This week we worked on u substitution, only this time it was with anti-differentiation. It works the same way a normal u sub would work, except you need to anti-deriv the function too. An example of this would be the integral of the function cot(7x)dx. First, I changed it cot to cos(7x)/sin(7x). Next, I chose a u. The u I chose was sin(7x). You would follow the same steps as u sub by taking the derivative and then solving for du to get du=7cos(7x)dx. However, we want to get cos(7x) to cancel out instead of 7cos(7x). In order to get cos(7x) I'd divide by 7 so (1/7)du=cos(7x)dx. Then you would go back to integral, and replace cos(7x)dx with u. So you'd get the integral of the function (1/7)(1/u)du. Anti-deriv this and you get (1/7)ln(u)+c. Plug back in sin(7x) for you to get the final answer (1/7)(sin7x)+c.
Another section we covered this week focused on slope fields. We did an activity using dot paper that helped in understanding the concept. Basically you plug a point into the equation to get the slope at that point. Then you would go on the dot paper and make a small line that had the slope that it equaled. For example, if the equation was dy/dx=x+y and the point you were evaluating was (-1,-1) you'd plug the point into the equation so you would get -1+-1 and get -2. You'd go to the point (-1,-1) and draw a small line with the slope of -2. If you do this for all the points you'll be able to see the shape of the graph of x+y.
Another section we covered this week focused on slope fields. We did an activity using dot paper that helped in understanding the concept. Basically you plug a point into the equation to get the slope at that point. Then you would go on the dot paper and make a small line that had the slope that it equaled. For example, if the equation was dy/dx=x+y and the point you were evaluating was (-1,-1) you'd plug the point into the equation so you would get -1+-1 and get -2. You'd go to the point (-1,-1) and draw a small line with the slope of -2. If you do this for all the points you'll be able to see the shape of the graph of x+y.