1. Use the - 8 definition of limit to show that lim(-4x + 1) = -7 2. Find the limit: lim 2x + 1 x-4 sin(5w) 3. Find the limit: lim W-0 W 1 -1 5. 4. Find the limit: lim X0 Find the limit: lim x+1 X X--1- x+1 x4-1 6. X-2 Find where the function is NOT continuous. Label each discontinuity as removable or non-removable. f(x) = 2+x x-4 7. Find the limit: lim x-7 X-7 8. Find the limit: lim (x3 x + x) 9. Find the vertical asymptotes, if any, of the function f(x) 4x = 4-x2 Use a limit to prove any vertical asymptotes. 10. Use the definition of derivative to find f'(x) for f(x) = x + x. 1 11. Find the derivative of f(x) = + tan(x). 12. Find the equation of the tangent line to y 13. Find the derivative of f(x) = tan(x) + 14. Find the derivative of f(x) = 2x + x dy x+4 = x+5 tan(x) at the point (1,1). 4 15. Find using implicit differentiation. x + 2xy + y = 16 dx 16. All the sides of a cube are expanding at the rate of 2 cm/sec. How fast is the volume of the cube expanding when each side is length 5 cm? 17. Find the absolute extrema of f(x) = x - 2x2 on the interval [-1,4]. 18. Let f (x) = x - x on the interval [0,1]. Which of the 3 theorems apply to this function on this interval? Rolle's Theorem, the Mean Value Theorem, or the Extreme Value Theorem? Note: more than 1 theorem may apply. 19. Find the critical numbers for f(x) = x3(x-5). Find the open intervals where the function is increasing and decreasing. Find all relative extrema and write them as points, (x, y). 1 2 20. Let f (x) = x-3x + 8x. Find all relative extrema using the Second Derivative Test. 3 21. Find all inflection points and determine the intervals of where the function is concave up/concave down. Let f(x) = sin(x) + cos(x), 0 x 2. 22. Find the horizontal asymptotes of: f(x) = -2x+1 4x-18 23. Analyze the graph of f(x) = x5 - 4x4 + 4x using the steps on the lecture notes for graphing (also see Exam 3 Part 2). With the information you derived sketch the graph. Clearly label and scale the axes, the intercepts, the local extrema, and all inflection points. 24. A rectangular area of 1600 ft is to be fenced off. Two opposite sides will be using fencing costing $1.00 per foot, and the remaining sides will use fencing costing $2.00 per foot. Find the dimensions of least cost. 25. Find the differential dy for y = 4x - 12. 26. Evaluate the indefinite integral: f x+2x+1 x dx 27. Solve the differential equation: f" (x) = -2, f'(0) = 3, f(1) = 1. 28. Evaluate the sum: (i-1) 29. Using a Riemann Sum, find the area using the curve y = 2x + 1 between the values x = 0 and x = 4.