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Area of a rectangle?
$A = lw$
Perimeter of a rectangle?
$P = 2l + 2w$
Volume of a cylinder?
$V = \pi r^2 h$
Surface area of a cylinder?
$A = 2\pi r^2 + 2\pi rh$
How to find critical points?
$f'(x) = 0$
Profit equation?
$P(x) = R(x) - C(x)$
First derivative test for local max?
If $f'(x)$ changes from positive to negative at $x=c$, then $f(c)$ is a local max.
First derivative test for local min?
If $f'(x)$ changes from negative to positive at $x=c$, then $f(c)$ is a local min.
Second derivative test for local max?
If $f'(c) = 0$ and $f''(c) < 0$, then $f(c)$ is a local max.
Second derivative test for local min?
If $f'(c) = 0$ and $f''(c) > 0$, then $f(c)$ is a local min.
Define objective function.
The function to be maximized or minimized in an optimization problem.
What are constraints in optimization?
Limitations or restrictions on the variables in an optimization problem.
Define critical points.
Points where the derivative of a function is zero or undefined.
What is a local maximum?
A point where the function's value is greater than or equal to all nearby points.
What is a local minimum?
A point where the function's value is less than or equal to all nearby points.
Define optimization.
The process of finding the maximum or minimum value of a function.
What is the first derivative test?
A method to determine if a critical point is a local max or min by analyzing the sign change of the first derivative.
What is the second derivative test?
A method to determine if a critical point is a local max or min by evaluating the sign of the second derivative.
What is a closed interval?
An interval that includes its endpoints.
Define endpoints.
The values at the beginning and end of a given interval.
How to maximize area with a fixed perimeter?
1. Define variables. 2. Write area and perimeter equations. 3. Express area in terms of one variable. 4. Find critical points. 5. Verify maximum.
Steps to minimize surface area with a fixed volume?
1. Define variables. 2. Write surface area and volume equations. 3. Express surface area in terms of one variable. 4. Find critical points. 5. Verify minimum.
How to solve a general optimization problem?
1. Identify objective function. 2. Establish constraints. 3. Formulate the optimization equation. 4. Find critical points. 5. Test critical points. 6. Consider endpoints.
How to find the maximum profit?
1. Define revenue and cost functions. 2. Formulate profit function: P(x) = R(x) - C(x). 3. Find critical points of P(x). 4. Verify maximum.
How to minimize the amount of material needed?
1. Define variables. 2. Write the equation for the amount of material. 3. Establish constraints. 4. Find critical points. 5. Test critical points.
How to maximize crop yield?
1. Define crop yield function. 2. Establish constraints. 3. Find critical points. 4. Verify maximum.
How to find the optimal dimensions for a container?
1. Define volume and surface area functions. 2. Establish constraints. 3. Express surface area in terms of one variable. 4. Find critical points. 5. Test critical points.
Steps for solving optimization problems on a closed interval?
1. Find critical points within the interval. 2. Evaluate the objective function at the critical points and endpoints. 3. Compare values to find max/min.
How to handle constraints in optimization?
1. Identify constraint equation. 2. Solve for one variable. 3. Substitute into objective function. 4. Optimize the resulting function.
How to interpret the results of an optimization problem?
1. State the optimal values of the variables. 2. Explain the meaning of the optimal value in the context of the problem. 3. Include units.