Quiz on interpreting the derivative: Instantaneous rate of change assessment

📚 Quick Study Guide

• ⏱️ The instantaneous rate of change represents the rate at which a function's output changes at a specific point.
• 📈 It's equivalent to the derivative of the function at that point.
• 📐 Geometrically, it's the slope of the tangent line to the function's graph at that point.
• 🧮 The derivative of a function $f(x)$ is denoted as $f'(x)$ or $\frac{dy}{dx}$.
• 📝 To find the instantaneous rate of change at $x=a$, calculate $f'(a)$.
• 💡 Common derivative rules include: Power Rule ($\frac{d}{dx}x^n = nx^{n-1}$), Constant Multiple Rule, Sum/Difference Rule.
• 🔍 Remember, differentiability implies continuity, but continuity doesn't imply differentiability.

Practice Quiz

1. Question 1: The position of a particle is given by $s(t) = 3t^2 - 4t + 1$. What is the instantaneous velocity at $t = 2$?
   1. A) 8
   2. B) 12
   3. C) 10
   4. D) 4
2. Question 2: Find the instantaneous rate of change of $f(x) = x^3 - 2x + 5$ at $x = 1$.
   1. A) 1
   2. B) 2
   3. C) 3
   4. D) 0
3. Question 3: A curve is defined by $y = \sqrt{x}$. What is the slope of the tangent line at $x = 4$?
   1. A) $\frac{1}{2}$
   2. B) $\frac{1}{4}$
   3. C) 1
   4. D) 2
4. Question 4: If $f(x) = \frac{1}{x}$, what is the instantaneous rate of change at $x = 2$?
   1. A) $\frac{1}{4}$
   2. B) $-\frac{1}{4}$
   3. C) $\frac{1}{2}$
   4. D) $-\frac{1}{2}$
5. Question 5: The temperature of a metal rod is given by $T(t) = 5t^2 + 2t$, where $t$ is in seconds. What is the rate of change of temperature at $t = 3$ seconds?
   1. A) 30
   2. B) 32
   3. C) 17
   4. D) 15
6. Question 6: Given $y = \sin(x)$, find the instantaneous rate of change at $x = 0$.
   1. A) 0
   2. B) 1
   3. C) -1
   4. D) $\frac{1}{2}$
7. Question 7: The volume of a sphere is $V(r) = \frac{4}{3}\pi r^3$. Find the rate of change of volume with respect to the radius when $r = 2$.
   1. A) $16\pi$
   2. B) $4\pi$
   3. C) $8\pi$
   4. D) $32\pi$