14.2.2 Understanding calculus applications

• The power rule states that the derivative of $x^{n}$ is nx^{n - 1}
• The derivative $\frac{dx}{dt}$ represents the rate of change of position with respect to time
• What is the derivative of $\cos(x)$?


  $- \sin(x)$
• Understanding the physical interpretation of derivatives is essential for applying calculus in physics.


  True
• What is the physical meaning of \frac{dF}{dx}</latex>?

  Force
• The derivative of $\cos(x)$ is $- \sin(x)$
• Steps to describe the relationship between integrals and derivatives:

  1️⃣ Integrals are inverse operations of derivatives

  2️⃣ The integral of velocity gives position

  3️⃣ The derivative of position gives velocity
• What does the derivative of position represent in physics?

  Velocity
• Derivatives in physics represent the rate of change of one quantity with respect to another


  True
• Match the type of integral with its physical meaning:

  $\int v\,dt$ ↔️ Position

  $\int a\,dt$ ↔️ Displacement

  $\int F\,dx$ ↔️ Work
• How do you calculate the area under the curve $y =$$x^{2}$ between the limits $a$ and $b$?


  $\int_{a}^{b} x^{2} \, dx$
• What does the derivative in physics represent?

  Rate of change
• What is the derivative of $e^{x}$?


  $e^{x}$
• What physical quantity does the derivative $\frac{dv}{dt}$ represent?


  Acceleration
• Derivatives in physics represent the rate of change of a quantity with respect to another.
• What does a derivative in physics represent?

  Rate of change
• The power rule states that the derivative of $x^{n}$ is $nx^{n - 1}$
• Match the derivative with its physical meaning:

  $\frac{dx}{dt}$ ↔️ Velocity

  $\frac{dv}{dt}$ ↔️ Acceleration

  $\frac{dF}{dx}$ ↔️ Force
• Match the derivative with its physical meaning:

  \frac{dx}{dt}</latex> ↔️ Velocity

  $\frac{dv}{dt}$ ↔️ Acceleration

  $\frac{dF}{dx}$ ↔️ Force
• The power rule states that the derivative of x^{n}</latex> is $nx^{n - 1}$


  True
• Match the type of derivative with its physical meaning:

  $\frac{dx}{dt}$ ↔️ Velocity

  $\frac{dv}{dt}$ ↔️ Acceleration

  $\frac{dF}{dx}$ ↔️ Force
• Steps to integrate common functions

  1️⃣ Identify the function

  2️⃣ Apply the integration rule

  3️⃣ Add the constant of integration $C$
• Match the type of derivative with its physical meaning:

  $\frac{dx}{dt}$ ↔️ Velocity

  $\frac{dv}{dt}$ ↔️ Acceleration

  $\frac{dF}{dx}$ ↔️ Force
• What is the derivative of \sin(x)</latex>?

  $\cos(x)$
• Order the following derivatives based on their physical meanings from lowest to highest order:

  1️⃣ $\frac{dx}{dt}$

  2️⃣ $\frac{dv}{dt}$

  3️⃣ $\frac{dF}{dx}$
• Match the type of derivative with its physical meaning:

  $\frac{dx}{dt}$ ↔️ Velocity

  $\frac{dv}{dt}$ ↔️ Acceleration

  $\frac{dF}{dx}$ ↔️ Force
• \frac{dx}{dt}</latex> represents the physical quantity called velocity
• The derivative of $e^{x}$ is $e^{x}$.


  True
• What does an integral in physics represent?

  Total change
• Why are derivatives important in physics?

  Analyze dynamic systems
• Match the function with its derivative:

  $x^{n}$ ↔️ $nx^{n - 1}$

  $e^{x}$ ↔️ $e^{x}$

  $\sin(x)$ ↔️ $\cos(x)$
• If position $x =$$t^{3}$, then velocity $v =$$\frac{dx}{dt} =$$3t^{2}$, illustrating the application of the power rule
• The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1} +$$C$, where $C$ represents the constant of integration
• Derivatives are used in physics to model and analyze static systems.

  False
• The derivative of $x^{5}$ is $5x^{4}$, representing the rate of change of position.


  True
• The derivative of velocity with respect to time is acceleration.


  True
• Order the following derivatives based on their physical meanings from lowest to highest order:

  1️⃣ $\frac{dx}{dt}$

  2️⃣ $\frac{dv}{dt}$

  3️⃣ $\frac{dF}{dx}$
• Acceleration is the rate of change of velocity with respect to time.

  True
• What is the derivative of $\sin(x)$?


  $\cos(x)$
• $\int F\,dx$ represents the total work done by a force