Problem: Write an article based on this summary: 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.

Answer: 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)=500q150q2{\displaystyle R(q)=500q-{
rac {1}{50}}q^{2}}. The first derivative, therefore, is: R(q)=500250q{\displaystyle R^{\prime }(q)=500-{
rac {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)=500250q{\displaystyle R^{\prime }(q)=500-{
rac {2}{50}}q} 0=500250q{\displaystyle 0=500-{
rac {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=500250q{\displaystyle 0=500-{
rac {2}{50}}q} 250q=500{\displaystyle {
rac {2}{50}}q=500} 150q=250{\displaystyle {
rac {1}{50}}q=250} q=50250{\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=500150q{\displaystyle p=500-{
rac {1}{50}}q} p=50015012,500{\displaystyle p=500-{
rac {1}{50}}12,500} p=500250{\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=pq{\displaystyle R=p*q}. The maximum revenue for this example, therefore, is: R=pq{\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.

Problem: Write an article based on this summary: Give your peony the light it loves. Use a grow light in addition to natural light. Water your peony. Feed your peony with a houseplant fertilizer. Prepare your plant for its resting period. Prune the stems and move the plant to a cool, dark locations.

Answer: Set the container outdoors in a protected location where the peony will be exposed to at least six to eight hours of direct sunlight. Peonies require lots of light to grow and bloom. If the peony is to be grown indoors, put it in front of a south- or west-facing window where it will get plenty of direct sunlight. A grow light will be necessary to supplement the natural light. Use a four-bulb fluorescent light fixture with two special full-spectrum 40-watt fluorescent bulbs and two 40-watt cool white bulbs. Set up the fixture so that the bulbs are about 6 inches (15.2cm) above the peony and leave it on for 12 to 14 hours each day. The light should be plugged into a timer that turns it on in the morning around sunrise and turns it off at the end of the day. Water the peony when the top inch of potting mix becomes dry. Pour the water evenly over the potting mix until the water drains freely from the bottom of the pot. When the new peony stems appear, begin giving it houseplant fertilizer every four weeks. It is important to use houseplant fertilizer as opposed to fertilizer for garden-grown peonies since it is being grown in a container. A water-soluble fertilizer is best. Always give it the fertilizer after a regular watering. Stop giving fertilizing around mid-summer. In late summer, water the peony less often. Let the soil dry completely before watering it again to encourage the peony to go dormant for the winter. Peonies must have a two- to three-month rest period. If the peony is being grown indoors, slowly reduce the number of hours of supplemental light it receives to coincide with the shorter days of autumn. If the peony is outdoors, leave it out until after a few hard frosts. When the leaves begin to turn yellow and die, use hand pruners to snip the stems off all the way down to the soil. Set the pot in an unheated garage or cool area in the basement. Bring it back out in the spring when the weather warms. Put it outdoors in a sunny spot or in front of the window and water it generously.

Problem: Write an article based on this summary: Check the engine. Look at your owner's manual. Identify the engine number.

Answer: Your vehicle's engine number should be stamped right onto the engine of your vehicle. Pop the hood of your vehicle or look at your motorcycle's engine from the side. You should see a sticker that clearly indicates the engine number. If you cannot find a sticker on your engine that lists your engine number, then check your owner's manual for this information. It should be located within the first few pages of the booklet. Your owner's manual may also include a picture of where to find the engine number on the engine block. The engine number is a six digit number that follows a three digit engine code. You may notice that the engine number includes three digits followed by six more digits. The first three digits are your vehicle's engine code and the last six digits are your vehicle's engine number.

Problem: Write an article based on this summary: Spray the stain with a gentle degreaser. Test your cleaning supplies in an unseen area before using on wood, enamel, or paint. Use an abrasive sponge or brush to remove the stain. Wipe clean with soap and water. Know that long developing stains may not come off at all.

Answer:
In order to fight off the gummy, sticky stains, you'll need something that cuts grease. You can use store-bought products like Goo-Gone, Dr. Bronner's, or Simple Green, or you can make your own. For strong stains, use 1 cup vinegar mixed with 4 cups water. Let the vinegar solution sit on the stain for a few minutes before wiping. You can add 1-2 tablespoons of ammonia for even more powerful cleaning. For smaller, gentler stains, mix 3 tablespoons baking soda with 1 cup water. These surfaces are much more prone to staining and warping if you're not careful. Some cleaners can strip away sealants, coatings, and paints if you're not careful. Always read the label for suggested surfaces to clean with. When in doubt, dab a bit of cleaning solution on a small area and check back 10 minutes later. If it is not stained or warped, you should be good to go. Some cleaners to consider include: Murphy's Oil Soap Dr. Bronner's Barkeep's Friend Warm water and 1-2 tablespoons trisodium phosphate Depending on the stain you can use different abrasives to remove the stain, from gentle sponges on walls to blades on glass and hard plastic. Don't dig into the stain. Simply work at the edges, trying to lift it from the surface, to prevent scratching. For walls, use the rough side of a sponge, or another gentle scrubber. For unpainted metal, a toothbrush or other hearty brush will work wonders. For painted metal, you need to use a gentle towel or microfiber cloth. Use the sharp side of a razor blade to scrape away the gummy residue from glass and plastic. With the blade at a roughly 45-degree angle to the surface, scrape away the loosened nicotine residue. If you're working on glass, use a glass cleaner like Windex to remove the last little bits of residue and wipe the surface clean. Otherwise, a few drops of dish soap and warm water should be fine. For serious stains, repeat the process, letting your degreaser soak into the stain for an extra 5-6 minutes. If the same stain has been building up for years, leading to a yellow, brownish tint to your items, they may never return to their original color. You can make improvements with gentle scrubbing and cleaning, but it may be time to repaint the walls or get newer objects if you can't fight off the yellow tinge.