Alexander's Midterm 2 Review (Chapter 3)-

Q1. Let y = 3x⁵ - 3x² + 4 + 2/x¹⁰. Find dy/d x:

Q2. Let y = sin(x). Find dy/dx:

Q3. Let y = cos(x). Find dy/dx:

Q4. Let y = tan(t). Find dy/dt:

Q5. Let x = e^t. Find dx/dt:

Q6. Let y = ln(x). Find dy/dx:

Q7. Let y=ln|x|. Find dy/dx:

Q8. Let y=tan(x). Find dy/dx:

Q9. Let y=sin(x³). Find dy/dx:

Q10. Let y = x²tan(x). Find dy/dx:

Q11. Let y = e^2xln(x). Find dy/dx:

Q12. Let y = ln(x)/x¹⁰. Find dy/dx:

Q13. Let y = sin(e^x/x⁵). Find dy/dx:

Q14. Suppose y² + x³y = e^2x. Find dy/dx:

Q15.Suppose (y⁵/x²) + ysin(x) = cos(x). Find dy/dx:

Q16. Suppose y = x^sin(x). Find dy/dx:

Q17. Suppose y = x^{(x^10)}. Find dy/dx:

 Q18. Let f and g be differentiable functions. Suppose y = f(x)^g(x). Find dy/dx in terms of f, g, f', g', and x.

Q19. Let f be  a  differentiable  function. Suppose y = f^-1(x). Find dy/dx in terms of f', f^-1 and x.

Q20. Let y = 8^x. Find dy/dx:

Q21. Let y = log₄(x). Find dy/dx:

Q22. Let y = log₂₀(x²). Find dy/dx:

Q23. Let y = sin^-1(x). Find dy/dx:

Q24. Let y = cos^-1(x). Find dy/dx:

Q25. Suppose I have a population of bunny rabbits.  At time t = 0 there are two bunny rabbits. I find that after t = 2 weeks there are now 8 bunny rabbits.  Assuming that the population of bunnies grows exponentially, how long will it take for there to be 100 bunny rabbits?

Q26. My mom always made me wrap up warm on cold winter's days.  So now I am a grownup I can do what I like and wear just a tee-shirt in the snow.  I catch a terrible cold.  On Sunday, December 1st, there is 1 cm³ of phlegm in my lungs. The next day there is 3 cm³ of phlegm in my lungs. On what day will my entire lungs be completely filled with phlegm, assuming their volume is 3000 cm3and that the phlegm is growing exponentially.

Q27. I start with 100 grams of a radioactive isotope. It has a half life of 3 weeks. How long will it take for there to be 40 grams of the isotope left?

 Q28. I do a silly thing. I decide it would be fun to jump into a lake of freezing snow-melt water. It is zero degrees. My heart stops instantly because of the cold and I die and sink to the bottom of the lake. Oh dear. :( My body temperature was 38^oC when I jumped in. After 1 minute ithas cooled to 20^oC. What is my body temperature after 10 minutes?

Q29. I blow up a spherical balloon at a rate of 10 cm³ per second.  What is the rate of change of the radius when the balloon has radius 5 cm?

Q30. Find the equation of the tangent to the curve y = x³ at the point (1, 1).  Use your answer to estimate, without using a calculator, the value of 1.07³.