5.7 Solving Optimization Problems

• What is the process of optimization in calculus?

  Finding maximum or minimum values
• Critical points are locations where the derivative is either zero or undefined
• Endpoints are the boundaries of the interval being considered in optimization problems.
• Steps to find the maximum or minimum value of a function

  1️⃣ Find the derivative of the function

  2️⃣ Set the derivative equal to zero and solve for critical points

  3️⃣ Evaluate the original function at critical points and endpoints

  4️⃣ Determine which point gives the maximum or minimum value
• What is the example given in the study material for optimization?

  Maximizing the area of a garden
• The perimeter of the rectangular garden is given by P = 2l + 2w = 100</latex>
• The width of the rectangular garden can be expressed as $w =$$50 - l$.
• What does the objective function represent in an optimization problem?

  The quantity to maximize or minimize
• The first step in identifying the objective function is to read the problem carefully.
• In the example of maximizing the volume of a rectangular prism, the objective function is $V =$$lwh$
• What does the constraint function do in an optimization problem?

  Restricts the values of variables
• The constraint function helps reduce the objective function to a single variable.
• The constraint function is a mathematical expression that restricts the values of variables
• What is the constraint function for a rectangular garden with a perimeter of 100 meters?

  2l + 2w = 100</latex>
• The constraint function simplifies the optimization process by reducing the objective function to a single variable.
• Optimization in calculus is used to find the maximum or minimum values of a function
• What are critical points in optimization?

  Derivative is zero or undefined
• To find the maximum or minimum value, you must evaluate the function at critical points and endpoints.
• Steps to find the maximum or minimum value of a function using optimization

  1️⃣ Find the derivative of the function

  2️⃣ Set the derivative equal to zero and solve

  3️⃣ Evaluate the function at critical points

  4️⃣ Evaluate the function at endpoints

  5️⃣ Determine which point yields the maximum or minimum value
• What is the formula for the area of a rectangle?

  $A =$$lw$
• The maximum area of a rectangular garden with a perimeter of 100 meters is 625
• What is an endpoint in the context of optimization?

  Boundary of the interval
• To find critical points, you must set the derivative of the function equal to zero.
• What is the purpose of the objective function in an optimization problem?

  Maximize or minimize a quantity
• Steps to identify and express the objective function

  1️⃣ Read the problem carefully

  2️⃣ Identify variables that influence the quantity

  3️⃣ Formulate an equation for the quantity
• The objective function for maximizing the volume of a rectangular prism is $V =$$lwh$
• What is the role of the constraint function in optimization?

  Restricts the values of variables
• How do you express the objective function in terms of a single variable?

  Use the constraint function
• Steps to express the objective function in terms of a single variable

  1️⃣ Solve the constraint function for one variable

  2️⃣ Substitute the solved expression into the objective function
• The reduced objective function for maximizing the area of a rectangular garden with a perimeter of 100 meters is $A =$$50l - l^{2}$
• Critical points of an objective function are locations where the derivative is zero or undefined.
• What are the steps to find critical points of an objective function?

  Find the derivative, set it to zero
• The critical point for the objective function $A =$$50l - l^{2}$ occurs at $l =$$25$, where the derivative is zero
• What is the perimeter of a rectangle with length $l$ and width $w$?


  $P =$$2l +$$2w$
• The area of a rectangle is given by $A =$$lw$
• If P = 100</latex>, then $w$ can be expressed as $w =$$50 -$l
• What is the area of the rectangle when expressed in terms of $l$?


  $A =$$50l - l^{2}$
• The derivative of $A =$$50l - l^{2}$ is $A' =$$50 - 2l$
• Setting $A' =$$0$ in $50 - 2l =$$0$ gives $l =$25
• What is the value of $w$ when $l =$$25$?


  w = 25</latex>