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Is it just multiplying everything in the 2nd last line by e^t ... ?

This is eerily similar to Q14 a) of the 2020 Extension 1 exam. I made a video working through the explanation, which you might find helpful when thinking in general about the benefit of the "equating of coefficients" technique. Enjoy!

Hmmm yes it all comes down to what the question writer meant by “corresponding” area. I’d just assumed it meant the same area from part a) but perhaps that’s not what they meant. Bloody confusing to use the word “corresponding” though if they meant just the area between the curve and the y-axis.

Ok video is ready to rock! Enjoy!

Can do! Hopefully I can get it done later tonight, but Handmaid's Tale is on so we'll see how I go ... Otherwise perhaps tomorrow night. But looking at username_2's solution, I'd say that all looks ok, except for the very last line where it looks like the π has inexplicably dropped off. I think...

This is a classic volumes of revolution question. Have you learned about the method of slices or the method of cylindrical shells? You could probably use either here ... It will ultimately result in an integration, but the key is understanding how to visualise what happens when you rotate the...

Maybe he is a long time listener first time caller 😂

The Rule of 72 is good stuff!! Here’s a video I made on a proof of it ...

I think you will find that the mere process of creating your cheat sheet will likely result in you not needing to use it 🧐

I think there is a mistake in the question. I actually had someone send me this question last year. Here's a video where I work through it in case it helps:

Ahhh yes I think I understand the approach now. It kind of speeds up the process since you can do a whole set of y' values in a single hit. So it's about picking a value for y' (i.e. m), plotting the resulting curve, and then overlaying the gradient lines on that curve. Fair enough.

No worries! Yeah the approach I used was just picking combinations of x and y, and then calculating y' to observe the slope of the tangent at that point. Naturally something like Desmos or Wolfram Alpha can do this over many combinations with little effort, but at the end of the day, the...

... and here's my thoughts on your second question. I may be missing something, but I think what your teacher has suggested is unnecessarily complex. Creating the slope field should be as simple as picking various x and y coordinates, plugging them into the formula for y' and then analysing the...

Here's my thoughts on your first question: