This week we did story-like problems. These problems were optimization problems. These were pretty confusing at first, but I feel like I quickly grasped the concepts. The steps to follow for optimaztioin problems are:
1. Understand
2. Develop a model: Include pictures
3. Graph the Function
4. Find te critical points and interpret the meaning
5. Solve the model
6. Interpret your soultion
The problems shown in the picture are fairly simple. The 1st one says Find 2 numbers whose sum is 20 and whose product is as large as possible. The first thing I'd do is understand the problem. Once I'm familiar with the wording and what exactly the problem is saying, I woul develop a model. For this one I developed the model a=x and b=20-x so to maximize the value it would be ab. So f(x)=x(20-x). I would distribute the x to get f(x)=20-x^2 and take the derivative to get f'(x)=20-2x. I would graph that and find the maximum which would be 10, so x=10. That would make a=10 and b=20-10, which would be 10 also.
The related rates problems were a bit more difficult than the optimization problems. They are in more terms of a real life situation. The steps to solving related rates problems are:
1. Draw and label a diagram (generally)
2. Find model for situation
3. Take derivative of model
4. Write down everything you know
5. Plug in known values (if you have more than one variable you made a mistake. Go back to step 1)
6. Solve for unknown and state solution in context of problem
One of the problems we did in class was there's a 15 foot ladder that starts with its base 5ft from the wall. The ladder slides away at a rate of 1.75 ft/s. After 3 seconds, how fast is the tip sliding from the wall? The first thing I did was draw a model. My picture was a right triangle, whose hypotenuse was 15 and the horizontal leg being x. Next I developed a model. I chose pathagorean theorm. My model was 15^2=x^2+y^2. I took the derivative of the model and got 0=2x*dx/dt+2y*dy/dt. Then I wrote down everything I know. I knew that x=10.25 because 5+1.75(3), and I knew that dx/dt=1.75 because that was given, and I knew that y=10.95 because that was given also, and I didn't know dy/dt because that's what we are solving for. I would then plug in the numbers I know so it would read 21.0dy/dt=-35.875. I'd solve for dy/dt and get dy/dt=-1.64.
1. Understand
2. Develop a model: Include pictures
3. Graph the Function
4. Find te critical points and interpret the meaning
5. Solve the model
6. Interpret your soultion
The problems shown in the picture are fairly simple. The 1st one says Find 2 numbers whose sum is 20 and whose product is as large as possible. The first thing I'd do is understand the problem. Once I'm familiar with the wording and what exactly the problem is saying, I woul develop a model. For this one I developed the model a=x and b=20-x so to maximize the value it would be ab. So f(x)=x(20-x). I would distribute the x to get f(x)=20-x^2 and take the derivative to get f'(x)=20-2x. I would graph that and find the maximum which would be 10, so x=10. That would make a=10 and b=20-10, which would be 10 also.
The related rates problems were a bit more difficult than the optimization problems. They are in more terms of a real life situation. The steps to solving related rates problems are:
1. Draw and label a diagram (generally)
2. Find model for situation
3. Take derivative of model
4. Write down everything you know
5. Plug in known values (if you have more than one variable you made a mistake. Go back to step 1)
6. Solve for unknown and state solution in context of problem
One of the problems we did in class was there's a 15 foot ladder that starts with its base 5ft from the wall. The ladder slides away at a rate of 1.75 ft/s. After 3 seconds, how fast is the tip sliding from the wall? The first thing I did was draw a model. My picture was a right triangle, whose hypotenuse was 15 and the horizontal leg being x. Next I developed a model. I chose pathagorean theorm. My model was 15^2=x^2+y^2. I took the derivative of the model and got 0=2x*dx/dt+2y*dy/dt. Then I wrote down everything I know. I knew that x=10.25 because 5+1.75(3), and I knew that dx/dt=1.75 because that was given, and I knew that y=10.95 because that was given also, and I didn't know dy/dt because that's what we are solving for. I would then plug in the numbers I know so it would read 21.0dy/dt=-35.875. I'd solve for dy/dt and get dy/dt=-1.64.