2 . [, f (z ) + 9 (x ) dx -2.5 1 1. f (2x) da 12. Suppose f is a function. Expand > f(2k + 1) as the sum of four terms to get rid of the ). 13. Use Newton's method to find a root of the function f(x) = x2 -26 starting with To = 5. What are :1 and 12 to 3 decimal places? 14. (a) Give an equation for the line tangent to y = cos(x) at z = =. (b) Find the approximate value of cos( + 0.1) given by this linearization. 2 For the following problems, use the fact that the derivative of f(x) = x . In(x) is f'(x) = In(x) + 1. 15. Find In(x) + 1 da using the Fundamental Theorem of Calculus. 16. Find the critical point(s) of r . In(x). 17. Use the second derivative test to determine whether the critical point(s) of r . In(x) is/are local maxima or local minima. 18. Find the absolute maxima and minima of r . In(x) on the interval [e-2, e2].19. If f(x) = x - sin(x), find the following: (a) f'(x) (b) f(x) 20. Let f(x) = 2x2 + 3x, a = 1, and b = 5. Find the r-value c with the property that f'(c) = f(6)-ffe). Which theorem guarantees this c exists? 21. A perfectly circular lake begins freezing starting from the shore. Ice develops evenly around the perimeter of the lake, so there is always a central circle of water. If the ice grows at a rate of 5 square meters per hour, how quickly is the radius of the unfrozen portion of the lake decreasing when the radius is 20 meters? 22. Given the following table of values for a function f on [0, 3], what are the left, midpoint, and right Rie- mann sums with 3 subintervals for the function? What is the right Riemann sum with 6 subintervals? 0 0.5 1 1.5 2 2.5 3 f(a) -2 -1.5 -0.5 0.5 1 2 4 23. Suppose f and g are differentiable functions and h(x) = f(x) - 3 . g(x). If f'(2) = 6 and g'(2) = 5, what is h'(2)? 24. Find the derivative of 5x2 + 1 with respect to r. 25. A rectangular card must have a total area of 20 in'. There is a printed area in the center with blank margins on the left and right of 1 inch each and top and bottom margins of = inch each. What dimensions of the card maximize its printed area? 26. What is the derivative of the function F() = / v d? 27. Evaluate 3(tan + x)?(sec? r + 1) do. 28. Find the net, oriented area under the curve y = 4x3 - 6x + 3 between r = 1 and r = 4. 100 29. What is the value of ) k? * = 11. The velocity of an object is given by v(t) = -6t + 5. (a) What is the acceleration of the object as a function of ? (b) What is the average velocity of the object on the interval [1, 3]? (c) What is the position of the object as a function of t if the object has position 3 when t = 1? 2. The graph of the function f is given below. 10 (a) What is lim f(x)? (b) Is f continuous at a = -2? (c) What is f'(-1)? -5 (d) What is ( f(x) da? (e) Is f differentiable at r = 1? (f) What is lim f(x)? -5 0 5 (g) Is f differentiable at r = 3? (h) Which is the largest? (Choose exactly one) 10 or Evaluate the following limits: 3. lim - 9 6. lim sin(x) - cos() - 4. lim 7. lim cos () c-+0 5. lim n(n + 1) 1-+00 2n2 For questions 8-11, suppose [ f() de = 4 and 9(x) = -2 and find the following: 8. f(z) + 3 dx 10. 7 9(x) dr