6.15 Finding Specific Antiderivatives Using Initial Conditions: Motion Applications

• What are initial conditions used for in motion problems?

  Determining specific antiderivatives
• What does $v(t_{0}) =$$v_{0}$ represent in motion problems?


  Initial velocity
• What is the constant of integration in a general antiderivative?

  An arbitrary constant
• What is the purpose of initial conditions in motion problems?

  Solving for specific antiderivatives
• <question_start>The initial condition for position is represented as $s(t_{0}) =$$s_{0}$, where $s_{0}$ is the initial position
• What is the arbitrary constant added to a general antiderivative called?

  Constant of integration
• <question_start>The general antiderivative of $\cos(x)$ is $\sin(x) +$ C
• \int f(x) dx = F(x) + C
• What does the general antiderivative represent?

  All possible functions
• The general antiderivative provides the set of all possible functions that could have been differentiated to obtain the original function.


  True
• What does s(t_{0}) = s_{0}</latex> represent in motion problems?

  Initial position
• Substituting the initial condition into the general antiderivative allows us to determine the unique function.

  True
• In motion problems, $t_{0}$ represents the initial time
• Match the variable with its description in motion problems:

  $s_{0}$ ↔️ Initial position

  $v_{0}$ ↔️ Initial velocity
• The general antiderivative of $x^{2}$ is \frac{x^{3}}{3} + C
• What is the constant $C$ called in indefinite integration?


  Constant of integration
• Steps to find the specific antiderivative using initial conditions

  1️⃣ Start with the general antiderivative

  2️⃣ Solve for the constant of integration $C$
• What role do initial conditions play in motion applications of integration?

  Solve for the constant $C$
• $s(0) =$$0^{2} +$$C \implies C =$$10$, which means $s(0) =$$10$ is the initial position
• The general antiderivative of $v(t)$ is expressed as $\int v(t) dt =$$F(t) +$$C$, where C represents the constant of integration
• What is the value of C if the velocity function is $v(t) =$$2t$ and $s(0) =$$10$?


  10
• Match the variable with its mathematical representation:

  Position ↔️ $\int v(t) dt =$$F(t) +$$C$

  Initial condition ↔️ $s(t_{0}) =$$s_{0}$

  Velocity ↔️ $v(t)$
• What is the value of C if the general antiderivative for position is $s(t) =$$\frac{1}{2}t^{2} +$$C$ and the initial condition is $s(0) =$$5$?


  5
• The specific antiderivative for $s(t) =$$\frac{1}{2}t^{2} +$$C$ with s(0) = 5</latex> is $s(t) =$$\frac{1}{2}t^{2} +$$5$, where C is equal to 5
• In the initial condition s(t_{0}) = s_{0}</latex>, $t_{0}$ represents the initial time
• What does an initial condition provide in motion problems?

  Known value of position
• <statement_start>Initial conditions such as $v(t_{0}) =$$v_{0}$ help find the specific antiderivative
• Match the term with its description:

  General antiderivative ↔️ Function whose derivative is $f(x)$

  Constant of integration ↔️ Arbitrary constant added to antiderivative

  Indefinite integral ↔️ Process to find antiderivatives
• What is the general antiderivative of $\cos(x)$?


  $\sin(x) +$$C$
• Steps to find the specific antiderivative using initial conditions:

  1️⃣ Find the general antiderivative of $v(t)$ or $a(t)$

  2️⃣ Substitute the initial condition into the general antiderivative

  3️⃣ Solve for the constant of integration $C$

  4️⃣ Replace $C$ in the general antiderivative to get the specific antiderivative
• <statement_start>Initial conditions provide known values of position, velocity, or acceleration at a particular time
• What is the specific position function for $v(t) =$$2t$ and s(0) = 3</latex>?


  $s(t) =$$t^{2} +$$3$
• <statement_start>To find the specific position function, we use the initial position to solve for the constant of integration
• What is the position function for $v(t) =$$2t$ and $s(0) =$$5$?


  $s(t) =$$t^{2} +$$5$
• What is the general antiderivative of $v(t) =$$2t$?


  $s(t) =$$t^{2} +$$C$
• What is the value of $C$ in the specific antiderivative s(t) = t^{2} + C</latex> given $s(0) =$$5$?


  $C =$$5$
• An initial condition allows you to find the constant of integration
• What initial condition is used to find velocity given acceleration in motion problems?

  $v(t_{0}) =$$v_{0}$
• What is the general antiderivative of $v(t) =$$2t$?


  $s(t) =$$t^{2} +$$C$
• What is the specific antiderivative of $v(t) =$$2t$ with s(0) = 3</latex>?


  $s(t) =$$t^{2} +$$3$