Calculus 1 problems: functions, integrals, series

Question

Im doing some older exams that my professor has provided, but I havent got the solutions for these. Can someone help confirm that the solutions Ive arrived at are correct?

Here are the problems: https://imgur.com/a/HWj4mAM

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Comments

The system notes that you have included a link within your question. This will usually trigger the system to hide your question from view until it’s approved. Although I haven’t removed the link, I’ve uploaded 4 questions from the link address.

Answer 1

Your answers seem to be correct.

1. The limit as x→0 of (eˣ-1)/x is 1 because eˣ≈1+x when x is small so (1+x-1)/x=x/x=1, and g(0)=1, so there is continuity at x=0.

2. Using the integration factor e^(x³/3) and substituting the initial conditions, the integration constant is -1, leading to ye^(x³/3)=e^(x³/3)-1, which is y=1-e^-(x³/3).

3. The absolute convergence test aᵣ₊₁/aᵣ<0 and 0<1 so aᵣ₊₁/aᵣ<1.

4. When a₁=x+1, |a₂| etc < x+1, making the radius 1.