The equation for the volume of a cube is: ![]() Reading the problem, we see that we want to maximize the volume, but solve for the height of the box. Step 1: Identify the equation we want to maximize. What height will produce a box with the maximum possible volume? We want to create a box with an open top and square base with a surface area of 300 square inches. Reading as many examples as you can and becoming more acquainted with the structure of these problems will help you get better at interpreting them. In each example, pay attention to the precise wording of the problem. Let’s work through several examples of optimization problems in order to gain a better understanding of the concept. Always do this first before solving any problem. The best way to prevent this confusion is to read the problem very carefully, draw picture representations of whatever you are trying to optimize, and label your equation and your constraint. These problems become difficult in AP® Calculus because students can become confused about which equation we are trying to optimize and which equation represents the constraint. A constraint can be an equation, and a constraint is always true in the concept of the problem. The types of optimization problems that we will be covering in this article involve something called a constraint. These are just some common, simple examples. We could be optimizing volume, area, distance, length, and many other quantities. There are many different types of optimization problems. Absolute extrema can be within the function or they can be at the ends of the interval we are searching for the extrema on. Absolute extrema are the overall maximum values or the overall minimum values. Local extrema are the peaks and troughs in an equation. We can have absolute extrema and local extrema. Extrema are the maximum or minimum values. Let’s get started.įirst, what is optimization? Optimization is when we are looking for the extrema of a function. ![]() Together, we will beat all of your fears and confusion. Reading this article will give you all the tools you need to solve optimization problems, including some examples that I will walk you through. Many AP® Calculus students struggle with optimization problems because they require a bit more critical thinking than a normal problem. How fast is the volume increasing when the radius is 10 cm?įirst we compute change in r, and then computer the equivalent change in V.One of the most challenging aspects of calculus is optimization. Solve for wanted rate of change or quantityĪ spherical balloon’s area is increasing at the constant rate of 5 cm/sec. Substitute in the known rates quantitiesĥ. We differentiate both sides with respect to time.Ĥ. How fast is the radius of the balloon changing when the radius is 10 cm?ģ. Volume questions are quite common examples of Related Rate Problems.Ī spherical balloon is being inflated so that its volume increases at a rate of 20 cm3/s. We know also know y from the original Pythagorean theorem. We know dx/dt = r, dl/dt = 0, and x = bī. I will leave it for a few steps for purpose of demonstrationĤ. We should automatically see at this point that since l is constant, dl/dt is 0. The moment we see a right triangle, we use the Pythagorean theorem Drawing this problem makes it easy to visualize.ī. What is the rate that the top of the ladder moves while sliding down the building when the base of the ladder is b meters from the building?Ī. It’s a bit off balance, and so beings to slide away from the building at a rate of r m/s. ![]() We have simpler sample problems following.)Ī ladder with a length of 1 meter is leaned up against a building. (Note: this is a more difficult than normal problems in its set up. We always plug in known values of variables after finding the derivative, never before finding the derivative. Solve for wanted rate of change or quantity Common Errors: Substitute in the known rates of change and/or known quantitiesĥ. Therefore, we differentiate both sides with respect to time.Ĥ. Rates are usually (for AP Calculus) in relation to time. This is often given in the problem, or is a relatively well-known relation (i.e., volume = length × width)ģ. Find the governing equation which relates the variables. This could be size, volume, distance, etc.Ģ. We must first identify the variables which are changing in the problem. No two problems are exactly the same, but these steps are a very good rubric for solving a wide variety of problems:ġ. There is a series of steps that generally point us in the direction of a solution to related rates problems. ![]() They come up on many AP Calculus tests and are an extremely common use of calculus. Related Rates problems are any problems where we are relating the rates of two (or more) variables.
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