In exercises 1-4, find each limit. Show intermediate steps. x2-4 ...

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In exercises 1-4, find each limit. Show intermediate steps. x2-4 - 1. lim x-*2 X2 + 5x-14 - 3x - 5x² 2. lim 2x² +4 3x 3. lim x 5+ 5 - X 4x - 9 4. lim x-> m x + 3 3x + 1 5. Find the equation of the line that is perpendicular to the tangent line to the curve, y= at the point (1,2). 4x - 2 In exercises 6-10, find the derivative of each function. Simplify your result in exercises 7 and 9. 6. y=sin53x = 7. V = e³* (3sin X - cosx) 8. Y = (x + 2) (xx + 3) 9. Y =x2x = 10. y = = sin" (3x) + In xsec X 11. Given (x) - After sketching a graph a. Find the local maximum and minimum values of f b. For which values of X is f concave upward? 12. A box with a square base is to have an open top. The area of the material in the box is to be 60 sq in. What should the dimensions be in order to make the volume as large as possible? 13. A 10-foot ladder is leaning against a wall. If a person pulls the base of the ladder away from the wall at the rate of 1 foot per second, how fast is the top going down the wall when the base of the ladder is 6 feet from the wall? In exercises 14-17, evaluate the integral. 14. x² 2 + x 3 dx 15. / cos³ X sin X dx (lnx)³ 16. dx 2 17. / x x - 1 dx 1 18. Find the area enclosed by the curves y=x2-2x - and y = =2x-x² 19. The region bounded by the parabola y=x², = the line X = 1, and the x-axis is rotated about the y-axis. Find the volume of the resulting solid. 20. A bacteria culture starts with 300 bacteria and after 3 hours there are 600 bacteria. Assume that the bacteria grow at a rate proportional to their size, which means you can use the function, y = Ae where Y is the number of bacteria and t is time in hours. a. Find the number of bacteria after 5 hours. b. When will the population reach 2,000?

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