Question

The derivative of a parabolic function becomes

• A. unit-impulse function
• B. ramp function
• C. gate function
• D. triangular function

Hint:

Keep this derivative chain in mind:
Parabola ($t^2/2$) $\xrightarrow{d/dt}$ Ramp ($t$) $\xrightarrow{d/dt}$ Step ($1$) $\xrightarrow{d/dt}$ Impulse ($\delta(t)$).

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
This question asks for the resulting mathematical function when we take the derivative of a parabolic function.

Step 2: Key Formula or Approach:
Let us define the standard signal functions used in system analysis:
- Parabolic function: $p(t) = \frac{t^2}{2} u(t)$
- Ramp function: $r(t) = t \cdot u(t)$
- Step function: $u(t)$
- Impulse function: $\delta(t)$

Step 3: Detailed Explanation:

• The derivative of the parabolic function with respect to time is:
\[ \frac{d}{dt} [p(t)] = \frac{d}{dt} \left[ \frac{t^2}{2} u(t) \right] \]

• For $t > 0$:
\[ \frac{d}{dt} \left( \frac{t^2}{2} \right) = t \]

• This is the expression for a ramp function $r(t) = t \cdot u(t)$.

• Further derivatives yield:
- Derivative of Ramp $\rightarrow$ Step function.
- Derivative of Step $\rightarrow$ Impulse function.

Step 4: Final Answer:
The derivative of a parabolic function is a ramp function, which corresponds to Option (B).