6.15 Finding Specific Antiderivatives Using Initial Conditions: Motion Applications

• An antiderivative of a function $f(x)$ is another function $F(x)$ such that $F'(x) =$$f(x)$. Integration is the process of finding the antiderivative
• What is the antiderivative of $f(x) =$$2x$?


  $F(x) =$$x^{2}$
• To understand integration, it's helpful to contrast it with differentiation
• Match the operation with its input and output:

  Differentiation ↔️ Function → Derivative

  Integration ↔️ Function → Antiderivative
• Understanding antiderivatives and integration is crucial for solving problems involving accumulation of change in AP Calculus BC.
• Initial conditions are values of the function or its derivatives at a specific point, usually $t =$$0$. They help determine the specific antiderivative by finding the value of the constant of integration
• If $F'(x) =$$2x$ and $F(0) =$$5$, what is F(x)</latex>?


  $F(x) =$$x^{2} +$$5$
• Integration is the process of finding the antiderivative
• What is the antiderivative of $f(x) =$$2x$?


  $F(x) =$$x^{2}$
• Understanding antiderivatives is essential for solving problems involving accumulation of change in AP Calculus BC.
• Initial conditions help determine the specific antiderivative by finding the value of the constant of integration
• What is the specific antiderivative of $F'(x) =$$2x$ if F(0) = 5</latex>?


  $F(x) =$$x^{2} +$$5$
• Steps to apply initial conditions to find specific antiderivatives

  1️⃣ Find the general antiderivative $F(x)$ of the given function $f(x)$.

  2️⃣ Use the initial condition to solve for the constant of integration, $C$.

  3️⃣ Substitute the value of $C$ back into the general antiderivative to obtain the specific antiderivative.
• Given f'(x) = 3x^{2}</latex> and $f(1) =$$4$, what is $f(x)$?


  $f(x) =$$x^{3} +$$3$
• An antiderivative of a function $f(x)$ is another function $F(x)$ such that $F'(x) =$$f(x)$ and integration is the process of finding the antiderivative
• Understanding antiderivatives and integration is crucial for solving problems involving accumulation of change in AP Calculus BC.
• Initial conditions are typically given at $t =$$0$ and help determine the specific antiderivative by solving for the constant of integration
• What is the specific antiderivative of F'(x) =2x</latex> if $F(0) =$$5$?


  $F(x) =$$x^{2} +$$5$
• Steps to apply initial conditions to find specific antiderivatives

  1️⃣ Find the general antiderivative $F(x)$ of the given function $f(x)$.

  2️⃣ Use the initial condition to solve for the constant of integration, $C$.

  3️⃣ Substitute the value of $C$ back into the general antiderivative to obtain the specific antiderivative.
• Given $f'(x) =$$3x^{2}$ and $f(1) =$$4$, what is $f(x)$?


  $f(x) =$$x^{3} +$$3$
• In motion problems, velocity is the derivative of position
• Integrating acceleration yields velocity in motion problems.
• If a particle's acceleration is $a(t) =$$6t$ and initial velocity is $v(0) =$$5$, what is the velocity function $v(t)$?


  $v(t) =$$3t^{2} +$$5$
• To find the position function $s(t)$, you need to integrate the velocity function.
• In motion problems, position, velocity, and acceleration are related through integration and differentiation.
• How are velocity and acceleration related to position in motion problems?

  Through integration
• What do you obtain by integrating acceleration?

  Velocity
• Integrating velocity yields position.
• A particle's acceleration isa(t) = 6t \, m / s^{2}</latex>. What is the initial velocity if $v(0) =$$5 \, m / s$?


  5 \, m / s
• Integrating $a(t) =$$6t$ results in $v(t) =$$3t^{2} +$$C$, where $C$ is the initial velocity
• What is the velocity function $v(t)$ for a particle with acceleration $a(t) =$$6t \, m / s^{2}$ and initial velocity $v(0) =$$5 \, m / s$?


  $v(t) =$$3t^{2} +$$5 \, m / s$
• The position function $s(t)$ for a particle with velocity $v(t) =$$3t^{2} +$$5 \, m / s$ is $s(t) =$$t^{3} +$$5t +$$D$, where $D$ is the initial position.
• The second derivative of position, $s''(t)$, is equal to acceleration
• Match the quantity with its definition:

  Acceleration ↔️ Rate of change of velocity

  Velocity ↔️ Rate of change of position

  Position ↔️ Location of the object
• What are initial conditions used for in motion problems?

  Determining the constant of integration
• Substituting the value of $C$ into the general antiderivative yields the specific antiderivative.
• The position function for a particle with $v(t) =$$2t^{2} - 6t +$$C_{1}$ and initial condition $s(0) =$$5 \, m$ is $s(t) =$$\frac{2}{3}t^{3} - 3t^{2} +$$C_{1}t +$$5 \, m$.
• What is the velocity function $v(t)$ for a particle with acceleration $a(t) =$$4t - 6 \, m / s^{2}$ and initial velocity $v(0) =$$3 \, m / s$?


  $v(t) =$$2t^{2} - 6t +$$3 \, m / s$
• What is the second step in solving problems involving specific antiderivatives and motion?

  Use initial conditions