Tutor profile: Jenna F.
what is an asymptote?
an asymptote is best explained as a line that represents an infinite problem in a function. The best example is to think about a water bottle and a group of people stranded in the desert. If there are only 2 stranded people, then each person can get one half of the bottle, or 1 out of 2 equal parts of the bottle. However, if there are 50 stranded people, then each person would get one-fiftieth of the bottle. That is such a small amount of water, but for people stranded in a desert, it is still something. so as this number of people increases, the amount each one of them gets becomes infinity smaller until you reach a point where the amount each person gets, although nothing, is basically nothing. This is what an asymptote represents. it shows an infinite issue in an equation, where the equation goes towards that value forever, but never really actually reaches it.
Sally is playing outside in the evening. She notices that her shadow is about 10 feet away. If Sally is 5 feet tall, what is the measure of the angle created by her shadow and the distance between her shadow and her head?
We will assume that sally is standing perfectly straight. This will allow us to use a triangle to represent this situation. The angle in question would be the angle opposite the sidelength of her height, and the angle adjacent to the sidelength of her shadow. This means we can set Tan θ ( θ representing the angle we care about) equal to the opposite sidelength divided by the adjacent sidelength. and to solve for the angle, we can find the arctan(5/10) which would equal about 26.56505118, or approximately 27 degrees.
What are the steps to solve a related rates question?
1. Draw a picture based on what they tell you Drawing a picture is very beneficial to understand what the question is trying to ask you. Creating a picture while reading the question helps you take words on a screen/piece of paper and imagine the real-life situation they are asking you about. 2. Organize the given information I tend to organize it using a little homemade 2x3 table, but whatever works best is all that is needed. Given information is not only the things they tell you, but the things they are asking for. It is very important to identify what the question is asking you to find. If you cant see the end of the path before you start, it becomes way harder to get there. 3. Find an equation that represents a relationship related to the problem. This can be anything from figuring out the radius and height of a cone is related, to using the Pythagorean theorem. If the question is asking you about the surface area, you need to write an equation that represents the surface area using the given information. 4. Take the derivative of the equation. I include this step because students usually get here and then forget what they need to do next. This is a calculus problem after all, and a lot of the time if you can not figure out what your next step is when you have an equation already, its time to take the derivative. 5. Using the derivative as a tool, see if you can plug things in to find the information you don't already know. doing this will help if you're not sure what order to do things. If you take the derivative of the equation you found, and you discover that to find what the question is asking for you need the rate of a different variable, you can now go do that and are one step closer to finding what you want. 6. Repeat steps 3-5 until you can solve for the information the question is asking for. Depending on the problem, you might only need to do steps 3-5 once, but more than likely you will have to find multiple derivatives to solve the question Done! Don't forget units :)