Analyzing motion along a line using derivatives:

• What is acceleration defined as in relation to velocity?

  The derivative of velocity
• What does the instantaneous rate of change represent graphically?

  Slope of the tangent line
• The instantaneous rate of change is the limit of the average rate of change as the interval approaches infinity.

  False
• What is the definition of position in motion analysis?

  The location of an object
• What is the mathematical expression for velocity in terms of derivatives?

  \frac{d}{dt}(x(t)) = v(t)</latex>
• The velocity at any time is the slope of the tangent line to the position function at that time


  True
• What is the formula for the average rate of change of a function $f(x)$ over the interval [a, b]?


  $\frac{f(b) - f(a)}{b - a}$
• Velocity is the first derivative of position with respect to time
• The velocity at any given time is the slope of the tangent line to the position function at that time

  True
• Acceleration is the rate of change of velocity over time.
• What does the formula $\lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$ represent?


  Slope of the tangent line
• The average rate of change is the slope of the secant
• Velocity is the rate of change of position
• The velocity at any given time is the slope of the tangent
• Acceleration is the rate of change of velocity
• What does the slope of the tangent line to the velocity function represent?

  Instantaneous acceleration
• Match the sign of velocity and acceleration with their meaning:

  Positive velocity ↔️ Object moves in the positive direction

  Negative acceleration ↔️ Object slows down

  Positive acceleration ↔️ Object speeds up

  Negative velocity ↔️ Object moves in the negative direction
• The acceleration of an object is the second derivative of its position
• What is the key to finding maximum or minimum values of velocity or position using derivatives?

  Analyze the signs of derivatives
• Velocity is the derivative of position
• The average rate of change formula is $\frac{f(b) - f(a)}{b - a}$
• What is the formula for acceleration in terms of derivatives?

  $\frac{d^{2}}{dt^{2}}(x(t)) =$$a(t)$
• What is the second derivative of position with respect to time?

  Acceleration
• Acceleration is the second derivative of position with respect to time

  True
• If the position function is $x(t) =$$3t^{2} +$$2t +$$1$, what is the velocity at time t?


  $v(t) =$$6t +$$2$
• In calculus, a rate of change describes how one quantity changes in relation to another
• The instantaneous rate of change is the limit of the average rate of change as the interval approaches zero


  True
• What is the mathematical expression for velocity as the derivative of position?

  $\frac{d}{dt}(x(t)) =$$v(t)$
• Velocity is the rate of change of an object's position over time.
• Match the concept with its definition and relationship to derivatives:

  Position ↔️ Location of an object at a given time; Starting point for analysis

  Velocity ↔️ Rate of change of position over time; First derivative of position

  Acceleration ↔️ Rate of change of velocity over time; Second derivative of position
• Match the type of rate of change with its formula:

  Average Rate ↔️ $\frac{f(b) - f(a)}{b - a}$

  Instantaneous Rate ↔️ $\lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$
• What are the three key concepts for analyzing motion along a line?

  Position, velocity, acceleration
• What is the mathematical expression for velocity in terms of position?

  $\frac{d}{dt}(x(t)) =$$v(t)$
• The derivative of the velocity function gives the acceleration.


  True
• Match the sign of velocity and acceleration with its effect on motion:

  Positive Velocity ↔️ Moving forward

  Negative Velocity ↔️ Moving backward

  Positive Acceleration ↔️ Increasing velocity

  Negative Acceleration ↔️ Decreasing velocity
• A positive velocity indicates movement in the negative direction.

  False
• What does a positive second derivative indicate about the object's motion?

  The object is speeding up
• Match the motion with the sign of velocity and acceleration:

  Positive velocity, positive acceleration ↔️ Object speeds up in the positive direction

  Negative velocity, negative acceleration ↔️ Object speeds up in the negative direction

  Positive velocity, negative acceleration ↔️ Object slows down in the positive direction

  Negative velocity, positive acceleration ↔️ Object slows down in the negative direction
• What are the three key concepts in the study of motion along a line?

  Position, velocity, and acceleration
• Analyzing the signs of velocity and acceleration helps determine the direction of motion.


  True