Article: In calculus, the derivative of any function is used to find the rate of change of that function. The maximum value of a given function occurs when the derivative equals zero. So, to maximize the revenue, find the first derivative of the revenue function.  Suppose the revenue function, in terms of number of units sold, is R(q)=500q−150q2{\displaystyle R(q)=500q-{\frac {1}{50}}q^{2}}. The first derivative, therefore, is: R′(q)=500−250q{\displaystyle R^{\prime }(q)=500-{\frac {2}{50}}q}  For a review of derivatives, see the wikiHow article on how to Take Derivatives. When the derivative is zero, the graph of the original function is at either a peak or a trough. This will be either the maximum or minimum value. For some higher level functions, there may be more than one solution to the zero derivative, but not a basic price-demand function.  R′(q)=500−250q{\displaystyle R^{\prime }(q)=500-{\frac {2}{50}}q} 0=500−250q{\displaystyle 0=500-{\frac {2}{50}}q} Use basic algebra to solve the derivative for the number of items to sell where the derivative is equal to zero. This will give you the number of items that will maximize the revenue.  0=500−250q{\displaystyle 0=500-{\frac {2}{50}}q} 250q=500{\displaystyle {\frac {2}{50}}q=500} 150q=250{\displaystyle {\frac {1}{50}}q=250} q=50∗250{\displaystyle q=50*250} q=12,500{\displaystyle q=12,500} Using the optimal number of sales from the derivative calculation, you can enter that value into the original price formula to find the optimal price.  p=500−150q{\displaystyle p=500-{\frac {1}{50}}q} p=500−15012,500{\displaystyle p=500-{\frac {1}{50}}12,500} p=500−250{\displaystyle p=500-250} p=250{\displaystyle p=250} After you have found the optimal number of sales and the optimal price, multiply them to find the maximum revenue. Recall that R=p∗q{\displaystyle R=p*q}. The maximum revenue for this example, therefore, is:  R=p∗q{\displaystyle R=p*q} R=(250)(12,500){\displaystyle R=(250)(12,500)} R=3,125,000{\displaystyle R=3,125,000} Based on these calculations, the optimal number of units to sell is 12,500, at the optimal price of $250 each. This will result in a maximum revenue, for this sample problem, of $3,125,000.
Question: What is a summary of what this article is about?
Find the first derivative of the revenue function. Set the derivative equal to 0. Solve for the number of items at the 0 value. Calculate the maximum price. Combine the results to calculate maximum revenue. Summarize the results.
Article: Washing your pet will kill the fleas that may be living on it. Prepare your pet for the bath by applying the flea soap near the pet’s ears, eyes, nose, mouth, and butt. When fleas feel water, they immediately look for safe places on the pet to hide out, such as around the neck, on the head, and near the anus. Do this before you get the pet wet. There may be a host of other fleas in various stages of their life cycle waiting to hop onto your pet, so be sure to use this method with other preventative measures, such as vacuuming, borate mixes, and cedar chips. and let the solution stand for at least 10 minutes. After you've applied the soap to the pet’s vulnerable areas, get the pet wet and lather its whole body up. Once the 10 minutes have passed, wash all of the soap out of the pet’s fur with warm water. Then, once the pet is dry, comb out the dead fleas. Take a shower immediately after washing your pet because some flea shampoos can cause irritation when left on human skin for prolonged periods of time. Flea products, such as Advantage or Frontline, are spot-treatments that are applied to pets monthly. Usually, these spot-treatments are dropped directly onto the skin of your pet, usually between the shoulder blades and at several spots down the spine. The treatment will then work its way through the skin (integumentary system). You can purchase pet-friendly flea products at your local pet store. You can use flea products in conjunction with a fine-toothed flea comb. Use a flea comb to pull off individual fleas from the pet's fur. This will prevent flea adaption to the treatments. You may find that some of the standard flea treatments are ineffective because the local flea population has adapted to a particular treatment (including the chemicals in flea collars). In this case, switch among other treatments and use an insect growth regulator (IGR) such as Nylar. For example, if a flea collar isn’t working, try scrubbing your pet with flea powder.  Your veterinarian will sell oral tablets that you can give your pet. These tablets attack the flea’s nervous system via the bloodstream and tissue of your dog or cat. While they kill all the fleas feeding on your pet, they do not prevent fleas. The use of an IGR will not allow propagation of succeeding generations and will prevent, in large measure, fleas adapting to the treatments. Flea tablets are given orally and take effect within several hours. Flea tablets kill off all of the fleas on an animal within about 60 minutes. However, the tablets are not preventative and will not make your pet immune to becoming infested with fleas a second time. Once the tablet has killed off the fleas, wash the animal to rid it of the dying fleas if you see the animal scratching. Some cats may become hyperactive, vocalize more, or show panting when given flea tablets. If this happens, try cutting the recommended dose, but you may have to discontinue use of tablets with sensitive cats. Since fleas feed on the blood of animals, you can treat fleas by manipulating what your pet eats. For example, mix vinegar into dogs' water. Mix a tablespoon of apple-cider vinegar into the dog's water bowl, or bathe the animal using water and vinegar. Don't try this with cats, as their pH is much more sensitive than dogs'.
Question: What is a summary of what this article is about?
Wash your pet every day with flea soap and water. Work in the soap Use a flea product on your cat or dog. Rotate types of treatments if the first is ineffective. Administer a flea tablet orally to rid your pet of a flea infestation quickly. Add deterrents to the pet's meal.