Math 2412 Review for Exam 2 1. Evaluate the following: (a) f (x) for f (x) = 3 . (b) f (x) for f (x) = 2x4 + 5x + 3. 5 1 (c) f (t) for f (t) = t2 2 + 4 . t t 2. Find the derivative of g(x) = esin x . 3. Find the derivative of f (x) = ln(sin x + cos x). 4. Find dy for eyx + 1 = x2 . dx 5. Find the derivative of y = e2x (2x 1)6 using logarithmic dierenti+ 5)2 (4 7x) (x3 ation. 6. Find dy dx for y = x=3 4 . x 7. Find the derivative of the following functions: (a) f (x) = x 2ex csc x (b) y = x2 8. Find the derivative of the following functions: ex 1 + x2 (b) y = (x + ln x)(2 3 x) (a) g(x) = 9. Find the derivative of the following functions: (a) y = 2 csc x + 5 sin x (b) f (x) = (x2 + 2x + 6)1 10. Find the derivative of the following functions: (a) g(x) = 4 tan1 (x2 + 3) (b) y = sin(tan x) d (csc x) = csc x cot x. dx d 1 (b) Prove the following rule cos1 x = . dx 1 x2 11. (a) Prove the following rule 12. A mass on a spring oscillates from equilibrium position according to the function: y(t) = 8 sin t, t 0. (a) Find the velocity and acceleration functions. (b) Find the position, velocity and acceleration of the mass at time t = . 4 13. If a ball is thrown from ground level vertically upward at a velocity of 80 ft/s, then the height of the ball t seconds later is: s(t) = 16t2 + 80t, t 0. (a) What is the height of the ball 1 second later? (b) How long does it take for the ball to return to the ground? (c) What is the velocity of the ball after 2 seconds? (d) When will the ball reach its maximum height (think about velocity at the peak)? 14. Find dy for: dx x2 y 2 + x sin y = 4. 15. Use logarithmic dierentiation to nd y for y = (x3 + 3x + 2)9 (4x5 + 3x + 4)3 (3x2 + 4)3 x2 + 3 16. (a) Find the derivative of y = xx . (b) Find the derivative of y = x (c) Find the derivative of y = x x . cos(x) . 17. A spherical balloon is being lled with a gas in such a way that when the 1 radius is 2 ft, the radius is increasing at the rate of ft/min. How fast 6 is the volume changing at this time? (Hint: The volume of a sphere is 4 V = r3 where r is the radius of the sphere) 3 18. Actually, know how to do all the homework from Section 3.9 (Related Rates). Assigned homework was the following problems from the section: 5,11,15,20,31,35,39,40