.
C) differentiable but not continuous.
D) neither continuous nor differentiable.
E) both continuous and differentiable.

▶️ Answer/Explanation

Solution

Correct Answer: E

Continuity:
\[ \lim_{x\rightarrow 3^-}f(x) = 3^2 = 9 \] \[ \lim_{x\rightarrow 3^+}f(x) = 6(3)-9 = 9 \] \[ f(3) = 6(3)-9 = 9 \] Since all three are equal, the function is continuous at \( x = 3 \).

Differentiability:
Left derivative: \( \frac{d}{dx}(x^2) = 2x \Rightarrow 2(3) = 6 \)
Right derivative: \( \frac{d}{dx}(6x-9) = 6 \)
Since both one-sided derivatives equal 6, the function is differentiable at \( x = 3 \).

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